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== 5.8#h(0.6em)Graphing Functions


  #figure(figph[People riding on chairlifts over a mountain slope.], alt: "People riding on chairlifts over a mountain slope.", caption: [The ski lifts and the mountain both have a slope.])


  
    
=== Learning Objectives


    After completing this module, you should be able to:


    + Graph functions using intercepts.
+ Compute slope.
+ Graph functions using slope and #math.equation(block: false, alt: "y")[$y$]-intercept.
+ Graph horizontal and vertical lines.
+ Interpret graphs of functions.
+ Model applications using slope and #math.equation(block: false, alt: "y")[$y$]-intercept.


    In this section, we will expand our knowledge of graphing by graphing linear functions. There are many real-world scenarios that can be represented by graphs of linear functions. Imagine a chairlift going up at a ski resort. The journey a skier takes travelling up the chairlift could be represented as a linear function with a positive slope. The journey a skier takes down the slopes could be represented by a linear function with a negative slope.


  
  
    
=== Graphing Functions Using Intercepts


    Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points. At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the #math.equation(block: false, alt: "x")[$x$]-axis and the #math.equation(block: false, alt: "y")[$y$]-axis. These points are called the #strong[intercepts] of a line. Let us review the graphs of the lines in .


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)]),
  alt: "Four coordinate planes. A line is plotted on each coordinate plane. The first coordinate plane is labeled, (a) 2x plus y equals 6. The line passes through the points (0, 6) and (3, 0). The second coordinate plane is labeled, (b) 3x minus 4y equals 12. The line passes through the points (0, negative 3) and (4, 0). The third coordinate plane is labeled, (c) x minus y equals 5. The line passes through the points (0, negative 5) and (5, 0). The fourth coordinate plane is labeled, (d) y equals negative 2x. The line passes through the points (negative 2, 4), (0, 0), and (2, negative 4).", caption: none,
)


    The table below lists where each of these lines crosses the #math.equation(block: false, alt: "x")[$x$]- and #math.equation(block: false, alt: "y")[$y$]-axis. Do you see a pattern? For each line, the #math.equation(block: false, alt: "y")[$y$]-coordinate of the point where the line crosses the #math.equation(block: false, alt: "x")[$x$]-axis is zero. The point where the line crosses the #math.equation(block: false, alt: "x")[$x$]-axis has the form #math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis")[$( a , 0 )$] and is called the #math.equation(block: false, alt: "x")[$x$]-intercept of the line. The #math.equation(block: false, alt: "x")[$x$]-intercept occurs when #math.equation(block: false, alt: "y")[$y$] is zero. In each line, the #math.equation(block: false, alt: "x")[$x$]-coordinate of the point where the line crosses the #math.equation(block: false, alt: "y")[$y$]-axis is zero. The point where the line crosses the #math.equation(block: false, alt: "y")[$y$]-axis has the form #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$] and is called the #math.equation(block: false, alt: "y")[$y$]-intercept of the line. The #math.equation(block: false, alt: "y")[$y$]-intercept occurs when #math.equation(block: false, alt: "x")[$x$] is zero.


    #figure(table(
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  [Figure (a)], [3], [#math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$]], [6], [#math.equation(block: false, alt: "open parenthesis 0 , 6 close parenthesis")[$( 0 , 6 )$]],
  [Figure (b)], [4], [#math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "open parenthesis 0 , minus 3 close parenthesis")[$( 0 , − 3 )$]],
  [Figure (c)], [5], [#math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$]], [#math.equation(block: false, alt: "minus 5")[$− 5$]], [#math.equation(block: false, alt: "open parenthesis 0 , minus 5 close parenthesis")[$( 0 , − 5 )$]],
  [Figure (d)], [0], [#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]], [0], [#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]],
  [General Figure], [#math.equation(block: false, alt: "a")[$a$]], [#math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis")[$( a , 0 )$]], [#math.equation(block: false, alt: "b")[$b$]], [#math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$]],
))


    #examplebox("Example 1")[Finding #math.equation(block: false, alt: "x")[$x$]- and #math.equation(block: false, alt: "y")[$y$]-Intercepts][
Find the #math.equation(block: false, alt: "x")[$x$]-intercept and #math.equation(block: false, alt: "y")[$y$]-intercept on the (a) and (b) graphs in .


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)]),
  alt: "Two x y coordinate planes labeled (a) and (b). In each coordinate plane, the x and y axes range from negative 8 to 8, in increments of 1. Graph (a) shows a line that passes through the following points, (negative 8, 6), (0, 2), (4, 0), and (8, negative 2). Graph (b) shows a line that passes through the following points, (0, negative 6), (2, 0), and (5, 6). Note: all values are approximate.", caption: none,
)


      
      #solutionbox[
In , the graph crosses the #math.equation(block: false, alt: "x")[$x$]-axis at the point #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]. The #math.equation(block: false, alt: "x")[$x$]-intercept is #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]. The graph crosses the #math.equation(block: false, alt: "y")[$y$]-axis at the point #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]. The #math.equation(block: false, alt: "y")[$y$]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]. In , the graph crosses the #math.equation(block: false, alt: "x")[$x$]-axis at the point #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]. The #math.equation(block: false, alt: "x")[$x$]-intercept is #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]. The graph crosses the #math.equation(block: false, alt: "y")[$y$]-axis at the point #math.equation(block: false, alt: "open parenthesis 0 , minus 6 close parenthesis")[$( 0 , − 6 )$]. The #math.equation(block: false, alt: "y")[$y$]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , minus 6 close parenthesis")[$( 0 , − 6 )$].
]
]


    
    #examplebox("Example 2")[Graphing a Function Using Intercepts][
Find the intercepts of #math.equation(block: false, alt: "2 x plus y equals 8")[$2 x + y = 8$]. Then graph the function using the intercepts.


      
      #solutionbox[
Let #math.equation(block: false, alt: "y equals 0")[$y = 0$] to find the #math.equation(block: false, alt: "x")[$x$]-intercept, and let #math.equation(block: false, alt: "x equals 0")[$x = 0$] to find the #math.equation(block: false, alt: "y")[$y$]-intercept.


        #figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 x plus y equals 8")[$2 x + y = 8$]], [], [#math.equation(block: false, alt: "2 x plus y equals 8")[$2 x + y = 8$]]),
  [To find the #math.equation(block: false, alt: "x")[$x$]-intercept, let #math.equation(block: false, alt: "y equals 0")[$y = 0$].], [#math.equation(block: false, alt: "2 x plus 0 equals 8")[$2 x + 0 = 8$]], [To find the #math.equation(block: false, alt: "y")[$y$]-intercept, let #math.equation(block: false, alt: "x equals 0")[$x = 0$].], [#math.equation(block: false, alt: "2 open parenthesis 0 close parenthesis plus y equals 8")[$2 ( 0 ) + y = 8$]],
  [Simplify.], [#math.equation(block: false, alt: "2 x, equals, 8; x, equals, 4")[$2 x & = & 8 \ x & = & 4$]], [Simplify.], [#math.equation(block: false, alt: "y equals 8")[$y = 8$]],
  [The #math.equation(block: false, alt: "x")[$x$]-intercept is:], [#math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]], [The #math.equation(block: false, alt: "y")[$y$]-intercept is:], [#math.equation(block: false, alt: "open parenthesis 0 , 8 close parenthesis")[$( 0 , 8 )$]],
))


        Plot the intercepts to get the graph in .


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})]),
  alt: "A line is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The line passes through the following points, (0, 8), (4, 0), and (8, negative 8).", caption: none,
)
]
]


    
  
  
    
=== Computing Slope


    When graphing 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. In mathematics, the measure of the steepness of a line is called the #strong[slope] of the line. To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices of the triangle and one side is horizontal and one side is vertical. Next, we measure or calculate 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].


    We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change, as shown in this illustration. Slope  is a rate of change.


