#set document(title: "4.4 Understand Slope of a Line", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 4.4#h(0.6em)Understand Slope of a Line #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Simplify: #math.equation(block: false, alt: "the fraction 1 minus 4 over 8 minus 2")[$frac(1 − 4, 8 − 2)$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Divide: #math.equation(block: false, alt: "the fraction 0 over 4 , the fraction 4 over 0")[$frac(0, 4) , frac(4, 0)$]. #linebreak() If you missed this problem, review . #solutionbox[ 0, undefined ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 15 over −3 , the fraction −15 over 3 , the fraction −15 over −3")[$frac(15, −3) , frac(−15, 3) , frac(−15, −3)$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "−5 , −5 , 5")[$−5 , −5 , 5$] ] ] When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat? In mathematics, the ‘tilt’ of a line is called the #emph[slope] of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill. In this section, we will explore the concept of slope. === Use Geoboards to Model Slope A #strong[geoboard] is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. A rubber band stretched between two pegs models a line. It starts at the book's example — from peg (1, 2) up to peg (4, 4): a run of 3 and a rise of 2, so the slope is m = rise/run = 2/3. Drag either peg to model a different line; the dashed gray leg is the run and the dashed red leg is the rise.{"activity":{"handles":\[{"id":"h-peg1","kind":"point","label":"First peg","x":"x1","y":"y1"},{"id":"h-peg2","kind":"point","label":"Second peg","x":"x2","y":"y2"}\],"label":"Geoboard pegs","prompt":"Drag either peg to a new spot. The dashed legs show the run (gray) and the rise (red); slope = rise over run.","requires":\["point-handles"\],"schema":1},"functions":\[{"color":"\#1f4e79","domain":null,"expression":"x1 + (x2 - x1)\*t","expression2":"y1 + (y2 - y1)\*t","id":"band","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"parametric","tMax":1,"tMin":0,"variable":"t","visible":true},{"color":"\#94a3b8","domain":null,"expression":"x1 + (x2 - x1)\*t","expression2":"y1","id":"runleg","inequality":"lt","lineDash":\[6,4\],"lineWidth":2,"mode":"parametric","tMax":1,"tMin":0,"variable":"t","visible":true},{"color":"\#b8442a","domain":null,"expression":"x2","expression2":"y1 + (y2 - y1)\*t","id":"riseleg","inequality":"lt","lineDash":\[6,4\],"lineWidth":2,"mode":"parametric","tMax":1,"tMin":0,"variable":"t","visible":true}\],"grid":{"majorSpacing":1,"showAxes":true,"showLabels":true},"parameters":\[{"id":"p-x1","label":"Peg 1 across","max":5,"min":0,"name":"x1","step":1,"value":1},{"id":"p-y1","label":"Peg 1 up","max":5,"min":0,"name":"y1","step":1,"value":2},{"id":"p-x2","label":"Peg 2 across","max":5,"min":0,"name":"x2","step":1,"value":4},{"id":"p-y2","label":"Peg 2 up","max":5,"min":0,"name":"y2","step":1,"value":4}\],"title":"Rubber band between two pegs: rise over run","version":1,"viewport":{"centerX":2.5,"centerY":2.5,"scale":45}}We’ll start by stretching a rubber band between two pegs as shown. The rubber band stretches from the peg at column 1, row 4 to the peg at column 4, row 2. This creates a right triangle with a vertical rise of 2 units and a horizontal run of 3 units, giving a slope of m = 2/3. Drag the endpoints to different pegs to see how the rise and run change the steepness of the line.{"functions":\[\],"geoboard":{"bands":\[{"closed":false,"color":"\#1f4e79","draggable":true,"id":"band1","readouts":\["slope"\],"vertices":\[\[0,1\],\[3,3\]\]}\],"cols":5,"draw":true,"rows":5},"grid":{"majorSpacing":1,"showAxes":false,"showLabels":false},"parameters":\[\],"title":"Modeling Slope on a Geoboard","viewport":{"centerX":2,"centerY":2,"scale":45}}#figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 4, row 2, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 4, row 2, forming a line.", caption: none) Doesn’t it look like a line? Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle, as shown in #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 4, and the peg in column 4, row 2, forming a right triangle. The 1, 2 peg is the vertex of the 90 degree angle, while the line between the 1, 4 and 4, 2 pegs forms the hypotenuse of the triangle.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 4, and the peg in column 4, row 2, forming a right triangle. The 1, 2 peg is the vertex of the 90 degree angle, while the line between the 1, 4 and 4, 2 pegs forms the hypotenuse of the triangle.", caption: none) We carefully make a 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal. To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the #strong[rise] and the horizontal distance is called the #strong[run], as shown. #figure(figph[In this illustration, there are two perpendicular lines with arrows. The first line extends straight upward and is labeled “rise”. The second arrow extends straight rightward and is labeled “run”.], alt: "In this illustration, there are two perpendicular lines with arrows. The first line extends straight upward and is labeled “rise”. The second arrow extends straight rightward and is labeled “run”.", caption: none) If our #strong[geoboard] and rubber band look just like the one shown in , the rise is 2. The rubber band goes up 2 units. (Each space is one unit.) #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 4, and the peg in column 4, row 2, forming a right triangle where the 1, 2 peg is the vertex of the 90 degree angle and the line between the 1, 4 peg and the 4, 2 peg forms the hypotenuse. The line between the 1, 2 peg and the 1, 4 peg is labeled “2”. The line between the 1, 2 peg and the 4, 2 peg is labeled “3”.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 4, and the peg in column 4, row 2, forming a right triangle where the 1, 2 peg is the vertex of the 90 degree angle and the line between the 1, 4 peg and the 4, 2 peg forms the hypotenuse. The line between the 1, 2 peg and the 1, 4 peg is labeled “2”. The line between the 1, 2 peg and the 4, 2 peg is labeled “3”.", caption: [The rise on this geoboard is 2, as the rubber band goes up two units.]) What is the run? The rubber band goes across 3 units. The run is 3. The book's rubber band stretches from the peg in column 1, row 5 to the peg in column 5, row 2 — a rise of 3 over a run of 4, so the slope is 3/4. Drag either end of the band to new pegs and watch the rise, the run, and the slope readout change together.{"functions":\[\],"geoboard":{"bands":\[{"closed":false,"color":"\#1f4e79","draggable":true,"id":"band1","readouts":\["slope"\],"vertices":\[\[0,0\],\[4,3\]\]}\],"cols":5,"draw":true,"rows":5},"grid":{"majorSpacing":1,"showAxes":false,"showLabels":false},"parameters":\[\],"title":"Slope on the geoboard: 3/4","viewport":{"centerX":2,"centerY":2,"scale":45}}The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter #math.equation(block: false, alt: "m")[$m$]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope of a Line] The #strong[slope of a line] is #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]. The #strong[rise] measures the vertical change and the #strong[run] measures the horizontal change between two points on the line. ] What is the slope of the line on the geoboard in ? The rubber band stretches from (1, 4) along a run of 2 and a rise of 1 to (3, 3), creating a slope of m = 1/2. Drag the endpoints to change the rise and run while keeping the slope constant at 1/2. Watch how the dashed legs of the right triangle update to show the current vertical and horizontal changes.{"functions":\[\],"geoboard":{"bands":\[{"closed":false,"color":"\#1f4e79","draggable":true,"id":"band1","readouts":\["slope"\],"vertices":\[\[0,1\],\[2,2\]\]}\],"cols":5,"draw":true,"rows":5},"grid":{"majorSpacing":1,"showAxes":false,"showLabels":false},"parameters":\[\],"title":"Model a line with slope 1/2","viewport":{"centerX":2,"centerY":2,"scale":45}}#math.equation(block: true, alt: "m, equals, the fraction rise over run; m, equals, the fraction 2 over 3")[$m & = & frac("rise", "run") \ m & = & frac(2, 3)$]The line has slope #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$]. This means that the line rises 2 units for every 3 units of run. When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. If the rise goes up it is positive and if it goes down it is negative. The run will go from left to right and be positive. The book's rubber band stretches from the peg in column 1, row 3 to the peg in column 4, row 4 — a rise of -1 over a run of 3, so the slope is -1/3. Drag either end of the band to new pegs and watch the rise, the run, and the slope readout change together.{"functions":\[\],"geoboard":{"bands":\[{"closed":false,"color":"\#1f4e79","draggable":true,"id":"band1","readouts":\["slope"\],"vertices":\[\[0,2\],\[3,1\]\]}\],"cols":5,"draw":true,"rows":5},"grid":{"majorSpacing":1,"showAxes":false,"showLabels":false},"parameters":\[\],"title":"Slope on the geoboard: -1/3","viewport":{"centerX":2,"centerY":2,"scale":45}}#examplebox("Example 1")[][ What is the slope of the line on the geoboard shown? #linebreak() #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line.", caption: none) #solutionbox[ Use the definition of slope: #math.equation(block: false, alt: "m equals the fraction rise over run .")[$m = frac("rise", "run") .$] Start at the left peg and count the spaces up and to the right to reach the second peg. #linebreak() #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 5 and the peg in column 5, row 2, forming a right triangle. The 1, 2 peg forms the vertex of the 90 degree angle and the line from the 1, 5 peg to the 5, 2 peg forms the hypotenuse of the triangle. The line from the 1, 2 peg to the 1, 5 peg is labeled “3”. The line from the 1, 2 peg to the 5, 2 peg is labeled “4”.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 5 and the peg in column 5, row 2, forming a right triangle. The 1, 2 peg forms the vertex of the 90 degree angle and the line from the 1, 5 peg to the 5, 2 peg forms the hypotenuse of the triangle. The line from the 1, 2 peg to the 1, 5 peg is labeled “3”. The line from the 1, 2 peg to the 5, 2 peg is labeled “4”.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([The rise is 3.], [#math.equation(block: false, alt: "m equals the fraction 3 over run")[$m = frac(3, "run")$]]), [The run is 4.], [#math.equation(block: false, alt: "m equals the fraction 3 over 4")[$m = frac(3, 4)$]], [], [The slope is #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$].], )) This means that the line rises 3 units for every 4 units of run. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ What is the slope of the line on the geoboard shown? #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 4, row 1, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 4, row 1, forming a line.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 3")[$frac(4, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ What is the slope of the line on the geoboard shown? #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 5, row 3, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 5, row 3, forming a line.