    #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  line((33.33pt, 33.33pt), (166.67pt, 33.33pt), stroke: 0.7pt + black)
  line((166.67pt, 33.33pt), (160.67pt, 35.83pt), (160.67pt, 30.83pt), close: true, fill: black, stroke: 0.7pt + black)
  content((100.00pt, 42.33pt), anchor: "south", text(size: 9pt)[Run])
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})]),
  alt: "An arrow pointing up and labeled Rise is connected at a right angle to an arrow pointing right and labelled Run.", caption: none,
)


    #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[
To calculate slope #math.equation(block: false, alt: "open parenthesis m close parenthesis")[$( m )$], use the formula


      #math.equation(block: true, alt: "m equals the fraction rise over run")[$m = frac(italic("rise"), italic("run"))$]
      where the rise measures the vertical change and the run measures the horizontal change.
]


    The concept of slope has many applications in the real world. In construction, the pitch of a roof, the slant of plumbing pipes, and the steepness of stairs are all applications of slope. As you ski or jog down a hill, you definitely experience slope.


    #examplebox("Example 3")[Finding the Slope from a Graph][
Find the slope of the line shown.


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  alt: "A line is plotted on an x y coordinate plane. The x-axis ranges from negative 1 to 9, in increments of 1. The y-axis ranges from negative 1 to 7, in increments of 1. The line passes through the following points, (0, 5), (3, 3), (6, 1), and (9, negative 1). Note: all values are approximate.", caption: none,
)


      
      #solutionbox[
#strong[Step 1:] Locate two points on the graph whose coordinates are integers, such as #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")[$( 3 , 3 )$]. 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")[$( 3 , 3 )$] as shown.


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  content((33.33pt, 170.00pt), anchor: "east", text(size: 8pt)[8])
})]),
  alt: "A line is plotted on an x y coordinate plane. The x-axis ranges from negative 2 to 10, in increments of 1. The y-axis ranges from negative 2 to 8, in increments of 1. The line passes through the following points, (0, 5), (3, 3), (6, 1), and (9, negative 1). A slope of the line is drawn connecting the points, (0, 3), (0, 5), and (3, 3). The vertical length between the points, (0, 3) and (0, 5) is labeled rise. The horizontal length between the points, (0, 3) and (3, 3) is labeled run. Note: all values are approximate.", caption: none,
)


        #strong[Step 2:] Count the rise; since it goes down, it is negative. The rise is −2.


        #strong[Step 3:] Count the run.	The run is 3.


        #strong[Step 4:] Use the slope formula #math.equation(block: false, alt: "m equals the fraction rise over run")[$m = frac(italic("rise"), italic("run"))$] substitute the values of the rise and run. #math.equation(block: false, alt: "m equals the fraction minus 2 over 3")[$m = frac(− 2, 3)$]


        The slope of the line is #math.equation(block: false, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$].


        The solution is #math.equation(block: false, alt: "y")[$y$] decreases by 2 units as #math.equation(block: false, alt: "x")[$x$] increases by 3 units.
]
]


    
    Sometimes we will need to find the slope of a line between two points when we don’t have a graph to measure the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing. First, we need to introduce some algebraic notation.


    We have seen that an ordered pair (#math.equation(block: false, alt: "x")[$x$], #math.equation(block: false, alt: "y")[$y$]) 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: "x")[$x$], #math.equation(block: false, alt: "y")[$y$]) be used to represent two different points? Mathematicians use subscripts to distinguish such points. For example, (#math.equation(block: false, alt: "x sub 1")[$x_(1)$], #math.equation(block: false, alt: "y sub 1")[$y_(1)$]) would be said aloud as “#math.equation(block: false, alt: "x")[$x$] sub 1, #math.equation(block: false, alt: "y")[$y$] sub 1” and (#math.equation(block: false, alt: "x sub 2")[$x_(2)$], #math.equation(block: false, alt: "y sub 2")[$y_(2)$]) read “#math.equation(block: false, alt: "x")[$x$] sub 2, #math.equation(block: false, alt: "y")[$y$] sub 2.” The “sub” is a short way of saying “subscript.” We will use (#math.equation(block: false, alt: "x sub 1")[$x_(1)$], #math.equation(block: false, alt: "y sub 1")[$y_(1)$]) to identify the first point and (#math.equation(block: false, alt: "x sub 2")[$x_(2)$], #math.equation(block: false, alt: "y sub 2")[$y_(2)$]) to identify the second point in our slope equation. If we had more than two points, (if we were finding more than one slope), we could use (#math.equation(block: false, alt: "x sub 3")[$x_(3)$], #math.equation(block: false, alt: "y sub 3")[$y_(3)$]), (#math.equation(block: false, alt: "x sub 4")[$x_(4)$], #math.equation(block: false, alt: "y sub 4")[$y_(4)$]), and so on.


    Let’s review 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")[$( 7 , 6 )$], as shown.


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  alt: "A line is plotted on an x y coordinate plane. The x and y axes range from negative 1 to 8, in increments of 1. The line passes through the following points, (2, 3) and (7, 6). A slope of the line is drawn connecting the points, (2, 3), (2, 6), and (7, 6). The vertical length between the points, (2, 3) and (2, 6), is labeled y sub 2 minus y sub 1 equals 6 minus 3 equals 3. The horizontal length between the points, (2, 6) and (7, 6) is labeled x sub 2 minus x sub 1 equals 7 minus 2 equals 5. Note: all values are approximate.", caption: none,
)


    On the graph, we count the rise of 3 and the run of 5. Notice on the graph that that (#math.equation(block: false, alt: "x sub 1")[$x_(1)$], #math.equation(block: false, alt: "y sub 1")[$y_(1)$]) is the point #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$] and (#math.equation(block: false, alt: "x sub 2")[$x_(2)$], #math.equation(block: false, alt: "y sub 2")[$y_(2)$]) is the point #math.equation(block: false, alt: "open parenthesis 7 , 6 close parenthesis")[$( 7 , 6 )$]. The rise can be found by subtracting the #math.equation(block: false, alt: "y")[$y$]-coordinates, 6 and 3, and the run can be found by subtracting the #math.equation(block: false, alt: "x")[$x$]-coordinates 7 and 2.


    #math.equation(block: true, alt: "m equals the fraction rise over run equals the fraction 6 minus 3 over 7 minus 2 equals the fraction 3 over 5")[$m = frac(italic("rise"), italic("run")) = frac(6 − 3, 7 − 2) = frac(3, 5)$]
    We have 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")[$m = frac(italic("rise"), italic("run"))$]. We can use this formula to find the slope of a line.


    #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[
To find the slope of the line between two points (#math.equation(block: false, alt: "x sub 1")[$x_(1)$], #math.equation(block: false, alt: "y sub 1")[$y_(1)$]) and (#math.equation(block: false, alt: "x sub 2")[$x_(2)$], #math.equation(block: false, alt: "y sub 2")[$y_(2)$]), use the formula


      #math.equation(block: true, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]
]


    #examplebox("Example 4")[Finding the Slope of the Line Using Points][
Use the slope formula to find the slope of the line through the points (−2, −3) and (−7, 4).


      
      #solutionbox[
We’ll call (−2, −3) point 1 and (−7, 4) point 2.


        #strong[Step 1:] 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")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]


        #strong[Step 2:] Substitute the values: #math.equation(block: false, alt: "m equals the fraction 4 minus open parenthesis minus 3 close parenthesis over minus 7 minus open parenthesis minus 2 close parenthesis")[$m = frac(4 − ( − 3 ), − 7 − ( − 2 ))$]


        #strong[Step 3:] Simplify: #math.equation(block: false, alt: "m equals the fraction 7 over minus 5 equals minus the fraction 7 over 5")[$m = frac(7, − 5) = − frac(7, 5)$]


        #strong[Step 4:] Verify the slope on the graph shown.


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]
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=== Graphing Functions Using Slope and #math.equation(block: false, alt: "y")[$y$]-Intercept


    We have graphed linear equations by plotting points and using intercepts. Once we see how an equation in slope-intercept form and its graph are related, we will have one more method we can use to graph lines. Review the graph of the equation #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$] in  and find its slope and #math.equation(block: false, alt: "y")[$y$]-intercept.


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)


    The vertical and horizontal lines in the graph show us the rise is 1 and the run is 2, respectively.


    Substituting into the slope formula: #math.equation(block: false, alt: "m equals the fraction rise over run equals the fraction 1 over 2")[$m = frac(italic("rise"), italic("run")) = frac(1, 2)$]


    The #math.equation(block: false, alt: "y")[$y$]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]. Look at the equation of this line.


    #math.equation(block: true, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$]
    Look at the slope and #math.equation(block: false, alt: "y")[$y$]-intercept.


    #math.equation(block: true, alt: "slope m equals the fraction 1 over 2 and y - intercept open parenthesis 0 , 3 close parenthesis")[$"slope" #h(0.28em) m #h(0.28em) = frac(1, 2) "and" #h(0.28em) y "-" "intercept" #h(0.28em) ( 0 , 3 )$]
    When a linear equation is solved for #math.equation(block: false, alt: "y")[$y$], the coefficient of the #math.equation(block: false, alt: "x")[$x$] term is the slope and the constant term is the #math.equation(block: false, alt: "y")[$y$]-coordinate of the #math.equation(block: false, alt: "y")[$y$]-intercept. We say that the equation #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$] is in #strong[slope-intercept form]. Sometimes the slope-intercept form is called the #math.equation(block: false, alt: "y")[$y$]-form.


    #examplebox("Example 5")[Finding the Slope and #math.equation(block: false, alt: "y")[$y$]-Intercept of a Line][
Identify the slope and #math.equation(block: false, alt: "y")[$y$]-intercept of the line from the equation:


        + #math.equation(block: false, alt: "y equals minus the fraction 4 over 7 x minus 2")[$y = − frac(4, 7) x − 2$]
+ #math.equation(block: false, alt: "x plus 3 y equals 9")[$x + 3 y = 9$]


      
      #solutionbox[
+ We compare our equation to the slope-intercept form of the equation. #linebreak() 
  #strong[Step 1:] Write the slope-intercept form of the equation of the line. #linebreak() 
  #math.equation(block: true, alt: "y equals m x plus b")[$y = m x + b$] #linebreak() 
  #strong[Step 2:] Write the equation of the line.  #linebreak() 
  #math.equation(block: true, alt: "y equals minus the fraction 4 over 7 x minus 2")[$y = − frac(4, 7) x − 2$] #linebreak() 
  #strong[Step 3:] Identify the slope.  #linebreak() 
  #math.equation(block: true, alt: "m equals minus the fraction 4 over 7")[$m = − frac(4, 7)$] #linebreak() 
  #strong[Step 4:] Identify the #math.equation(block: false, alt: "y")[$y$]-intercept.  #linebreak() 
  #math.equation(block: true, alt: "y - intercept is open parenthesis 0 , minus 2 close parenthesis")[$y "-" "intercept is" #h(0.28em) ( 0 , − 2 )$] #linebreak()
+ When an equation of a line is not given in slope-intercept form, our first step will be to solve the equation for #math.equation(block: false, alt: "y")[$y$]. #linebreak() 
  #strong[Step 1:] Solve for #math.equation(block: false, alt: "y")[$y$]. #linebreak() 
  #math.equation(block: true, alt: "x plus 3 y equals 9")[$x + 3 y = 9$] #linebreak() 
  #strong[Step 2:] Subtract #math.equation(block: false, alt: "x")[$x$] from each side. #linebreak() 
  #math.equation(block: true, alt: "3 y equals minus x plus 9")[$3 y = − x + 9$] #linebreak() 
  #strong[Step 3:] Divide both sides by 3. #linebreak() 
  #math.equation(block: true, alt: "the fraction 3 y over 3 equals the fraction minus x plus 9 over 3")[$frac(3 y, 3) = frac(− x + 9, 3)$] #linebreak() 
  #strong[Step 4:] Simplify. #linebreak() 
  #math.equation(block: true, alt: "y equals minus the fraction 1 over 3 x plus 3")[$y = − frac(1, 3) x + 3$] #linebreak() 
  #strong[Step 5:] Write the slope-intercept form of the equation of the line. #linebreak() 
  #math.equation(block: true, alt: "y equals m x plus b")[$y = m x + b$] #linebreak() 
  #strong[Step 6:] Write the equation of the line. #linebreak() 
  #math.equation(block: true, alt: "y equals minus the fraction 1 over 3 x plus 3")[$y = − frac(1, 3) x + 3$] #linebreak() 
  #strong[Step 7:] Identify the slope. #linebreak() 
  #math.equation(block: true, alt: "m equals minus the fraction 1 over 3")[$m = − frac(1, 3)$] #linebreak() 
  #strong[Step 8:] Identify the #math.equation(block: false, alt: "y")[$y$]-intercept. #linebreak() 
  #math.equation(block: true, alt: "y - intercept is open parenthesis 0 , 3 close parenthesis")[$y "-" "intercept is" #h(0.28em) ( 0 , 3 )$]
]
]


    
    #examplebox("Example 6")[Graphing the Slope and #math.equation(block: false, alt: "y")[$y$]-Intercept][
Graph the line of the equation #math.equation(block: false, alt: "y equals minus x plus 4")[$y = − x + 4$] using its slope and #math.equation(block: false, alt: "y")[$y$]-intercept.


      
      #solutionbox[
The equation is in slope-intercept form #math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$].


        #math.equation(block: true, alt: "y equals minus x plus 4 .")[$y = − x + 4 "."$]
        #strong[Step 1:] Identify the slope and #math.equation(block: false, alt: "y")[$y$]-intercept. #linebreak()  #math.equation(block: false, alt: "m equals minus 1")[$m = − 1$], #math.equation(block: false, alt: "y")[$y$]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$].


        #strong[Step 2:] Plot the #math.equation(block: false, alt: "y")[$y$]-intercept on the coordinate system .


        + Identify the rise over the run.  #linebreak() #math.equation(block: false, alt: "m equals minus 1 equals the fraction minus 1 over 1")[$m = − 1 = frac(− 1, 1)$]
+ Count out the rise and run to mark the second point.  #linebreak() rise #math.equation(block: false, alt: "minus 1")[$− 1$], run 1