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] ] ] #examplebox("Example 2")[][ What is the slope of the line on the geoboard shown? #linebreak() #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line.", caption: none) #solutionbox[ Use the definition of slope: #math.equation(block: false, alt: "m equals the fraction rise over run .")[$m = frac("rise", "run") .$] Start at the left peg and count the units down and to the right to reach the second peg. #linebreak() #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3, the peg in column 1, row 4 and the peg in column 4, row 4, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 4, 4 peg forms the hypotenuse of the triangle. The line from the 1, 3 peg to the 1, 4 peg is labeled “negative 1”. The line from the 1, 4 peg to the 4, 4 peg is labeled “3”.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3, the peg in column 1, row 4 and the peg in column 4, row 4, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 4, 4 peg forms the hypotenuse of the triangle. The line from the 1, 3 peg to the 1, 4 peg is labeled “negative 1”. The line from the 1, 4 peg to the 4, 4 peg is labeled “3”.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([The rise is −1.], [#math.equation(block: false, alt: "equals the fraction −1 over run")[$= frac(−1, "run")$]]), [The run is 3.], [#math.equation(block: false, alt: "m equals the fraction −1 over 3; m equals minus the fraction 1 over 3")[$m = frac(−1, 3) \ m = − frac(1, 3)$]], [], [The slope is #math.equation(block: false, alt: "minus the fraction 1 over 3")[$− frac(1, 3)$].], )) This means that the line drops 1 unit for every 3 units of run. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ What is the slope of the line on the geoboard? #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2 and the peg in column 4, row 4, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2 and the peg in column 4, row 4, forming a line.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ What is the slope of the line on the geoboard? #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1 and the peg in column 4, row 5, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1 and the peg in column 4, row 5, forming a line.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 over 3")[$− frac(4, 3)$] ] ] Notice that in the slope is positive and in the slope is negative. Do you notice any difference in the two lines shown in (a) and (b)? #figure(figph[The figure shows two grids of evenly spaced pegs, one labeled (a) and one labeled (b). There are 5 columns and 5 rows of pegs in each grid. In the (a) grid, a rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line. Below this grid is the slope of a line defined as m equals 3 fourths. In the (b) grid, a rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line. Below this grid is the slope of a line defined as m equals negative 1 third.], alt: "The figure shows two grids of evenly spaced pegs, one labeled (a) and one labeled (b). There are 5 columns and 5 rows of pegs in each grid. In the (a) grid, a rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line. Below this grid is the slope of a line defined as m equals 3 fourths. In the (b) grid, a rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line. Below this grid is the slope of a line defined as m equals negative 1 third.", caption: none) We ‘read’ a line from left to right just like we read words in English. As you read from left to right, the line in (a) is going up; it has #strong[positive slope]. The line in (b) is going down; it has #strong[negative slope]. The geoboard rubber band stretches from (1, 2) along a run of 4 and a rise of -1 to the peg at (5, 3). This creates a slope of m = rise/run = -1/4. Drag the endpoints to see how changing the vertical drop affects the steepness of the downhill line.{"functions":\[\],"geoboard":{"bands":\[{"closed":false,"color":"\#1f4e79","draggable":true,"id":"band1","readouts":\["slope"\],"vertices":\[\[0,3\],\[4,2\]\]}\],"cols":5,"draw":true,"rows":5},"grid":{"majorSpacing":1,"showAxes":false,"showLabels":false},"parameters":\[\],"title":"Model a line with slope -1/4","viewport":{"centerX":2,"centerY":2,"scale":45}}#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Positive and Negative Slopes] #figure(figph[The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled “Positive slope”. The line on the right is a diagonal line that drops from left to right. It is labeled “Negative slope”.], alt: "The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled “Positive slope”. The line on the right is a diagonal line that drops from left to right. It is labeled “Negative slope”.", caption: none) ] #examplebox("Example 3")[][ Use a geoboard to model a line with slope #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]. #solutionbox[ To model a line on a geoboard, we need the rise and the run. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]]), [Replace #math.equation(block: false, alt: "m")[$m$] with #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$].], [#math.equation(block: false, alt: "the fraction 1 over 2 equals the fraction rise over run")[$frac(1, 2) = frac("rise", "run")$]], )) So, the rise is 1 and the run is 2. Start at a peg in the lower left of the geoboard. Stretch the rubber band up 1 unit, and then right 2 units. #linebreak() #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3, the peg in column 1, row 4 and the peg in column 3, row 3, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 3, 3 peg forms the hypotenuse of the triangle. The line from the 1, 3 peg to the 1, 4 peg is labeled “1”. The line from the 1, 3 peg to the 3, 3 peg is labeled “2”.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3, the peg in column 1, row 4 and the peg in column 3, row 3, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 3, 3 peg forms the hypotenuse of the triangle. The line from the 1, 3 peg to the 1, 4 peg is labeled “1”. The line from the 1, 3 peg to the 3, 3 peg is labeled “2”.", caption: none) The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the slope #math.equation(block: false, alt: "m equals the fraction 1 over 3")[$m = frac(1, 3)$]. Draw a picture to show your results. #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 3, the peg in column 2, row 4 and the peg in column 5, row 3, forming a right triangle. The 2, 3 peg forms the vertex of the 90 degree angle and the line from the 2, 4 peg to the 5, 3 peg forms the hypotenuse of the triangle.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 3, the peg in column 2, row 4 and the peg in column 5, row 3, forming a right triangle. The 2, 3 peg forms the vertex of the 90 degree angle and the line from the 2, 4 peg to the 5, 3 peg forms the hypotenuse of the triangle.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the slope #math.equation(block: false, alt: "m equals the fraction 3 over 2")[$m = frac(3, 2)$]. Draw a picture to show your results. #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1, the peg in column 1, row 4 and the peg in column 3, row 1, forming a right triangle. The 1, 1 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 3, 1 peg forms the hypotenuse of the triangle.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1, the peg in column 1, row 4 and the peg in column 3, row 1, forming a right triangle. The 1, 1 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 3, 1 peg forms the hypotenuse of the triangle.", caption: none) ] ] #examplebox("Example 4")[][ Use a geoboard to model a line with slope #math.equation(block: false, alt: "the fraction −1 over 4.")[$frac(−1, 4) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]]), [Replace #math.equation(block: false, alt: "m")[$m$] with #math.equation(block: false, alt: "the fraction −1 over 4")[$frac(−1, 4)$].], [#math.equation(block: false, alt: "the fraction −1 over 4 equals the fraction rise over run")[$frac(−1, 4) = frac("rise", "run")$]], )) So, the rise is #math.equation(block: false, alt: "−1")[$−1$] and the run is 4. Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down. We stretch the rubber band down 1 unit, then go to the right 4 units, as shown. #linebreak() #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 3 and the peg in column 5, row 3, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 2 peg to the 5, 3 peg forms the hypotenuse of the triangle. The line from the 1, 2 peg to the 1, 3 peg is labeled “negative 1”. The line from the 1, 3 peg to the 5, 3 peg is labeled “4”.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 3 and the peg in column 5, row 3, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 2 peg to the 5, 3 peg forms the hypotenuse of the triangle. The line from the 1, 2 peg to the 1, 3 peg is labeled “negative 1”. The line from the 1, 3 peg to the 5, 3 peg is labeled “4”.", caption: none) The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is #math.equation(block: false, alt: "the fraction −1 over 4")[$frac(−1, 4)$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the slope #math.equation(block: false, alt: "m equals the fraction −2 over 3")[$m = frac(−2, 3)$]. Draw a picture to show your results. #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 3, the peg in column 2, row 5 and the peg in column 3, row 5, forming a right triangle. The 2, 5 peg forms the vertex of the 90 degree angle and the line from the 2, 3 peg to the 3, 5 peg forms the hypotenuse of the triangle.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 3, the peg in column 2, row 5 and the peg in column 3, row 5, forming a right triangle. The 2, 5 peg forms the vertex of the 90 degree angle and the line from the 2, 3 peg to the 3, 5 peg forms the hypotenuse of the triangle.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the slope #math.equation(block: false, alt: "m equals the fraction −1 over 3")[$m = frac(−1, 3)$]. Draw a picture to show your results. #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1, the peg in column 1, row 2 and the peg in column 4, row 2, forming a right triangle. The 1, 2 peg forms the vertex of the 90 degree angle and the line from the 1, 1 peg to the 4, 2 peg forms the hypotenuse of the triangle.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1, the peg in column 1, row 2 and the peg in column 4, row 2, forming a right triangle. The 1, 2 peg forms the vertex of the 90 degree angle and the line from the 1, 1 peg to the 4, 2 peg forms the hypotenuse of the triangle.", caption: none) ] ] === Use #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] to Find the Slope of a Line from its Graph Now, we’ll look at some graphs on the #math.equation(block: false, alt: "x y")[$x y$]-coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards. To find the slope, we must count out the rise and the run. But where do we start? We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run. #examplebox("Example 5")[How to Use #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] to Find the Slope of a Line from its Graph][ Find the slope of the line shown. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1).", caption: none) #solutionbox[ #figure(figph[This table has three columns and four rows. The first row says, “Step 1. Locate two points on the graph whose coordinates are integers. Mark (0, negative 3) and (5, 1).” To the right is a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 6. The y-axis of the plane runs from negative 4 to 2. The points (0, negative 3) and (5, 1) are plotted.], alt: "This table has three columns and four rows. The first row says, “Step 1. Locate two points on the graph whose coordinates are integers. Mark (0, negative 3) and (5, 1).” To the right is a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 6. The y-axis of the plane runs from negative 4 to 2. The points (0, negative 3) and (5, 1) are plotted.", caption: none) #figure(figph[The second row says, “Step 2. Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point. Starting at (0, negative 3), sketch a right triangle to (5, 1).” In the graph on the right, an additional point is plotted at (0, 1). The three points form a right triangle, with the line from (0, negative 3) to (5, 1) forming the hypotenuse and the lines from (0, negative 3) to (0, 1) and (0, 1) to (5, 1) forming the legs.], alt: "The second row says, “Step 2. Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point. Starting at (0, negative 3), sketch a right triangle to (5, 1).” In the graph on the right, an additional point is plotted at (0, 1). The three points form a right triangle, with the line from (0, negative 3) to (5, 1) forming the hypotenuse and the lines from (0, negative 3) to (0, 1) and (0, 1) to (5, 1) forming the legs.", caption: none) #figure(figph[The third row then says, “Step 3. Count the rise and the run on the legs of the triangle.” The rise is 4 and the run is 5.], alt: "The third row then says, “Step 3. Count the rise and the run on the legs of the triangle.” The rise is 4 and the run is 5.", caption: none) #figure(figph[The fourth row says, “Step 4. Take the ratio of the rise to run to find the slope. Use the slope formula. Substitute the values of the rise and run.” To the right is the slope formula, m equals rise divided by run. The slope of the line is 4 divided by 5, or four fifths. This means that y increases 4 units as x increases 5 units.], alt: "The fourth row says, “Step 4. Take the ratio of the rise to run to find the slope. Use the slope formula. Substitute the values of the rise and run.” To the right is the slope formula, m equals rise divided by run. The slope of the line is 4 divided by 5, or four fifths. This means that y increases 4 units as x increases 5 units.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line shown. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 8 to 1 and the y-axis runs from negative 1 to 4. A line passes through the points (negative 5, 1) and (0, 3).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 8 to 1 and the y-axis runs from negative 1 to 4. A line passes through the points (negative 5, 1) and (0, 3).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 5")[$frac(2, 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line shown. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 4. A line passes through the points (0, negative 1) and (4, 2).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 4. A line passes through the points (0, negative 1) and (4, 2).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find the slope of a line from its graph using #math.equation(block: false, alt: "m equals the fraction rise over run .")[$m = frac("rise", "run") .$]] + Locate two points on the line whose coordinates are integers. + Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point. + Count the rise and the run on the legs of the triangle. + Take the ratio of rise to run to find the slope, #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]. ] #examplebox("Example 6")[][ Find the slope of the line shown. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 9 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 5), (3, 3), and (6, 1).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 9 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 5), (3, 3), and (6, 1).", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Locate two points on the graph whose coordinates are integers.], [#math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$]]), [Which point is on the left?], [#math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$]], [Starting at #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$], sketch a right triangle to #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$].], [#figure(figph[A line on a Cartesian coordinate plane demonstrating the calculation of slope using 'rise' and 'run'. The line descends, showing a rise of -2 (from y=5 to y=3) and a run of 3 (from x=0 to x=3).], alt: "A line on a Cartesian coordinate plane demonstrating the calculation of slope using 'rise' and 'run'. The line descends, showing a rise of -2 (from y=5 to y=3) and a run of 3 (from x=0 to x=3).", caption: none)], [Count the rise—it is negative.], [The rise is #math.equation(block: false, alt: "−2")[$−2$].], [Count the run.], [The run is 3.], [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [Substitute the values of the rise and run.], [#math.equation(block: false, alt: "m equals the fraction −2 over 3")[$m = frac(−2, 3)$]], [Simplify.], [#math.equation(block: false, alt: "m equals minus the fraction 2 over 3")[$m = − frac(2, 3)$]], [], [The slope of the line is #math.equation(block: false, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$].], )) So #math.equation(block: false, alt: "y")[$y$] decreases by 2 units as #math.equation(block: false, alt: "x")[$x$] increases by 3 units. What if we used the points #math.equation(block: false, alt: "open parenthesis −3 , 7 close parenthesis")[$( −3 , 7 )$] and #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis")[$( 6 , 1 )$] to find the slope of the line? #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 7) and (6, 1). An additional point is plotted at (negative 3, 1). The three points form a right triangle, with the line from (negative 3, 7) to (6, 1) forming the hypotenuse and the lines from (negative 3, 7) to negative 1, 7) and from (negative 1, 7) to (6, 1) forming the legs.], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 7) and (6, 1). An additional point is plotted at (negative 3, 1). The three points form a right triangle, with the line from (negative 3, 7) to (6, 1) forming the hypotenuse and the lines from (negative 3, 7) to negative 1, 7) and from (negative 1, 7) to (6, 1) forming the legs.", caption: none) The rise would be #math.equation(block: false, alt: "−6")[$−6$] and the run would be 9. Then #math.equation(block: false, alt: "m equals the fraction −6 over 9")[$m = frac(−6, 9)$], and that simplifies to #math.equation(block: false, alt: "m equals minus the fraction 2 over 3")[$m = − frac(2, 3)$]. Remember, it does not matter which points you use—the slope of the line is always the same. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line shown. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 6 to 1. A line passes through the points (0, negative 2) and (3, negative 6).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 6 to 1. A line passes through the points (0, negative 2) and (3, negative 6).", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 over 3")[$− frac(4, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line shown. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 3 to 6 and the y-axis runs from negative 3 to 2. A line passes through the points (0, 1) and (5, negative 2).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 3 to 6 and the y-axis runs from negative 3 to 2. A line passes through the points (0, 1) and (5, negative 2).", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 5")[$− frac(3, 5)$] ] ] In the last two examples, the lines had #emph[y]-intercepts with integer values, so it was convenient to use the #emph[y]-intercept as one of the points to find the slope. In the next example, the #emph[y]-intercept is a fraction. Instead of using that point, we’ll look for two other points whose coordinates are integers. This will make the slope calculations easier. #examplebox("Example 7")[][ Find the slope of the line shown. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from 0 to 8 and the y-axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6).], alt: "The graph shows the x y coordinate plane. The x-axis runs from 0 to 8 and the y-axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6).", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Locate two points on the graph whose coordinates are integers.], [#math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 7 , 6 close parenthesis")[$( 7 , 6 )$]]), [Which point is on the left?], [#math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$]], [Starting at #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$], sketch a right triangle to #math.equation(block: false, alt: "open parenthesis 7 , 6 close parenthesis")[$( 7 , 6 )$].], [#figure(figph[A coordinate plane with a line through points (2,3) and (7,6). A right triangle visually represents 'rise' (3 units) and 'run' (5 units) to illustrate slope calculation.], alt: "A coordinate plane with a line through points (2,3) and (7,6). A right triangle visually represents 'rise' (3 units) and 'run' (5 units) to illustrate slope calculation.", caption: none)], [Count the rise.], [The rise is 3.], [Count the run.], [The run is 5.], [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [Substitute the values of the rise and run.], [#math.equation(block: false, alt: "m equals the fraction 3 over 5")[$m = frac(3, 5)$]], [], [The slope of the line is #math.equation(block: false, alt: "the fraction 3 over 5")[$frac(3, 5)$].], )) This means that #math.equation(block: false, alt: "y")[$y$] increases 3 units as #math.equation(block: false, alt: "x")[$x$] increases 5 units. When we used geoboards to introduce the concept of slope, we said that we would always start with the point on the left and count the rise and the run to get to the point on the right. That way the run was always positive and the rise determined whether the slope was positive or negative. What would happen if we started with the point on the right? Let’s use the points #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 7 , 6 close parenthesis")[$( 7 , 6 )$] again, but now we’ll start at #math.equation(block: false, alt: "open parenthesis 7 , 6 close parenthesis")[$( 7 , 6 )$]. #figure(figph[The graph shows the x y coordinate plane. The x -axis runs from 0 to 8. The y -axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6). An additional point is plotted at (7, 3). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (7, 3) and from (7, 3) to (7, 6) forming the legs.], alt: "The graph shows the x y coordinate plane. The x -axis runs from 0 to 8. The y -axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6). An additional point is plotted at (7, 3). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (7, 3) and from (7, 3) to (7, 6) forming the legs.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([Count the rise.], [The rise is #math.equation(block: false, alt: "−3")[$−3$].]), [Count the run. It goes from right to left, so it is negative.], [The run is #math.equation(block: false, alt: "−5")[$−5$].], [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]], [Substitute the values of the rise and run.], [#math.equation(block: false, alt: "m equals the fraction −3 over −5")[$m = frac(−3, −5)$]], [], [The slope of the line is #math.equation(block: false, alt: "the fraction −3 over −5")[$frac(−3, −5)$].], )) It does not matter where you start—the slope of the line is always the same. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line shown. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 4 to 2 and the y-axis runs from negative 6 to 2. A line passes through the points (negative 3, 4) and (1, 1).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 4 to 2 and the y-axis runs from negative 6 to 2. A line passes through the points (negative 3, 4) and (1, 1).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 4")[$frac(5, 4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line shown. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 4 and the y-axis runs from negative 2 to 3. A line passes through the points (1, negative 1) and (3, 2).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 4 and the y-axis runs from negative 2 to 3. A line passes through the points (1, negative 1) and (3, 2).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 2")[$frac(3, 2)$] ] ] === Find the Slope of Horizontal and Vertical Lines Do you remember what was special about horizontal and vertical lines? Their equations had just one variable. #math.equation(block: true, alt: "Horizontal line y equals b, Vertical line x equals a; y -coordinates are the same., x -coordinates are the same.")[$"Horizontal line" #h(0.5em) y = b & & & "Vertical line" #h(0.5em) x = a \ \ "y" "-coordinates are the same." & & & "x" "-coordinates are the same."$]So how do we find the slope of the horizontal line #math.equation(block: false, alt: "y equals 4")[$y = 4$]? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens when we do this. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 4) and (3, 4).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 4) and (3, 4).", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([What is the rise?], [The rise is #math.equation(block: false, alt: "0")[$0$].]), [Count the run.], [The run is #math.equation(block: false, alt: "3")[$3$].], [What is the slope?], [#math.equation(block: false, alt: "m equals the fraction rise over run; m equals the fraction 0 over 3; m equals 0")[$m = frac("rise", "run") \ m = frac(0, 3) \ m = 0$]], [], [The slope of the horizontal line #math.equation(block: false, alt: "y equals 4")[$y = 4$] is #math.equation(block: false, alt: "0")[$0$].], )) All horizontal lines have slope 0. When the #emph[y]-coordinates are the same, the rise is 0. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope of a Horizontal Line] The slope of a horizontal line, #math.equation(block: false, alt: "y equals b")[$y = b$], is 0. ] The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away. Now, we’ll consider a vertical line, the line #math.equation(block: false, alt: "x equals 3")[$x = 3$]. #figure(figph[The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 2. A line passes through the points (3, 0) and (3, 2).], alt: "The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 2. A line passes through the points (3, 0) and (3, 2).", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([What is the rise?], [The rise is #math.equation(block: false, alt: "2")[$2$].]), [Count the run.], [The run is #math.equation(block: false, alt: "0")[$0$].], [What is the slope?], [#math.equation(block: false, alt: "m equals the fraction rise over run; m equals the fraction 2 over 0")[$m = frac("rise", "run") \ m = frac(2, 0)$]], )) But we can’t divide by 0. Division by 0 is not defined. So we say that the slope of the vertical line #math.equation(block: false, alt: "x equals 3")[$x = 3$] is undefined. The slope of any vertical line is undefined. When the #emph[x]-coordinates of a line are all the same, the run is 0. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope of a Vertical Line] The slope of a vertical line, #math.equation(block: false, alt: "x equals a")[$x = a$], is undefined. ] #examplebox("Example 8")[][ Find the slope of each line: ⓐ #math.equation(block: false, alt: "x equals 8")[$x = 8$] ⓑ #math.equation(block: false, alt: "y equals −5")[$y = −5$]. #solutionbox[ + ⓐ #math.equation(block: false, alt: "x equals 8")[$x = 8$] #linebreak() This is a vertical line. #linebreak() Its slope is undefined. #linebreak() #linebreak() + ⓑ #math.equation(block: false, alt: "y equals −5")[$y = −5$] #linebreak() This is a horizontal line. #linebreak() It has slope 0. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #math.equation(block: false, alt: "x equals −4.")[$x = −4 .$] #solutionbox[ undefined ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of the line: #math.equation(block: false, alt: "y equals 7.")[$y = 7 .$] #solutionbox[ 0 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Quick Guide to the Slopes of Lines] #figure(figph[This figure shows four lines with arrows. The first line rises up and runs to the right. It has a positive slope. The second line falls down and runs to the right. It has a negative slope. The third line is neither rises nor falls, extending horizontally in either direction. It has a slope of zero. The fourth line is completely vertical, one end rising up and the other rising down, running neither to the left nor right. It has an undefined slope.], alt: "This figure shows four lines with arrows. The first line rises up and runs to the right. It has a positive slope. The second line falls down and runs to the right. It has a negative slope. The third line is neither rises nor falls, extending horizontally in either direction. It has a slope of zero. The fourth line is completely vertical, one end rising up and the other rising down, running neither to the left nor right. It has an undefined slope.", caption: none) ] Remember, we ‘read’ a line from left to right, just like we read written words in English. === Use the Slope Formula to find the Slope of a Line Between Two Points Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation. We have seen that an ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] be used to represent two different points? Mathematicians use subscripts to distinguish the points. #math.equation(block: true, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis, read ‘ x sub 1, y sub 1’; open parenthesis x sub 2 , y sub 2 close parenthesis, read ‘ x sub 2, y sub 2’")[$( x_(1) , y_(1) ) & & "read ‘" #h(0.2em) x #h(0.2em) "sub 1," #h(0.2em) y #h(0.2em) "sub 1’" \ ( x_(2) , y_(2) ) & & "read ‘" #h(0.2em) x #h(0.2em) "sub 2," #h(0.2em) y #h(0.2em) "sub 2’"$]The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class? We will use #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis")[$( x_(1) , y_(1) )$] to identify the first point and #math.equation(block: false, alt: "open parenthesis x sub 2 , y sub 2 close parenthesis")[$( x_(2) , y_(2) )$] to identify the second point. If we had more than two points, we could use #math.equation(block: false, alt: "open parenthesis x sub 3 , y sub 3 close parenthesis")[$( x_(3) , y_(3) )$], #math.equation(block: false, alt: "open parenthesis x sub 4 , y sub 4 close parenthesis")[$( x_(4) , y_(4) )$], and so on. Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 7 , 6 close parenthesis")[$( 7 , 6 )$]. #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from 0 to 7. A line passes through the points (2, 3) and (7, 6), which are plotted and labeled. The ordered pair (2, 3) is labeled (x subscript 1, y subscript 1). The ordered pair (7, 6) is labeled (x subscript 2, y subscript 2). An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The first leg, from (2, 3) to (2, 6) is labeled y subscript 2 minus y subscript 1, 6 minus 3, and 3. The second leg, from (2, 3) to (7, 6), is labeled x subscript 2 minus x subscript 1, y minus 2, and 5.], alt: "The graph shows the x y coordinate plane. The x and y-axes run from 0 to 7. A line passes through the points (2, 3) and (7, 6), which are plotted and labeled. The ordered pair (2, 3) is labeled (x subscript 1, y subscript 1). The ordered pair (7, 6) is labeled (x subscript 2, y subscript 2). An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The first leg, from (2, 3) to (2, 6) is labeled y subscript 2 minus y subscript 1, 6 minus 3, and 3. The second leg, from (2, 3) to (7, 6), is labeled x subscript 2 minus x subscript 1, y minus 2, and 5.", caption: none) Since we have two points, we will use subscript notation, #math.equation(block: false, alt: "open parenthesis 2 , x sub 1 , 3 y sub 1 close parenthesis")[$( limits(2 ",")^(x_(1) ,) limits(3)^(y_(1)) )$]#math.equation(block: false, alt: "open parenthesis 7 , 6 x sub 2 , y sub 2 close parenthesis")[$( limits(7 "," 6)^(x_(2) , y_(2)) )$]. On the graph, we counted the rise of 3 and the run of 5. Notice that the rise of 3 can be found by subtracting the #emph[y]-coordinates 6 and 3. #math.equation(block: true, alt: "3 equals 6 minus 3")[$3 = 6 − 3$]And the run of 5 can be found by subtracting the #emph[x]-coordinates 7 and 2. #math.equation(block: true, alt: "5 equals 7 minus 2")[$5 = 7 − 2$]We know #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]. So #math.equation(block: false, alt: "m equals the fraction 3 over 5")[$m = frac(3, 5)$]. We rewrite the rise and run by putting in the coordinates #math.equation(block: false, alt: "m equals the fraction 6 minus 3 over 7 minus 2")[$m = frac(6 − 3, 7 − 2)$]. But 6 is #math.equation(block: false, alt: "y sub 2")[$y_(2)$], the #emph[y]-coordinate of the second point and 3 is #math.equation(block: false, alt: "y sub 1")[$y_(1)$], the #emph[y]-coordinate of the first point. So we can rewrite the slope using subscript notation. #math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over 7 minus 2")[$m = frac(y_(2) − y_(1), 7 − 2)$] Also, 7 is #math.equation(block: false, alt: "x sub 2")[$x_(2)$], the #emph[x]-coordinate of the second point and 2 is #math.equation(block: false, alt: "x sub 1")[$x_(1)$], the #emph[x]-coordinate of the first point. So, again, we rewrite the slope using subscript notation. #math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$] We’ve shown that #math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$] is really another version of #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]. We can use this formula to find the slope of a line when we have two points on the line. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Slope Formula] The slope of the line between two points #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis")[$( x_(1) , y_(1) )$] and #math.equation(block: false, alt: "open parenthesis x sub 2 , y sub 2 close parenthesis")[$( x_(2) , y_(2) )$] is #math.equation(block: true, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]This is the #strong[slope formula]. The slope is: #math.equation(block: true, alt: "y of the second point minus y of the first point; over; x of the second point minus x of the first point.")[$y #h(0.2em) "of the second point minus" #h(0.2em) y #h(0.2em) "of the first point" \ "over" \ \ x #h(0.2em) "of the second point minus" #h(0.2em) x #h(0.2em) "of the first point."$] ] #examplebox("Example 9")[][ Use the #strong[slope formula] to find the slope of the line between the points #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$] and #math.equation(block: false, alt: "open parenthesis 4 , 5 close parenthesis")[$( 4 , 5 )$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([We'll call #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$] point \#1 and #math.equation(block: false, alt: "open parenthesis 4 , 5 close parenthesis")[$( 4 , 5 )$] point \#2.], [#math.equation(block: false, alt: "open parenthesis 1 , 2 x sub 1 , y sub 1 close parenthesis open parenthesis 4 , 5 x sub 2 , y sub 2 close parenthesis")[$( limits(display(1 "," 2))^(x_(1) , y_(1)) ) #h(0.5em) attach(( limits(4 "," 5)^(x_(2) , y_(2)) ))$]]), [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$].], [Substitute the values.], [], [#math.equation(block: false, alt: "y")[$y$] of the second point minus #math.equation(block: false, alt: "y")[$y$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 5 minus 2 over x sub 2 minus x sub 1")[$m = frac(5 − 2, x_(2) − x_(1))$].], [#math.equation(block: false, alt: "x")[$x$] of the second point minus #math.equation(block: false, alt: "x")[$x$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 5 minus 2 over 4 minus 1")[$m = frac(5 − 2, 4 − 1)$].], [Simplify the numerator and the denominator.], [#math.equation(block: false, alt: "m equals the fraction 3 over 3")[$m = frac(3, 3)$].], [Simplify.], [#math.equation(block: false, alt: "m equals 1")[$m = 1$].], )) Let’s confirm this by counting out the slope on a graph using #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]. #figure(figph[The graph shows the x y-coordinate plane. The x and y-axes of the plane run from 0 to 7. A line passes through the points (1, 2) and (4, 5), which are plotted. An additional point is plotted at (1, 5). The three points form a right triangle, with the line from (1, 2) to (4, 5) forming the hypotenuse and the lines from (1, 2) to (1, 5) and from (1, 5) to (4, 5) forming the legs. The leg from (1, 2) to (1, 5) is labeled “rise” and the leg from (1, 5) to (4, 5) is labeled “run”.], alt: "The graph shows the x y-coordinate plane. The x and y-axes of the plane run from 0 to 7. A line passes through the points (1, 2) and (4, 5), which are plotted. An additional point is plotted at (1, 5). The three points form a right triangle, with the line from (1, 2) to (4, 5) forming the hypotenuse and the lines from (1, 2) to (1, 5) and from (1, 5) to (4, 5) forming the legs. The leg from (1, 2) to (1, 5) is labeled “rise” and the leg from (1, 5) to (4, 5) is labeled “run”.", caption: none) It doesn’t matter which point you call point \#1 and which one you call point \#2. The slope will be the same. Try the calculation yourself. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the slope formula to find the slope of the line through the points: #math.equation(block: false, alt: "open parenthesis 8 , 5 close parenthesis")[$( 8 , 5 )$] and #math.equation(block: false, alt: "open parenthesis 6 , 3 close parenthesis")[$( 6 , 3 )$]. #solutionbox[ 1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the slope formula to find the slope of the line through the points: #math.equation(block: false, alt: "open parenthesis 1 , 5 close parenthesis")[$( 1 , 5 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 9 close parenthesis")[$( 5 , 9 )$]. #solutionbox[ 1 ] ] #examplebox("Example 10")[][ Use the slope formula to find the slope of the line through the points #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$] and #math.equation(block: false, alt: "open parenthesis −7 , 4 close parenthesis")[$( −7 , 4 )$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([We'll call #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$] point \#1 and #math.equation(block: false, alt: "open parenthesis −7 , 4 close parenthesis")[$( −7 , 4 )$] point \#2.], [#math.equation(block: false, alt: "open parenthesis −2 , −3 x sub 1 , y sub 1 close parenthesis open parenthesis −7 , 4 x sub 2 , y sub 2 close parenthesis")[$( limits(display(−2 "," −3))^(x_(1) , y_(1)) ) #h(0.5em) attach(( limits(−7 "," 4)^(x_(2) , y_(2)) ))$]]), [Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$].], [Substitute the values.], [], [#math.equation(block: false, alt: "y")[$y$] of the second point minus #math.equation(block: false, alt: "y")[$y$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 4 minus open parenthesis −3 close parenthesis over x sub 2 minus x sub 1")[$m = frac(4 − ( −3 ), x_(2) − x_(1))$].], [#math.equation(block: false, alt: "x")[$x$] of the second point minus #math.equation(block: false, alt: "x")[$x$] of the first point], [#math.equation(block: false, alt: "m equals the fraction 4 minus open parenthesis −3 close parenthesis over −7 minus open parenthesis −2 close parenthesis")[$m = frac(4 − ( −3 ), −7 − ( −2 ))$].], [Simplify.], [#math.equation(block: false, alt: "m equals the fraction 7 over −5; m equals minus the fraction 7 over 5")[$m = frac(7, −5) \ m = − frac(7, 5)$]], )) Let’s verify this slope on the graph shown. #figure(figph[The graph shows the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 2 and the y-axis of the plane runs from negative 6 to 5. A line passes through the points (negative 7, 4) and (negative 2, negative 3), which are plotted and labeled. An additional point is plotted at (negative 7, negative 3). The three points form a right triangle, with the line from (negative 7, 4) to (negative 2, negative 3) forming the hypotenuse and the lines from (negative 7, 4) to (negative 7, negative 3) and from (negative 7, negative 3) to (negative 2, negative 3) forming the legs. The leg from (negative 7, 4) to (negative 7, negative 3) is labeled “rise” and the leg from (negative 7, negative 3) to (negative 2, negative 3) is labeled “run”.], alt: "The graph shows the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 2 and the y-axis of the plane runs from negative 6 to 5. A line passes through the points (negative 7, 4) and (negative 2, negative 3), which are plotted and labeled. An additional point is plotted at (negative 7, negative 3). The three points form a right triangle, with the line from (negative 7, 4) to (negative 2, negative 3) forming the hypotenuse and the lines from (negative 7, 4) to (negative 7, negative 3) and from (negative 7, negative 3) to (negative 2, negative 3) forming the legs. The leg from (negative 7, 4) to (negative 7, negative 3) is labeled “rise” and the leg from (negative 7, negative 3) to (negative 2, negative 3) is labeled “run”.", caption: none) #math.equation(block: true, alt: "m, equals, the fraction rise over run; m, equals, the fraction −7 over 5; m, equals, minus the fraction 7 over 5")[$m & = & frac("rise", "run") \ m & = & frac(−7, 5) \ m & = & − frac(7, 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the slope formula to find the slope of the line through the points: #math.equation(block: false, alt: "open parenthesis −3 , 4 close parenthesis")[$( −3 , 4 )$] and #math.equation(block: false, alt: "open parenthesis 2 , −1 close parenthesis .")[$( 2 , −1 ) .$] #solutionbox[ #math.equation(block: true, alt: "−1")[$−1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the slope formula to find the slope of the line through the pair of points: #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$] and #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$]. #solutionbox[ 10 ] ] === Graph a Line Given a Point and the Slope Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines. One other method we can use to graph lines is called the #strong[point–slope method]. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line. The book graphs the line through (1, −1) with slope m = ¾: starting at the marked point, count a run of 4 to reach (5, −1), then a rise of 3 up to (5, 2) on the line. The dashed legs draw that run-and-rise triangle live — the run leg stays 4, the rise leg stretches to 4m. Drag m to 2 and the rise becomes 8; drag m negative and the rise leg points downward, so the line falls left to right; at m = 0 the rise vanishes and the line is horizontal.{"functions":\[{"color":"\#1f4e79","domain":null,"expression":"m\*(x - 1) - 1","expression2":"","id":"line","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true},{"color":"\#94a3b8","domain":\[1,5\],"expression":"-1","expression2":"","id":"runleg","inequality":"lt","lineDash":\[6,4\],"lineWidth":1.75,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true},{"color":"\#94a3b8","domain":null,"expression":"5","expression2":"-1 + 4\*m\*t","id":"riseleg","inequality":"lt","lineDash":\[6,4\],"lineWidth":1.75,"mode":"parametric","tMax":1,"tMin":0,"variable":"t","visible":true},{"color":"\#b8442a","domain":null,"expression":"1 + 0.15\*cos(t)","expression2":"-1 + 0.15\*sin(t)","id":"point","inequality":"lt","lineDash":\[\],"lineWidth":3,"mode":"parametric","tMax":6.2832,"tMin":0,"variable":"t","visible":true}\],"grid":{"majorSpacing":1,"showAxes":true,"showLabels":true},"parameters":\[{"id":"p-m","label":"Slope m","max":4,"min":-4,"name":"m","step":0.25,"value":0.75}\],"title":"Line through (1, \\u22121) with slope m","version":1,"viewport":{"centerX":0,"centerY":0,"scale":40}}#examplebox("Example 11")[How To Graph a Line Given a Point and The Slope][ Graph the line passing through the point #math.equation(block: false, alt: "open parenthesis 1 , −1 close parenthesis")[$( 1 , −1 )$] whose slope is #math.equation(block: false, alt: "m equals the fraction 3 over 4")[$m = frac(3, 4)$]. #solutionbox[ #figure(figph[This table has three columns and four rows. The first row says, “Step 1. Plot the given point. Plot (1, negative 1).” To the right is a graph of the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis of the plane runs from negative 3 to 4. The point (0, negative 1) is plotted.], alt: "This table has three columns and four rows. The first row says, “Step 1. Plot the given point. Plot (1, negative 1).” To the right is a graph of the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis of the plane runs from negative 3 to 4. The point (0, negative 1) is plotted.", caption: none) #figure(figph[The second row says, “Step 2. Use the slope formula m equals rise divided by run to identify the rise and the run.” The rise and run are 3 and 4, so m equals 3 divided by 4.], alt: "The second row says, “Step 2. Use the slope formula m equals rise divided by run to identify the rise and the run.” The rise and run are 3 and 4, so m equals 3 divided by 4.", caption: none) #figure(figph[The third row says “Step 3. Starting at the given point, count out the rise and run to mark the second point.” We start at (1, negative 1) and count the rise and run. Up three units and right 4 units. In the graph on the right, an additional two points are plotted: (1, 2), which is 3 units up from (1, negative 1), and (5, 2), which is 3 units up and 4 units right from (1, negative 1).], alt: "The third row says “Step 3. Starting at the given point, count out the rise and run to mark the second point.” We start at (1, negative 1) and count the rise and run. Up three units and right 4 units. In the graph on the right, an additional two points are plotted: (1, 2), which is 3 units up from (1, negative 1), and (5, 2), which is 3 units up and 4 units right from (1, negative 1).", caption: none) #figure(figph[The fourth row says “Step 4. Connect the points with a line.” On the graph to the right, a line is drawn through the points (1, negative 1) and (5, 2). This line is also the hypotenuse of the right triangle formed by the three points, (1, negative 1), (1, 2) and (5, 2).], alt: "The fourth row says “Step 4. Connect the points with a line.” On the graph to the right, a line is drawn through the points (1, negative 1) and (5, 2). This line is also the hypotenuse of the right triangle formed by the three points, (1, negative 1), (1, 2) and (5, 2).