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  content((57.50pt, 80.00pt), anchor: "north", text(size: 8pt)[−5])
  line((69.00pt, 85.00pt), (69.00pt, 82.00pt), stroke: 0.5pt + black)
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  line((80.50pt, 85.00pt), (80.50pt, 82.00pt), stroke: 0.5pt + black)
  content((80.50pt, 80.00pt), anchor: "north", text(size: 8pt)[−3])
  line((92.00pt, 85.00pt), (92.00pt, 82.00pt), stroke: 0.5pt + black)
  content((92.00pt, 80.00pt), anchor: "north", text(size: 8pt)[−2])
  line((103.50pt, 85.00pt), (103.50pt, 82.00pt), stroke: 0.5pt + black)
  content((103.50pt, 80.00pt), anchor: "north", text(size: 8pt)[−1])
  line((126.50pt, 85.00pt), (126.50pt, 82.00pt), stroke: 0.5pt + black)
  content((126.50pt, 80.00pt), anchor: "north", text(size: 8pt)[1])
  line((138.00pt, 85.00pt), (138.00pt, 82.00pt), stroke: 0.5pt + black)
  content((138.00pt, 80.00pt), anchor: "north", text(size: 8pt)[2])
  line((149.50pt, 85.00pt), (149.50pt, 82.00pt), stroke: 0.5pt + black)
  content((149.50pt, 80.00pt), anchor: "north", text(size: 8pt)[3])
  line((161.00pt, 85.00pt), (161.00pt, 82.00pt), stroke: 0.5pt + black)
  content((161.00pt, 75.50pt), anchor: "north", text(size: 8pt)[4])
  line((172.50pt, 85.00pt), (172.50pt, 82.00pt), stroke: 0.5pt + black)
  content((172.50pt, 80.00pt), anchor: "north", text(size: 8pt)[5])
  line((184.00pt, 85.00pt), (184.00pt, 82.00pt), stroke: 0.5pt + black)
  content((184.00pt, 80.00pt), anchor: "north", text(size: 8pt)[6])
  line((195.50pt, 85.00pt), (195.50pt, 82.00pt), stroke: 0.5pt + black)
  content((195.50pt, 80.00pt), anchor: "north", text(size: 8pt)[7])
  line((207.00pt, 85.00pt), (207.00pt, 82.00pt), stroke: 0.5pt + black)
  content((207.00pt, 80.00pt), anchor: "north", text(size: 8pt)[8])
  line((218.50pt, 85.00pt), (218.50pt, 82.00pt), stroke: 0.5pt + black)
  content((218.50pt, 80.00pt), anchor: "north", text(size: 8pt)[9])
  line((230.00pt, 85.00pt), (230.00pt, 82.00pt), stroke: 0.5pt + black)
  content((230.00pt, 80.00pt), anchor: "north", text(size: 8pt)[10])
  line((112.00pt, 0.00pt), (115.00pt, 0.00pt), stroke: 0.5pt + black)
  content((110.00pt, 0.00pt), anchor: "east", text(size: 8pt)[−10])
  line((112.00pt, 8.50pt), (115.00pt, 8.50pt), stroke: 0.5pt + black)
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  line((112.00pt, 17.00pt), (115.00pt, 17.00pt), stroke: 0.5pt + black)
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  line((112.00pt, 25.50pt), (115.00pt, 25.50pt), stroke: 0.5pt + black)
  content((110.00pt, 25.50pt), anchor: "east", text(size: 8pt)[−7])
  line((112.00pt, 34.00pt), (115.00pt, 34.00pt), stroke: 0.5pt + black)
  content((110.00pt, 34.00pt), anchor: "east", text(size: 8pt)[−6])
  line((112.00pt, 42.50pt), (115.00pt, 42.50pt), stroke: 0.5pt + black)
  content((110.00pt, 42.50pt), anchor: "east", text(size: 8pt)[−5])
  line((112.00pt, 51.00pt), (115.00pt, 51.00pt), stroke: 0.5pt + black)
  content((110.00pt, 51.00pt), anchor: "east", text(size: 8pt)[−4])
  line((112.00pt, 59.50pt), (115.00pt, 59.50pt), stroke: 0.5pt + black)
  content((110.00pt, 59.50pt), anchor: "east", text(size: 8pt)[−3])
  line((112.00pt, 68.00pt), (115.00pt, 68.00pt), stroke: 0.5pt + black)
  content((110.00pt, 68.00pt), anchor: "east", text(size: 8pt)[−2])
  line((112.00pt, 76.50pt), (115.00pt, 76.50pt), stroke: 0.5pt + black)
  content((110.00pt, 76.50pt), anchor: "east", text(size: 8pt)[−1])
  line((112.00pt, 93.50pt), (115.00pt, 93.50pt), stroke: 0.5pt + black)
  content((110.00pt, 93.50pt), anchor: "east", text(size: 8pt)[1])
  line((112.00pt, 102.00pt), (115.00pt, 102.00pt), stroke: 0.5pt + black)
  content((110.00pt, 102.00pt), anchor: "east", text(size: 8pt)[2])
  line((112.00pt, 110.50pt), (115.00pt, 110.50pt), stroke: 0.5pt + black)
  content((110.00pt, 110.50pt), anchor: "east", text(size: 8pt)[3])
  line((112.00pt, 119.00pt), (115.00pt, 119.00pt), stroke: 0.5pt + black)
  content((105.50pt, 119.00pt), anchor: "east", text(size: 8pt)[4])
  line((112.00pt, 127.50pt), (115.00pt, 127.50pt), stroke: 0.5pt + black)
  content((110.00pt, 127.50pt), anchor: "east", text(size: 8pt)[5])
  line((112.00pt, 136.00pt), (115.00pt, 136.00pt), stroke: 0.5pt + black)
  content((110.00pt, 136.00pt), anchor: "east", text(size: 8pt)[6])
  line((112.00pt, 144.50pt), (115.00pt, 144.50pt), stroke: 0.5pt + black)
  content((110.00pt, 144.50pt), anchor: "east", text(size: 8pt)[7])
  line((112.00pt, 153.00pt), (115.00pt, 153.00pt), stroke: 0.5pt + black)
  content((110.00pt, 153.00pt), anchor: "east", text(size: 8pt)[8])
  line((112.00pt, 161.50pt), (115.00pt, 161.50pt), stroke: 0.5pt + black)
  content((110.00pt, 161.50pt), anchor: "east", text(size: 8pt)[9])
  line((112.00pt, 170.00pt), (115.00pt, 170.00pt), stroke: 0.5pt + black)
  content((110.00pt, 170.00pt), anchor: "east", text(size: 8pt)[10])
})]),
  alt: "A line is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The line passes through the following points, (negative 6, 10), (0, 4), (1, 3), (4, 0), and (9, negative 5). Note: all values are approximate.", caption: none,
)
]
]


    
  
  
    
=== Graphing Horizontal and Vertical Lines


    Some linear equations have only one variable. They may have just #math.equation(block: false, alt: "x")[$x$] without the #math.equation(block: false, alt: "y")[$y$], or just #math.equation(block: false, alt: "y")[$y$] without an #math.equation(block: false, alt: "x")[$x$]. This changes how we make a table of values to get the points to plot. Let us consider the equation #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$]. This equation has only one variable, #math.equation(block: false, alt: "x")[$x$]. The equation says that #math.equation(block: false, alt: "x")[$x$] is always equal to #math.equation(block: false, alt: "minus 3")[$− 3$], so its value does not depend on #math.equation(block: false, alt: "y")[$y$]. No matter what the value of #math.equation(block: false, alt: "y")[$y$] is, the value of #math.equation(block: false, alt: "x")[$x$] is always #math.equation(block: false, alt: "minus 3")[$− 3$]. To make a table of values, write #math.equation(block: false, alt: "minus 3")[$− 3$] in for all the #math.equation(block: false, alt: "x")[$x$]-values. Then choose any values for #math.equation(block: false, alt: "y")[$y$]. Since #math.equation(block: false, alt: "x")[$x$] does not depend on #math.equation(block: false, alt: "y")[$y$], you can choose any numbers you like. But to fit the points on our coordinate graph, we will use 1, 2, and 3 for the #math.equation(block: false, alt: "y")[$y$]-coordinates in the table below.


    #figure(table(
  columns: 3, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "x equals minus 3")[$x = − 3$]], [], [],
  [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [(#math.equation(block: false, alt: "x")[$x$], #math.equation(block: false, alt: "y")[$y$])],
  [−3], [1], [#math.equation(block: false, alt: "open parenthesis minus 3 , 1 close parenthesis")[$( − 3 , 1 )$]],
  [#math.equation(block: false, alt: "minus 3")[$− 3$]], [2], [#math.equation(block: false, alt: "open parenthesis minus 3 , 2 close parenthesis")[$( − 3 , 2 )$]],
  [#math.equation(block: false, alt: "minus 3")[$− 3$]], [3], [#math.equation(block: false, alt: "open parenthesis minus 3 , 3 close parenthesis")[$( − 3 , 3 )$]],
))


    Plot the points from the table and connect them with a straight line . Notice that we have graphed a vertical line.


    #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  line((65.71pt, 0.00pt), (65.71pt, 170.00pt), stroke: 0.7pt + black)
  content((56.71pt, 127.50pt), anchor: "south", text(size: 9pt)[x = -3])
  circle((65.71pt, 121.43pt), radius: 3.0pt, fill: black, stroke: none)
  circle((65.71pt, 109.29pt), radius: 3.0pt, fill: black, stroke: none)
  circle((65.71pt, 97.14pt), radius: 3.0pt, fill: black, stroke: none)
  line((0.00pt, 85.00pt), (234.00pt, 85.00pt), stroke: 0.7pt + black, mark: (end: ">", fill: black, size: 0.18))
  line((115.00pt, 0.00pt), (115.00pt, 174.00pt), stroke: 0.7pt + black, mark: (end: ">", fill: black, size: 0.18))
  line((0.00pt, 85.00pt), (0.00pt, 82.00pt), stroke: 0.5pt + black)
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  line((32.86pt, 85.00pt), (32.86pt, 82.00pt), stroke: 0.5pt + black)
  content((32.86pt, 80.00pt), anchor: "north", text(size: 8pt)[−5])
  line((49.29pt, 85.00pt), (49.29pt, 82.00pt), stroke: 0.5pt + black)
  content((49.29pt, 80.00pt), anchor: "north", text(size: 8pt)[−4])
  line((65.71pt, 85.00pt), (65.71pt, 82.00pt), stroke: 0.5pt + black)
  content((65.71pt, 80.00pt), anchor: "north", text(size: 8pt)[−3])
  line((82.14pt, 85.00pt), (82.14pt, 82.00pt), stroke: 0.5pt + black)
  content((82.14pt, 80.00pt), anchor: "north", text(size: 8pt)[−2])
  line((98.57pt, 85.00pt), (98.57pt, 82.00pt), stroke: 0.5pt + black)
  content((98.57pt, 80.00pt), anchor: "north", text(size: 8pt)[−1])
  line((131.43pt, 85.00pt), (131.43pt, 82.00pt), stroke: 0.5pt + black)
  content((131.43pt, 80.00pt), anchor: "north", text(size: 8pt)[1])
  line((147.86pt, 85.00pt), (147.86pt, 82.00pt), stroke: 0.5pt + black)
  content((147.86pt, 80.00pt), anchor: "north", text(size: 8pt)[2])
  line((164.29pt, 85.00pt), (164.29pt, 82.00pt), stroke: 0.5pt + black)
  content((164.29pt, 80.00pt), anchor: "north", text(size: 8pt)[3])
  line((180.71pt, 85.00pt), (180.71pt, 82.00pt), stroke: 0.5pt + black)
  content((180.71pt, 80.00pt), anchor: "north", text(size: 8pt)[4])
  line((197.14pt, 85.00pt), (197.14pt, 82.00pt), stroke: 0.5pt + black)
  content((197.14pt, 80.00pt), anchor: "north", text(size: 8pt)[5])
  line((213.57pt, 85.00pt), (213.57pt, 82.00pt), stroke: 0.5pt + black)
  content((213.57pt, 80.00pt), anchor: "north", text(size: 8pt)[6])
  line((230.00pt, 85.00pt), (230.00pt, 82.00pt), stroke: 0.5pt + black)
  content((230.00pt, 80.00pt), anchor: "north", text(size: 8pt)[7])
  line((112.00pt, 0.00pt), (115.00pt, 0.00pt), stroke: 0.5pt + black)
  content((110.00pt, 0.00pt), anchor: "east", text(size: 8pt)[−7])
  line((112.00pt, 12.14pt), (115.00pt, 12.14pt), stroke: 0.5pt + black)
  content((110.00pt, 12.14pt), anchor: "east", text(size: 8pt)[−6])
  line((112.00pt, 24.29pt), (115.00pt, 24.29pt), stroke: 0.5pt + black)
  content((110.00pt, 24.29pt), anchor: "east", text(size: 8pt)[−5])
  line((112.00pt, 36.43pt), (115.00pt, 36.43pt), stroke: 0.5pt + black)
  content((110.00pt, 36.43pt), anchor: "east", text(size: 8pt)[−4])
  line((112.00pt, 48.57pt), (115.00pt, 48.57pt), stroke: 0.5pt + black)
  content((110.00pt, 48.57pt), anchor: "east", text(size: 8pt)[−3])
  line((112.00pt, 60.71pt), (115.00pt, 60.71pt), stroke: 0.5pt + black)
  content((110.00pt, 60.71pt), anchor: "east", text(size: 8pt)[−2])
  line((112.00pt, 72.86pt), (115.00pt, 72.86pt), stroke: 0.5pt + black)
  content((110.00pt, 72.86pt), anchor: "east", text(size: 8pt)[−1])
  line((112.00pt, 97.14pt), (115.00pt, 97.14pt), stroke: 0.5pt + black)
  content((110.00pt, 97.14pt), anchor: "east", text(size: 8pt)[1])
  line((112.00pt, 109.29pt), (115.00pt, 109.29pt), stroke: 0.5pt + black)
  content((110.00pt, 109.29pt), anchor: "east", text(size: 8pt)[2])
  line((112.00pt, 121.43pt), (115.00pt, 121.43pt), stroke: 0.5pt + black)
  content((110.00pt, 121.43pt), anchor: "east", text(size: 8pt)[3])
  line((112.00pt, 133.57pt), (115.00pt, 133.57pt), stroke: 0.5pt + black)
  content((110.00pt, 133.57pt), anchor: "east", text(size: 8pt)[4])
  line((112.00pt, 145.71pt), (115.00pt, 145.71pt), stroke: 0.5pt + black)
  content((110.00pt, 145.71pt), anchor: "east", text(size: 8pt)[5])
  line((112.00pt, 157.86pt), (115.00pt, 157.86pt), stroke: 0.5pt + black)
  content((110.00pt, 157.86pt), anchor: "east", text(size: 8pt)[6])
  line((112.00pt, 170.00pt), (115.00pt, 170.00pt), stroke: 0.5pt + black)
  content((110.00pt, 170.00pt), anchor: "east", text(size: 8pt)[7])
})]),
  alt: "A line is plotted on an x y coordinate plane. The x and y axes range from negative 7 to 7, in increments of 1. The line representing x equals negative 3 is vertical and it passes through the following points, (negative 3, 3), (negative 3, 2), and (negative 3, 1). Note: all values are approximate.", caption: [Graph of #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$]],
)


    What is the slope? If we take the two points #math.equation(block: false, alt: "open parenthesis minus 3 , 3 close parenthesis")[$( − 3 , 3 )$] and #math.equation(block: false, alt: "open parenthesis minus 3 , 1 close parenthesis")[$( − 3 , 1 )$] then the rise is 2 and the run is 0.


    Using the slope formula we get: #math.equation(block: false, alt: "m equals the fraction rise over run equals the fraction 2 over 0")[$m = frac(italic("rise"), italic("run")) = frac(2, 0)$]


    The slope is #strong[undefined] since division by zero is undefined. We say that the slope of the vertical line #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$] is undefined. The slope of any #strong[vertical line] #math.equation(block: false, alt: "x equals a")[$x = a$] (where #math.equation(block: false, alt: "a")[$a$] is any number) will be undefined.


    What if the equation has #math.equation(block: false, alt: "y")[$y$] but no #math.equation(block: false, alt: "x")[$x$]? Let’s graph the equation #math.equation(block: false, alt: "y equals 4")[$y = 4$]. This time the #math.equation(block: false, alt: "y")[$y$]-value is a constant, so in this equation, #math.equation(block: false, alt: "y")[$y$] does not depend on #math.equation(block: false, alt: "x")[$x$]. Fill in 4 for all the #math.equation(block: false, alt: "y")[$y$] values in the table below and then choose any values for #math.equation(block: false, alt: "x")[$x$]. We will use 0, 2, and 4 for the #math.equation(block: false, alt: "x")[$x$]-coordinates.


    #figure(table(
  columns: 3, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "y equals 4")[$y = 4$]], [], [],
  [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [(#math.equation(block: false, alt: "x")[$x$], #math.equation(block: false, alt: "y")[$y$])],
  [0], [4], [#math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$]],
  [2], [4], [#math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$]],
  [4], [4], [#math.equation(block: false, alt: "open parenthesis 4 , 4 close parenthesis")[$( 4 , 4 )$]],
))


    In , we have graphed a horizontal line passing through the #math.equation(block: false, alt: "y")[$y$]-axis at 4.