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line passing through the point #math.equation(block: false, alt: "open parenthesis 2 , −2 close parenthesis")[$( 2 , −2 )$] with the slope #math.equation(block: false, alt: "m equals the fraction 4 over 3")[$m = frac(4, 3)$]. #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 4, negative 10) and (2, negative 2).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 4, negative 10) and (2, negative 2).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line passing through the point #math.equation(block: false, alt: "open parenthesis −2 , 3 close parenthesis")[$( −2 , 3 )$] with the slope #math.equation(block: false, alt: "m equals the fraction 1 over 4")[$m = frac(1, 4)$]. #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 2, 3) and (10, 6).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 2, 3) and (10, 6).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph a line given a point and the slope.] + Plot the given point. + Use the slope formula #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] to identify the rise and the run. + Starting at the given point, count out the rise and run to mark the second point. + Connect the points with a line. ] #examplebox("Example 12")[][ Graph the line with #emph[y]-intercept 2 whose slope is #math.equation(block: false, alt: "m equals minus the fraction 2 over 3")[$m = − frac(2, 3)$]. #solutionbox[ Plot the given point, the #emph[y]-intercept, #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]. #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (0, 2) is plotted.], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (0, 2) is plotted.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([Identify the rise and the run.], [#math.equation(block: false, alt: "m equals minus the fraction 2 over 3")[$m = − frac(2, 3)$]]), [], [#math.equation(block: false, alt: "the fraction rise over run equals the fraction −2 over 3")[$frac("rise", "run") = frac(−2, 3)$]], [], [#math.equation(block: false, alt: "rise equals −2")[$"rise" = −2$]], [], [#math.equation(block: false, alt: "run equals 3")[$"run" = 3$]], )) Count the rise and the run. Mark the second point. #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The points (0, 2), (0, 0), and (3,0) are plotted and labeled. The line from (0, 2) to (0, 0) is labeled “down 2” and the line from (0, 0) to (3, 0) is labeled “right 3”.], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The points (0, 2), (0, 0), and (3,0) are plotted and labeled. The line from (0, 2) to (0, 0) is labeled “down 2” and the line from (0, 0) to (3, 0) is labeled “right 3”.", caption: none) Connect the two points with a line. #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (0, 2) and (3,0).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (0, 2) and (3,0).", caption: none) You can check your work by finding a third point. Since the slope is #math.equation(block: false, alt: "m equals minus the fraction 2 over 3")[$m = − frac(2, 3)$], it can be written as #math.equation(block: false, alt: "m equals the fraction 2 over −3")[$m = frac(2, −3)$]. Go back to #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] and count out the rise, 2, and the run, #math.equation(block: false, alt: "−3")[$−3$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line with the #emph[y]-intercept 4 and slope #math.equation(block: false, alt: "m equals minus the fraction 5 over 2.")[$m = − frac(5, 2) .$] #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 4) and passes through the point (4, negative 6).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 4) and passes through the point (4, negative 6).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line with the #emph[x]-intercept #math.equation(block: false, alt: "−3")[$−3$] and slope #math.equation(block: false, alt: "m equals minus the fraction 3 over 4")[$m = − frac(3, 4)$]. #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 3, 0) and passes through the point (1, negative 3).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 3, 0) and passes through the point (1, negative 3).", caption: none) ] ] #examplebox("Example 13")[][ Graph the line passing through the point #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] whose slope is #math.equation(block: false, alt: "m equals 4.")[$m = 4 .$] #solutionbox[ Plot the given point. #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (negative 1, negative 3) is plotted and labeled.], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (negative 1, negative 3) is plotted and labeled.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([Identify the rise and the run.], [#math.equation(block: false, alt: "m equals 4")[$m = 4$]]), [Write 4 as a fraction.], [#math.equation(block: false, alt: "the fraction rise over run equals the fraction 4 over 1")[$frac("rise", "run") = frac(4, 1)$]], [], [#math.equation(block: false, alt: "rise equals 4 , run equals 1")[$"rise" = 4 , "run" = 1$]], )) Count the rise and run and mark the second point. #figure(figph[This figure shows how to graph the line passing through the point (negative 1, negative 3) whose slope is 4. The first step is to identify the rise and run. The rise is 4 and the run is 1. 4 divided by 1 is 4, so the slope is 4. Next we count the rise and run and mark the second point. To the right is a graph of the x y-coordinate plane. The x and y-axes run from negative 5 to 5. We start at the plotted point (negative 1, negative 3) and count the rise, 4. We reach the point negative 1, 1, which we plot. We then count the run from this point, which is 1. We reach the point (0, 1), which is plotted. The last step is to connect the two points with a line. We draw a line which passes through the points (negative 1, negative 3) and (0, 1).], alt: "This figure shows how to graph the line passing through the point (negative 1, negative 3) whose slope is 4. The first step is to identify the rise and run. The rise is 4 and the run is 1. 4 divided by 1 is 4, so the slope is 4. Next we count the rise and run and mark the second point. To the right is a graph of the x y-coordinate plane. The x and y-axes run from negative 5 to 5. We start at the plotted point (negative 1, negative 3) and count the rise, 4. We reach the point negative 1, 1, which we plot. We then count the run from this point, which is 1. We reach the point (0, 1), which is plotted. The last step is to connect the two points with a line. We draw a line which passes through the points (negative 1, negative 3) and (0, 1).", caption: none) Connect the two points with a line. #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (-1, -3) and (1,0).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (-1, -3) and (1,0).", caption: none) You can check your work by finding a third point. Since the slope is #math.equation(block: false, alt: "m equals 4")[$m = 4$], it can be written as #math.equation(block: false, alt: "m equals the fraction −4 over −1")[$m = frac(−4, −1)$]. Go back to #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] and count out the rise, #math.equation(block: false, alt: "−4")[$−4$], and the run, #math.equation(block: false, alt: "−1")[$−1$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line with the point #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and slope #math.equation(block: false, alt: "m equals 3")[$m = 3$]. #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (negative 1, 4).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (negative 1, 4).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the line with the point #math.equation(block: false, alt: "open parenthesis 4 , −2 close parenthesis")[$( 4 , −2 )$] and slope #math.equation(block: false, alt: "m equals −2")[$m = −2$]. #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (4, negative 2) and (5, negative 4).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (4, negative 2) and (5, negative 4).", caption: none) ] ] === Solve Slope Applications At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now. #examplebox("Example 14")[][ The ‘pitch’ of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown? #figure(figph[This figure shows a house with a sloped roof. The roof on one half of the building is labeled "pitch of the roof". There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled "rise equals 9 feet". There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled "run equals 18 feet".], alt: "This figure shows a house with a sloped roof. The roof on one half of the building is labeled \"pitch of the roof\". There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled \"rise equals 9 feet\". There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled \"run equals 18 feet\".", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$]]), [Substitute the values for rise and run.], [#math.equation(block: false, alt: "m equals the fraction 9 over 18")[$m = frac(9, 18)$]], [Simplify.], [#math.equation(block: false, alt: "m equals the fraction 1 over 2")[$m = frac(1, 2)$]], [The slope of the roof is #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$].], [], [], [The roof rises 1 foot for every 2 feet of horizontal run.