    #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  line((0.00pt, 133.57pt), (230.00pt, 133.57pt), stroke: 0.7pt + black)
  content((172.50pt, 142.57pt), anchor: "south", text(size: 9pt)[y = 4])
  circle((115.00pt, 133.57pt), radius: 3.0pt, fill: black, stroke: none)
  circle((147.86pt, 133.57pt), radius: 3.0pt, fill: black, stroke: none)
  circle((180.71pt, 133.57pt), radius: 3.0pt, fill: black, stroke: none)
  line((0.00pt, 85.00pt), (234.00pt, 85.00pt), stroke: 0.7pt + black, mark: (end: ">", fill: black, size: 0.18))
  line((115.00pt, 0.00pt), (115.00pt, 174.00pt), stroke: 0.7pt + black, mark: (end: ">", fill: black, size: 0.18))
  line((0.00pt, 85.00pt), (0.00pt, 82.00pt), stroke: 0.5pt + black)
  content((0.00pt, 80.00pt), anchor: "north", text(size: 8pt)[−7])
  line((16.43pt, 85.00pt), (16.43pt, 82.00pt), stroke: 0.5pt + black)
  content((16.43pt, 80.00pt), anchor: "north", text(size: 8pt)[−6])
  line((32.86pt, 85.00pt), (32.86pt, 82.00pt), stroke: 0.5pt + black)
  content((32.86pt, 80.00pt), anchor: "north", text(size: 8pt)[−5])
  line((49.29pt, 85.00pt), (49.29pt, 82.00pt), stroke: 0.5pt + black)
  content((49.29pt, 80.00pt), anchor: "north", text(size: 8pt)[−4])
  line((65.71pt, 85.00pt), (65.71pt, 82.00pt), stroke: 0.5pt + black)
  content((65.71pt, 80.00pt), anchor: "north", text(size: 8pt)[−3])
  line((82.14pt, 85.00pt), (82.14pt, 82.00pt), stroke: 0.5pt + black)
  content((82.14pt, 80.00pt), anchor: "north", text(size: 8pt)[−2])
  line((98.57pt, 85.00pt), (98.57pt, 82.00pt), stroke: 0.5pt + black)
  content((98.57pt, 80.00pt), anchor: "north", text(size: 8pt)[−1])
  line((131.43pt, 85.00pt), (131.43pt, 82.00pt), stroke: 0.5pt + black)
  content((131.43pt, 80.00pt), anchor: "north", text(size: 8pt)[1])
  line((147.86pt, 85.00pt), (147.86pt, 82.00pt), stroke: 0.5pt + black)
  content((147.86pt, 80.00pt), anchor: "north", text(size: 8pt)[2])
  line((164.29pt, 85.00pt), (164.29pt, 82.00pt), stroke: 0.5pt + black)
  content((164.29pt, 80.00pt), anchor: "north", text(size: 8pt)[3])
  line((180.71pt, 85.00pt), (180.71pt, 82.00pt), stroke: 0.5pt + black)
  content((180.71pt, 80.00pt), anchor: "north", text(size: 8pt)[4])
  line((197.14pt, 85.00pt), (197.14pt, 82.00pt), stroke: 0.5pt + black)
  content((197.14pt, 80.00pt), anchor: "north", text(size: 8pt)[5])
  line((213.57pt, 85.00pt), (213.57pt, 82.00pt), stroke: 0.5pt + black)
  content((213.57pt, 80.00pt), anchor: "north", text(size: 8pt)[6])
  line((230.00pt, 85.00pt), (230.00pt, 82.00pt), stroke: 0.5pt + black)
  content((230.00pt, 80.00pt), anchor: "north", text(size: 8pt)[7])
  line((112.00pt, 0.00pt), (115.00pt, 0.00pt), stroke: 0.5pt + black)
  content((110.00pt, 0.00pt), anchor: "east", text(size: 8pt)[−7])
  line((112.00pt, 12.14pt), (115.00pt, 12.14pt), stroke: 0.5pt + black)
  content((110.00pt, 12.14pt), anchor: "east", text(size: 8pt)[−6])
  line((112.00pt, 24.29pt), (115.00pt, 24.29pt), stroke: 0.5pt + black)
  content((110.00pt, 24.29pt), anchor: "east", text(size: 8pt)[−5])
  line((112.00pt, 36.43pt), (115.00pt, 36.43pt), stroke: 0.5pt + black)
  content((110.00pt, 36.43pt), anchor: "east", text(size: 8pt)[−4])
  line((112.00pt, 48.57pt), (115.00pt, 48.57pt), stroke: 0.5pt + black)
  content((110.00pt, 48.57pt), anchor: "east", text(size: 8pt)[−3])
  line((112.00pt, 60.71pt), (115.00pt, 60.71pt), stroke: 0.5pt + black)
  content((110.00pt, 60.71pt), anchor: "east", text(size: 8pt)[−2])
  line((112.00pt, 72.86pt), (115.00pt, 72.86pt), stroke: 0.5pt + black)
  content((110.00pt, 72.86pt), anchor: "east", text(size: 8pt)[−1])
  line((112.00pt, 97.14pt), (115.00pt, 97.14pt), stroke: 0.5pt + black)
  content((110.00pt, 97.14pt), anchor: "east", text(size: 8pt)[1])
  line((112.00pt, 109.29pt), (115.00pt, 109.29pt), stroke: 0.5pt + black)
  content((110.00pt, 109.29pt), anchor: "east", text(size: 8pt)[2])
  line((112.00pt, 121.43pt), (115.00pt, 121.43pt), stroke: 0.5pt + black)
  content((110.00pt, 121.43pt), anchor: "east", text(size: 8pt)[3])
  line((112.00pt, 133.57pt), (115.00pt, 133.57pt), stroke: 0.5pt + black)
  content((105.50pt, 133.57pt), anchor: "east", text(size: 8pt)[4])
  line((112.00pt, 145.71pt), (115.00pt, 145.71pt), stroke: 0.5pt + black)
  content((110.00pt, 145.71pt), anchor: "east", text(size: 8pt)[5])
  line((112.00pt, 157.86pt), (115.00pt, 157.86pt), stroke: 0.5pt + black)
  content((110.00pt, 157.86pt), anchor: "east", text(size: 8pt)[6])
  line((112.00pt, 170.00pt), (115.00pt, 170.00pt), stroke: 0.5pt + black)
  content((110.00pt, 170.00pt), anchor: "east", text(size: 8pt)[7])
})]),
  alt: "A line is plotted on an x y coordinate plane. The x and y axes range from negative 7 to 7, in increments of 1. The line representing y equals 4 is horizontal and it passes through the following points, (0, 4), (2, 4), and (4, 4). Note: all values are approximate.", caption: [Graph of #math.equation(block: false, alt: "y equals 4")[$y = 4$]],
)


    What is the slope? If we take the two points #math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$] and #math.equation(block: false, alt: "open parenthesis 4 , 4 close parenthesis")[$( 4 , 4 )$] then the rise is 0 and the run is 2. Using the slope formula, we get #math.equation(block: false, alt: "m equals the fraction rise over run equals the fraction 0 over 2 equals 0")[$m = frac(italic("rise"), italic("run")) = frac(0, 2) = 0$]. The slope of the horizontal line #math.equation(block: false, alt: "y equals 4")[$y = 4$] is 0. The slope of any #strong[horizontal line] #math.equation(block: false, alt: "y equals b")[$y = b$] (where #math.equation(block: false, alt: "b")[$b$] is any number) will be 0. When the #math.equation(block: false, alt: "y")[$y$]-coordinates are the same, the rise is 0.


    #examplebox("Example 7")[Graphing A Vertical Line][
Graph: #math.equation(block: false, alt: "x equals 2")[$x = 2$].


      
      #solutionbox[
The equation has only one variable, #math.equation(block: false, alt: "x")[$x$], and #math.equation(block: false, alt: "x")[$x$] is always equal to 2. We create a table where #math.equation(block: false, alt: "x")[$x$] is always 2 and then put in any values for #math.equation(block: false, alt: "y")[$y$]. The graph is a vertical line passing through the #math.equation(block: false, alt: "x")[$x$]-axis at 2 .