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use , substituting the rise = 14 and run = 24. #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 12")[$frac(7, 12)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use , substituting rise = 15 and run = 36. #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 12")[$frac(5, 12)$] ] ] #examplebox("Example 15")[][ Have you ever thought about the sewage pipes going from your house to the street? They must slope down #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] inch per foot in order to drain properly. What is the required slope? #figure(figph[This figure is a right triangle. One leg is negative one quarter inch and the other leg is one foot.], alt: "This figure is a right triangle. One leg is negative one quarter inch and the other leg is one foot.", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Use the slope formula.], [#math.equation(block: false, alt: "m equals the fraction rise over run; m equals the fraction minus the fraction 1 over 4 inch over 1 foot; m equals the fraction minus the fraction 1 over 4 inch over 12 inches")[$m = frac("rise", "run") \ m = frac(− frac(1, 4) "inch", "1 foot") \ m = frac(− frac(1, 4) "inch", "12 inches")$]]), [Simplify.], [#math.equation(block: false, alt: "m equals minus the fraction 1 over 48")[$m = − frac(1, 48)$]], [], [The slope of the pipe is #math.equation(block: false, alt: "minus the fraction 1 over 48")[$− frac(1, 48)$].], )) The pipe drops 1 inch for every 48 inches of horizontal run. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of a pipe that slopes down #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] inch per foot. #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 36")[$− frac(1, 36)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the slope of a pipe that slopes down #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] inch per yard. #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 48")[$− frac(1, 48)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Access these online resources for additional instruction and practice with understanding slope of a line. - #link("https://openstax.org/l/25Geoboard")[Practice Slope with a Virtual Geoboard] - #link("https://openstax.org/l/25VirtualGeo")[Small, Medium, and Large Virtual Geoboards] - #link("https://openstax.org/l/25APGeoboard")[Explore Area and Perimeter with a Geoboard] ] === Key Concepts - #strong[Find the Slope of a Line from its Graph using] #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] + Locate two points on the line whose coordinates are integers. + Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point. + Count the rise and the run on the legs of the triangle. + Take the ratio of rise to run to find the slope. #linebreak() - #strong[Graph a Line Given a Point and the Slope] + Plot the given point. + Use the slope formula #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] to identify the rise and the run. + Starting at the given point, count out the rise and run to mark the second point. + Connect the points with a line. #linebreak() #linebreak() - #strong[Slope of a Horizontal Line] - The slope of a horizontal line, #math.equation(block: false, alt: "y equals b")[$y = b$], is 0. - #strong[Slope of a vertical line] - The slope of a vertical line, #math.equation(block: false, alt: "x equals a")[$x = a$], is undefined ==== Practice Makes Perfect #strong[Use Geoboards to Model Slope] In the following exercises, find the slope modeled on each geoboard. This interactive geoboard lets you explore slope by stretching rubber bands between pegs. Drag the endpoints to create lines with different rises and runs, and watch the slope value update in real time. Use this space to test your understanding of how vertical and horizontal changes affect the steepness and direction of a line.{"functions":\[\],"geoboard":{"bands":\[\],"cols":5,"draw":true,"rows":5},"grid":{"majorSpacing":1,"showAxes":false,"showLabels":false},"parameters":\[\],"title":"Practice Modeling Slope on a Geoboard","viewport":{"centerX":2,"centerY":2,"scale":45}}#figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 5, row 2, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 5, row 2, forming a line.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] ] #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 4 and the peg in column 5, row 2, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 4 and the peg in column 5, row 2, forming a line.", caption: none) #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 4, row 2, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 4, row 2, forming a line.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 3")[$frac(2, 3)$] ] #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 3, row 4 and the peg in column 5, row 1, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 3, row 4 and the peg in column 5, row 1, forming a line.", caption: none) #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 1 and the peg in column 4, row 4, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 1 and the peg in column 4, row 4, forming a line.", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction −3 over 2 equals minus the fraction 3 over 2")[$frac(−3, 2) = − frac(3, 2)$] ] #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 5, row 4, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 5, row 4, forming a line.", caption: none) #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1 and the peg in column 5, row 4, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1 and the peg in column 5, row 4, forming a line.", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] ] #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 2 and the peg in column 4, row 5, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 2 and the peg in column 4, row 5, forming a line.", caption: none) In the following exercises, model each slope. Draw a picture to show your results. #math.equation(block: true, alt: "the fraction 2 over 3")[$frac(2, 3)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 5 and the peg in column 5, row 3, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 5 and the peg in column 5, row 3, forming a line.", caption: none) ] #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 5, row 3, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 5, row 3, forming a line.", caption: none) ] #math.equation(block: true, alt: "the fraction 4 over 3")[$frac(4, 3)$] #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 3, row 5, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 3, row 5, forming a line.", caption: none) ] #math.equation(block: true, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] #solutionbox[ #figure(figph[The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2 and the peg in column 4, row 4, forming a line.], alt: "The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2 and the peg in column 4, row 4, forming a line.", caption: none) ] #math.equation(block: true, alt: "minus the fraction 3 over 2")[$− frac(3, 2)$] #strong[Use] #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac("rise", "run")$] #strong[to find the Slope of a Line from its Graph] In the following exercises, find the slope of each line shown. #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 10, negative 8), (0, negative 4), and (10, 0).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 10, negative 8), (0, negative 4), and (10, 0).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 5")[$frac(2, 5)$] ] #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, negative 8) and (2, negative 2).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, negative 8) and (2, negative 2).", caption: none) #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 4, negative 6) and (4, 4).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 4, negative 6) and (4, 4).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 4")[$frac(5, 4)$] ] #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, negative 2) and passes through the point (3, 3).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, negative 2) and passes through the point (3, 3).", caption: none) #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 3) and (3, 1).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 3) and (3, 1).", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 3")[$− frac(1, 3)$] ] #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 4) and (2, 2).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 4) and (2, 2).", caption: none) #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, 6) and passes through the point (4, 3).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, 6) and passes through the point (4, 3).", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] ] #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the point (negative 3, 1) and intercepts the y-axis at (0, negative 1).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the point (negative 3, 1) and intercepts the y-axis at (0, negative 1).", caption: none) #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (2, 4).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (2, 4).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] ] #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 1) and (2, 3).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 1) and (2, 3).", caption: none) #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 6) and (1, 1).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 6) and (1, 1).", caption: none) #solutionbox[ #math.equation(block: true, alt: "−3")[$−3$] ] #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the point (negative 1, 3) and intercepts the x-axis at (3, 0).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the point (negative 1, 3) and intercepts the x-axis at (3, 0).", caption: none) #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 6) and (1, 4).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 6) and (1, 4).", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] ] #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 1, 3) and (1, 2).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 1, 3) and (1, 2).", caption: none) #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 1).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 1).", caption: none) #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$] ] #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (4, 2) and (7, 3).