        #figure(table(
  columns: 3, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "x equals 2")[$x = 2$]], [], [],
  [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [(#math.equation(block: false, alt: "x")[$x$], #math.equation(block: false, alt: "y")[$y$])],
  [2], [1], [#math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$]],
  [2], [2], [#math.equation(block: false, alt: "open parenthesis 2 , 2 close parenthesis")[$( 2 , 2 )$]],
  [2], [3], [#math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$]],
))


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  line((112.00pt, 157.86pt), (115.00pt, 157.86pt), stroke: 0.5pt + black)
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  line((112.00pt, 170.00pt), (115.00pt, 170.00pt), stroke: 0.5pt + black)
  content((110.00pt, 170.00pt), anchor: "east", text(size: 8pt)[7])
})]),
  alt: "A line is plotted on an x y coordinate plane. The x and y axes range from negative 7 to 7, in increments of 1. The line representing x equals 2 is vertical and it passes through the following points, (2, 1), (2, 2), and (2, 3). Note: all values are approximate.", caption: none,
)
]
]


    
    #examplebox("Example 8")[Graphing A Horizontal Line][
Graph: #math.equation(block: false, alt: "y equals minus 1")[$y = − 1$].


      
      #solutionbox[
The equation #math.equation(block: false, alt: "y equals minus 1")[$y = − 1$] has only one variable, #math.equation(block: false, alt: "y")[$y$]. The value of #math.equation(block: false, alt: "y")[$y$] is constant. All the ordered pairs in the next table have the same #math.equation(block: false, alt: "y")[$y$]-coordinate. The graph is a horizontal line passing through the #math.equation(block: false, alt: "y")[$y$]-axis at −1 .


        #figure(table(
  columns: 3, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "y equals minus 1")[$y = − 1$]], [], [],
  [#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]], [(#math.equation(block: false, alt: "x")[$x$], #math.equation(block: false, alt: "y")[$y$])],
  [0], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "open parenthesis 0 , minus 1 close parenthesis")[$( 0 , − 1 )$]],
  [3], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "open parenthesis 3 , minus 1 close parenthesis")[$( 3 , − 1 )$]],
  [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "open parenthesis minus 3 , minus 1 close parenthesis")[$( − 3 , − 1 )$]],
))


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  import cetz.draw: *
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  line((112.00pt, 145.71pt), (115.00pt, 145.71pt), stroke: 0.5pt + black)
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  line((112.00pt, 170.00pt), (115.00pt, 170.00pt), stroke: 0.5pt + black)
  content((110.00pt, 170.00pt), anchor: "east", text(size: 8pt)[7])
})]),
  alt: "A line is plotted on an x y coordinate plane. The x and y axes range from negative 7 to 7, in increments of 1. The line representing y equals negative 1 is horizontal and it passes through the following points, (negative 3, negative 1), (0, negative 1), and (3, negative 1). Note: all values are approximate.", caption: none,
)
]
]


    
    The table below summarizes all the methods we have used to graph lines.


    #figure(figph[A table titled, Methods to Graph Lines. The table has four columns. The first column is titled, Point Plotting. It shows a blank x y table. Text reads, Find three points. Plot the points. Make sure they line up, then draw the line. The second column is titled, Slope-Intercept. It shows the equation y equals m x plus b. Text reads, Find the slope and y-intercept. Start at the y-intercept, then count the slope to get a second point. The third column is titled, Intercepts. It shows an x y table with 0 in the first row under x and 0 in the second row under y. Text reads, Find the intercepts and a third point. Plot the points, make sure they line up, then draw the line. The fourth column is titled, Recognize Vertical and Horizontal Lines. Text reads, The equation has only one variable. x equals a, vertical. y equals b, horizontal.], alt: "A table titled, Methods to Graph Lines. The table has four columns. The first column is titled, Point Plotting. It shows a blank x y table. Text reads, Find three points. Plot the points. Make sure they line up, then draw the line. The second column is titled, Slope-Intercept. It shows the equation y equals m x plus b. Text reads, Find the slope and y-intercept. Start at the y-intercept, then count the slope to get a second point. The third column is titled, Intercepts. It shows an x y table with 0 in the first row under x and 0 in the second row under y. Text reads, Find the intercepts and a third point. Plot the points, make sure they line up, then draw the line. The fourth column is titled, Recognize Vertical and Horizontal Lines. Text reads, The equation has only one variable. x equals a, vertical. y equals b, horizontal.", caption: none)


  
  
    
=== Interpreting Graphs of Functions


    An important yet often overlooked area in algebra involves interpreting graphs. Oftentimes in math classes, students are given mathematical functions and can make graphs to represent them. But the interpretation of graphs is a more applicable skill to the real world. Being able to “read” a graph—understanding its domain and range, what the intercepts mean, and what the slope (or curve) means— that's a real-world skill.


    #examplebox("Example 9")[Interpreting a Graph][
In  the #math.equation(block: false, alt: "x")[$x$]-axis on the graph represents the 120-minute bike ride Juan went on. The #math.equation(block: false, alt: "y")[$y$]-axis represents how far away he was from his home.


        #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  content((0.00pt, 69.50pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 10pt)[Juan′s Bike Ride])
  line((0.00pt, 42.00pt), (39.94pt, 12.98pt), stroke: 0.7pt + black)
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  circle((0.00pt, 42.00pt), radius: 10.50pt, fill: white, stroke: 0.7pt + black)
  content((0.00pt, 42.00pt), text(size: 9pt, fill: black)[0,0])
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  content((-38.27pt, -48.13pt), anchor: "north-east", text(size: 9pt)[90,12])
  circle((-39.94pt, 12.98pt), radius: 3pt, fill: black, stroke: 0.7pt + black)
  content((-54.06pt, 16.96pt), anchor: "east", text(size: 9pt)[135,0])
})]),
  alt: "A graph titled, Juan's Bike Ride. The horizontal axis representing time in minutes ranges from 0 to 150, in increments of 15. The vertical axis representing the distance from home in miles ranges from 0 to 15, in increments of 3. The graph shows a line that passes through the points, (0, 0), (30, 6), (60, 6), (90, 12), and (135, 0).", caption: none,
)


        + Interpret the #math.equation(block: false, alt: "x")[$x$]- and #math.equation(block: false, alt: "y")[$y$]-intercept.
+ For each segment, find the slope.
+ Create an interpretation of this graph (i.e., make up a story that goes with it).


      
      #solutionbox[
+ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is the #math.equation(block: false, alt: "x")[$x$]- and #math.equation(block: false, alt: "y")[$y$]-intercept and represents Juan at home before his bike ride. The distance from home is 0 miles and 0 minutes have passed.
+ In the first 30 minutes, the slope is #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] and indicates Juan is traveling 1 mile for every 5 minutes. Between 30 and 60 minutes, the slope is 0 and indicates that he’s not riding the bike (the distance is not increasing). Then between 60 and 90 minutes, the slope is #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] again. Finally, after 90 minutes the slope is #math.equation(block: false, alt: "minus the fraction 4 over 15 ,")[$− frac(4, 15) ,$] meaning Juan is getting 4 miles closer to home every 15 minutes.
+ Answers will vary. Juan left his house for a bike ride. After 30 minutes, he was 6 miles from home and he stopped for ice cream at his local ice-cream truck. He enjoyed his ice cream for 30 minutes. He then jumped back on his bike and rode to his friend’s house. He arrived there 30 minutes later. His friend’s house was 12 miles from his home. His friend was not home so he immediately turned around and quickly rode home in 45 minutes.
]
]


    
  
  
    
=== Modeling Applications Using Slope and #math.equation(block: false, alt: "y")[$y$]-Intercept


    Many real-world applications are modeled by linear equations. We will review a few applications here so you can understand how equations written in slope-intercept form relate to real-world situations. Usually when a linear equation model uses real-world data, different letters are used for the variables instead of using only #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$]. The variable names often remind us of what quantities are being measured. Also, we often need to extend the axes in our rectangular coordinate system to bigger positive and negative numbers to accommodate the data in the application.


    #examplebox("Example 10")[Converting Temperature][
The equation #math.equation(block: false, alt: "F equals the fraction 9 over 5 C plus 32")[$F = frac(9, 5) C + 32$] is used to convert temperatures from degrees Celsius (#math.equation(block: false, alt: "C")[$C$]) to degrees Fahrenheit (#math.equation(block: false, alt: "F")[$F$]).