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (4, 2) and (7, 3).", caption: none) #strong[Find the Slope of Horizontal and Vertical Lines] In the following exercises, find the slope of each line. #math.equation(block: true, alt: "y equals 3")[$y = 3$] #solutionbox[ 0 ] #math.equation(block: true, alt: "y equals 1")[$y = 1$] #math.equation(block: true, alt: "x equals 4")[$x = 4$] #solutionbox[ undefined ] #math.equation(block: true, alt: "x equals 2")[$x = 2$] #math.equation(block: true, alt: "y equals −2")[$y = −2$] #solutionbox[ 0 ] #math.equation(block: true, alt: "y equals −3")[$y = −3$] #math.equation(block: true, alt: "x equals −5")[$x = −5$] #solutionbox[ undefined ] #math.equation(block: true, alt: "x equals −4")[$x = −4$] #strong[Use the Slope Formula to find the Slope of a Line between Two Points] In the following exercises, use the slope formula to find the slope of the line between each pair of points. #math.equation(block: true, alt: "open parenthesis 1 , 4 close parenthesis , open parenthesis 3 , 9 close parenthesis")[$( 1 , 4 ) , ( 3 , 9 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 2")[$frac(5, 2)$] ] #math.equation(block: true, alt: "open parenthesis 2 , 3 close parenthesis , open parenthesis 5 , 7 close parenthesis")[$( 2 , 3 ) , ( 5 , 7 )$] #math.equation(block: true, alt: "open parenthesis 0 , 3 close parenthesis , open parenthesis 4 , 6 close parenthesis")[$( 0 , 3 ) , ( 4 , 6 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] ] #math.equation(block: true, alt: "open parenthesis 0 , 1 close parenthesis , open parenthesis 5 , 4 close parenthesis")[$( 0 , 1 ) , ( 5 , 4 )$] #math.equation(block: true, alt: "open parenthesis 2 , 5 close parenthesis , open parenthesis 4 , 0 close parenthesis")[$( 2 , 5 ) , ( 4 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 2")[$− frac(5, 2)$] ] #math.equation(block: true, alt: "open parenthesis 3 , 6 close parenthesis , open parenthesis 8 , 0 close parenthesis")[$( 3 , 6 ) , ( 8 , 0 )$] #math.equation(block: true, alt: "open parenthesis −3 , 3 close parenthesis , open parenthesis 4 , −5 close parenthesis")[$( −3 , 3 ) , ( 4 , −5 )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 8 over 7")[$− frac(8, 7)$] ] #math.equation(block: true, alt: "open parenthesis −2 , 4 close parenthesis , open parenthesis 3 , −1 close parenthesis")[$( −2 , 4 ) , ( 3 , −1 )$] #math.equation(block: true, alt: "open parenthesis −1 , −2 close parenthesis , open parenthesis 2 , 5 close parenthesis")[$( −1 , −2 ) , ( 2 , 5 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 3")[$frac(7, 3)$] ] #math.equation(block: true, alt: "open parenthesis −2 , −1 close parenthesis , open parenthesis 6 , 5 close parenthesis")[$( −2 , −1 ) , ( 6 , 5 )$] #math.equation(block: true, alt: "open parenthesis 4 , −5 close parenthesis , open parenthesis 1 , −2 close parenthesis")[$( 4 , −5 ) , ( 1 , −2 )$] #solutionbox[ #math.equation(block: true, alt: "−1")[$−1$] ] #math.equation(block: true, alt: "open parenthesis 3 , −6 close parenthesis , open parenthesis 2 , −2 close parenthesis")[$( 3 , −6 ) , ( 2 , −2 )$] #strong[Graph a Line Given a Point and the Slope] In the following exercises, graph each line with the given point and slope. #math.equation(block: true, alt: "open parenthesis 1 , −2 close parenthesis")[$( 1 , −2 )$]; #math.equation(block: true, alt: "m equals the fraction 3 over 4")[$m = frac(3, 4)$] #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, negative 2) and (5, 1).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, negative 2) and (5, 1).", caption: none) ] #math.equation(block: true, alt: "open parenthesis 1 , −1 close parenthesis")[$( 1 , −1 )$]; #math.equation(block: true, alt: "m equals the fraction 2 over 3")[$m = frac(2, 3)$] #math.equation(block: true, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$]; #math.equation(block: true, alt: "m equals minus the fraction 1 over 3")[$m = − frac(1, 3)$] #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (2, 5) and (5, 4).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (2, 5) and (5, 4).", caption: none) ] #math.equation(block: true, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]; #math.equation(block: true, alt: "m equals minus the fraction 1 over 2")[$m = − frac(1, 2)$] #math.equation(block: true, alt: "open parenthesis −3 , 4 close parenthesis")[$( −3 , 4 )$]; #math.equation(block: true, alt: "m equals minus the fraction 3 over 2")[$m = − frac(3, 2)$] #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 4) and (negative 1, 1).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 4) and (negative 1, 1).", caption: none) ] #math.equation(block: true, alt: "open parenthesis −2 , 5 close parenthesis")[$( −2 , 5 )$]; #math.equation(block: true, alt: "m equals minus the fraction 5 over 4")[$m = − frac(5, 4)$] #math.equation(block: true, alt: "open parenthesis −1 , −4 close parenthesis")[$( −1 , −4 )$]; #math.equation(block: true, alt: "m equals the fraction 4 over 3")[$m = frac(4, 3)$] #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 1, negative 4) and intercepts the x-axis at (2, 0).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 1, negative 4) and intercepts the x-axis at (2, 0).", caption: none) ] #math.equation(block: true, alt: "open parenthesis −3 , −5 close parenthesis")[$( −3 , −5 )$]; #math.equation(block: true, alt: "m equals the fraction 3 over 2")[$m = frac(3, 2)$] #emph[y]-intercept 3; #math.equation(block: false, alt: "m equals minus the fraction 2 over 5")[$m = − frac(2, 5)$] #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 3) and passes through the point (5, 1).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 3) and passes through the point (5, 1).", caption: none) ] #emph[y]-intercept 5; #math.equation(block: false, alt: "m equals minus the fraction 4 over 3")[$m = − frac(4, 3)$] #emph[x]-intercept #math.equation(block: false, alt: "−2")[$−2$]; #math.equation(block: false, alt: "m equals the fraction 3 over 4")[$m = frac(3, 4)$] #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 3).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 3).", caption: none) ] #emph[x]-intercept #math.equation(block: false, alt: "−1")[$−1$]; #math.equation(block: false, alt: "m equals the fraction 1 over 5")[$m = frac(1, 5)$] #math.equation(block: true, alt: "open parenthesis −3 , 3 close parenthesis")[$( −3 , 3 )$]; #math.equation(block: true, alt: "m equals 2")[$m = 2$] #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 3) and (negative 2, 5).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 3) and (negative 2, 5).", caption: none) ] #math.equation(block: true, alt: "open parenthesis −4 , 2 close parenthesis")[$( −4 , 2 )$]; #math.equation(block: true, alt: "m equals 4")[$m = 4$] #math.equation(block: true, alt: "open parenthesis 1 , 5 close parenthesis")[$( 1 , 5 )$]; #math.equation(block: true, alt: "m equals −3")[$m = −3$] #solutionbox[ #figure(figph[The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, 5) and (2, 2).], alt: "The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, 5) and (2, 2).", caption: none) ] #math.equation(block: true, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$]; #math.equation(block: true, alt: "m equals −1")[$m = −1$] ==== Everyday Math #strong[Slope of a roof.] An easy way to determine the slope of a roof is to set one end of a 12 inch level on the roof surface and hold it level. Then take a tape measure or ruler and measure from the other end of the level down to the roof surface. This will give you the slope of the roof. Builders, sometimes, refer to this as pitch and state it as an “#math.equation(block: false, alt: "x")[$x$] 12 pitch” meaning #math.equation(block: false, alt: "the fraction x over 12")[$frac(x, 12)$], where #math.equation(block: false, alt: "x")[$x$] is the measurement from the roof to the level—the rise. It is also sometimes stated as an “#math.equation(block: false, alt: "x")[$x$]-in-12 pitch”. + ⓐ What is the slope of the roof in this picture? + ⓑ What is the pitch in construction terms? #linebreak() #figure(figph[This figure shows one side of a sloped roof of a house. The rise of the roof is labeled “4 inches” and the run of the roof is labeled “12 inches”.], alt: "This figure shows one side of a sloped roof of a house. The rise of the roof is labeled “4 inches” and the run of the roof is labeled “12 inches”.", caption: none) #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] ⓑ 4 12 pitch or 4-in-12 pitch ] The slope of the roof shown here is measured with a 12” level and a ruler. What is the slope of this roof? #figure(figph[This figure shows one side of a sloped roof of a house. The rise of the roof is measured with a ruler and shown to be 7 inches. The run of the roof is measured with a twelve inch level and shown to be 12 inches.], alt: "This figure shows one side of a sloped roof of a house. The rise of the roof is measured with a ruler and shown to be 7 inches. The run of the roof is measured with a twelve inch level and shown to be 12 inches.", caption: none) #strong[Road grade.] A local road has a grade of 6%. The grade of a road is its slope expressed as a percent. Find the slope of the road as a fraction and then simplify. What rise and run would reflect this slope or grade? #solutionbox[ #math.equation(block: false, alt: "the fraction 3 over 50")[$frac(3, 50)$]; rise = 3, run = 50 ] #strong[Highway grade.] A local road rises 2 feet for every 50 feet of highway. + ⓐ What is the slope of the highway? + ⓑ The grade of a highway is its slope expressed as a percent. What is the grade of this highway? #strong[Wheelchair ramp.] The rules for wheelchair ramps require a maximum 1-inch rise for a 12-inch run. + ⓐ How long must the ramp be to accommodate a 24-inch rise to the door? + ⓑ Create a model of this ramp. #solutionbox[ ⓐ 288 inches (24 feet) ⓑ Models will vary. ] #strong[Wheelchair ramp.] A 1-inch rise for a 16-inch run makes it easier for the wheelchair rider to ascend a ramp. + ⓐ How long must a ramp be to easily accommodate a 24-inch rise to the door? + ⓑ Create a model of this ramp. ==== Writing Exercises What does the sign of the slope tell you about a line? #solutionbox[ When the slope is a positive number the line goes up from left to right. When the slope is a negative number the line goes down from left to right. ] How does the graph of a line with slope #math.equation(block: false, alt: "m equals the fraction 1 over 2")[$m = frac(1, 2)$] differ from the graph of a line with slope #math.equation(block: false, alt: "m equals 2")[$m = 2$]? Why is the slope of a vertical line “undefined”? #solutionbox[ A vertical line has 0 run and since division by 0 is undefined the slope is undefined. ] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This table has seven rows and four columns. The first row is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the next row reads “use geoboards to model slope.” The third row reads “use m equals rise divided by run to find the slope of a line from its graph.” The fourth row reads “find the slope of horizontal and vertical lines.” The fifth row reads “use the slope formula to find the slope of a line between two points.” The sixth row reads “graph a line given a point and the slope.” The last row reads “solve slope applications.” The remaining columns are blank.], alt: "This table has seven rows and four columns. The first row is a header row and it labels each column. The first column is labeled \"I can …\", the second \"Confidently\", the third “With some help” and the last \"No–I don’t get it\". In the “I can…” column the next row reads “use geoboards to model slope.” The third row reads “use m equals rise divided by run to find the slope of a line from its graph.” The fourth row reads “find the slope of horizontal and vertical lines.” The fifth row reads “use the slope formula to find the slope of a line between two points.” The sixth row reads “graph a line given a point and the slope.” The last row reads “solve slope applications.” The remaining columns are blank.", caption: none) ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?