        + Find the Fahrenheit temperature for a Celsius temperature of 0°.
+ Find the Fahrenheit temperature for a Celsius temperature of 20°.
+ Interpret the slope and #math.equation(block: false, alt: "F")[$F$]-intercept of the equation.
+ Graph the equation.


      
      #solutionbox[
+ Find the Fahrenheit temperature for a Celsius temperature of 0°.
  #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find #math.equation(block: false, alt: "F")[$F$] when #math.equation(block: false, alt: "C equals 0")[$C = 0$].], [#math.equation(block: false, alt: "F equals the fraction 9 over 5 open parenthesis 0 close parenthesis plus 32")[$F = frac(9, 5) ( 0 ) + 32$]]),
  [Simplify.], [#math.equation(block: false, alt: "F equals 32")[$F = 32$]],
))
+ Find the Fahrenheit temperature for a Celsius temperature of 20°.
  #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find #math.equation(block: false, alt: "F")[$F$] when #math.equation(block: false, alt: "C equals 20")[$C = 20$].], [#math.equation(block: false, alt: "F equals the fraction 9 over 5 open parenthesis 20 close parenthesis plus 32")[$F = frac(9, 5) ( 20 ) + 32$]]),
  [Simplify.], [#math.equation(block: false, alt: "F equals 36 plus 32")[$F = 36 + 32$]],
  [Simplify.], [#math.equation(block: false, alt: "F equals 68")[$F = 68$]],
))
+ Interpret the slope and #math.equation(block: false, alt: "F")[$F$]-intercept of the equation.
  Even though this equation uses #math.equation(block: false, alt: "F")[$F$] and #math.equation(block: false, alt: "C")[$C$], it is still in slope-intercept form.

#math.equation(block: true, alt: "y equals m x plus b; F equals m C plus b; F equals the fraction 9 over 5 C plus 32")[$y = m x + b \ F = m C + b \ F = frac(9, 5) C + 32$]

The slope #math.equation(block: false, alt: "the fraction 9 over 5")[$frac(9, 5)$] means that the temperature Fahrenheit (#math.equation(block: false, alt: "F")[$F$]) increases 9 degrees when the temperature Celsius (#math.equation(block: false, alt: "C")[$C$]) increases 5 degrees.

The #math.equation(block: false, alt: "F")[$F$]-intercept means that when the temperature is 0° on the Celsius scale, it is 32° on the Fahrenheit scale.
+ Graph the equation.
  We will need to use a larger scale than our usual. Start at the #math.equation(block: false, alt: "F")[$F$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 32 close parenthesis")[$( 0 , 32 )$], and then count out the rise of 9 and the run of 5 to get a second point as shown.

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  content((115.00pt, -19.00pt), anchor: "north", text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 9pt)[C])
  content((-32.00pt, 85.00pt), angle: 90deg, anchor: "south", text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 9pt)[F])
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  alt: "A line is plotted on a coordinate plane. The horizontal axis representing C ranges from negative 40 to 40, in increments of 10. The vertical axis representing F ranges from negative 40 to 60, in increments of 10. The line passes through the points, (negative 30, negative 37), (negative 17, 0), (0, 33), (5, 42), and (20, 70). Note: all values are approximate.", caption: none,
)
]
]


    
    #examplebox("Example 11")[Calculating Driving Costs][
Sam drives a delivery van. The equation #math.equation(block: false, alt: "C equals 0.5 d plus 60")[$C = 0.5 d + 60$] models the relation between his weekly cost, #math.equation(block: false, alt: "C")[$C$], in dollars and the number of miles, #math.equation(block: false, alt: "d")[$d$], that he drives.


        + Find Sam’s cost for a week when he drives 0 miles.
+ Find the cost for a week when he drives 250 miles.
+ Interpret the slope and #math.equation(block: false, alt: "C")[$C$]-intercept of the equation.
+ Graph the equation.


      
      #solutionbox[
+ Find Sam’s cost for a week when he drives 0 miles.
  #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find #math.equation(block: false, alt: "C")[$C$] when #math.equation(block: false, alt: "d")[$d$] = 0.], [#math.equation(block: false, alt: "C equals 0.5 open parenthesis 0 close parenthesis plus 60")[$C = 0.5 ( 0 ) + 60$]]),
  [Simplify.], [#math.equation(block: false, alt: "C equals 60")[$C = 60$]],
))

Sam’s costs are \$60 when he drives 0 miles.
+ Find the cost for a week when he drives 250 miles.
  #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find #math.equation(block: false, alt: "C")[$C$] when #math.equation(block: false, alt: "d equals 250")[$d = 250$].], [#math.equation(block: false, alt: "C equals 0.5 open parenthesis 250 close parenthesis plus 60")[$C = 0.5 ( 250 ) + 60$]]),
  [Simplify.], [#math.equation(block: false, alt: "C equals 185")[$C = 185$]],
))

Sam’s costs are \$185 when he drives 250 miles.
+ Interpret the slope and #math.equation(block: false, alt: "C")[$C$]-intercept of the equation.
  #math.equation(block: true, alt: "y equals m x plus b; C equals 0.5 d plus 60")[$y = m x + b \ C = 0.5 d + 60$]

The slope, 0.5, means that the weekly cost, #math.equation(block: false, alt: "C")[$C$], increases by \$0.50 when the number of miles driven, #math.equation(block: false, alt: "d")[$d$], increases by 1. The #math.equation(block: false, alt: "C")[$C$]-intercept means that when the number of miles driven is 0, the weekly cost is \$60.
+ Graph the equation .
  We’ll need to use a larger scale than usual. Start at the #math.equation(block: false, alt: "C")[$C$]-intercept (0, 60). To count out the slope #math.equation(block: false, alt: "m equals 0.5")[$m = 0.5$], we rewrite it as an equivalent fraction that will make our graphing easier.

#math.equation(block: true, alt: "m equals 0.5 equals the fraction 5 over 10 equals the fraction 50 over 100")[$m = 0.5 = frac(5, 10) = frac(50, 100)$]So to graph the next point go up 50 from the intercept of 60 and then to the right 100. The second point will be (100, 110).

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)
]
]


    
  
  
  
  
    
=== Key Terms


    - intercepts of a line
- slope
- slope-intercept form


  
  
    
=== Key Concepts


    - Every linear function can be graphically represented by a unique line that shows all the solutions of the equation.
- The points where the graph of a line intersects the #math.equation(block: false, alt: "x")[$x$]-axis and #math.equation(block: false, alt: "y")[$y$]-axis are called the intercepts of the line.
- Most lines will have one #math.equation(block: false, alt: "x")[$x$]-intercept and one #math.equation(block: false, alt: "y")[$y$]-intercept. Only if the line is straight vertical (no #math.equation(block: false, alt: "y")[$y$]-intercept) or straight horizontal (no #math.equation(block: false, alt: "x")[$x$]-intercept) will it not have both intercepts. Note that a line that is straight vertical is not a function, but a line that is straight horizontal is a function.
- Since any two points determine a straight line, any linear function can be graphed if both intercepts are known.
- The slope of a linear function is the ratio of the vertical change divided by the horizontal change. It is often referred to as #math.equation(block: false, alt: "the fraction r i s e over r u n")[$frac(r i s e, r u n)$].
- A formula for finding the slope of linear functions is #math.equation(block: false, alt: "the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1 ,")[$frac(y_(2) − y_(1), x_(2) − x_(1)) ,$] for any two points of the linear function #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")[$( x_(2) , y_(2) )$].


  
  
    
=== Formulas


    - To calculate slope #math.equation(block: false, alt: "open parenthesis m close parenthesis")[$( m )$], use the formula #linebreak() 
   #math.equation(block: false, alt: "m equals the fraction r i s e over r u n")[$m = frac(r i s e, r u n)$], #linebreak()  where the rise measures the vertical change and the run measures the horizontal change.
- To find 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")[$( x_(2) , y_(2) )$], use the formula #linebreak()  #math.equation(block: false, alt: "m equals the fraction y 2 minus y 1 over x 2 minus x 1")[$m = frac(y 2 − y 1, x 2 − x 1)$]