#set document(title: "11.4 Graphing an Equation", 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"))


    
== 11.4#h(0.6em)Graphing an Equation


    We can graph equations written in the form #math.equation(block: false, alt: "y equals")[$y =$] (expression in x). The graphing keys are located on the top row of the keypad. There are two steps to graphing an equation:


    + Entering the equation
+ Setting the graphing window


    
      
=== Standard Window


      The standard window displays values from #math.equation(block: false, alt: "minus 10")[$− 10$] to #math.equation(block: false, alt: "10")[$10$] on both axes.


      #examplebox("Example 1")[][
+ Press Y= and enter #math.equation(block: false, alt: "2 X minus 5")[$2 X − 5$] after #math.equation(block: false, alt: "Y sub 1 equals")[$Y_(1) =$] by keying in
                #math.equation(block: true, alt: "2 X , T , θ , n minus 5, Use the X , T , θ , n to enter X .")[$2 " " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display(X "," T "," θ "," n)$] " " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display(−)$] " " 5 &  &  & "Use the " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display(X "," T "," θ "," n)$] " " " to enter " X .$]
+ Press  ZOOM #math.equation(block: false, alt: "6")[$6$] to set the standard window, and the graph will appear (see Figure).#figure(figph[GC graphing screens], alt: "GC graphing screens", caption: none)


        You can press 2nd WINDOW to see the settings for the standard window. #math.equation(block: false, alt: "X s c l equals 1")[$X s c l = 1$] means that the tick marks on the #math.equation(block: false, alt: "x")[$x$]-axis are spaced #math.equation(block: false, alt: "1")[$1$] unit apart.


        Press 2nd MODE to Quit the graph and return to the Home screen, where we enter computations. From the Home screen, press GRAPH to return to the graph.
]


    
    
      
=== Tracing


      The calculator can display the coordinates of selected points on the graph. Press the TRACE key to see a "bug" blinking on the graph. The coordinates of the bug are displayed at the bottom of the screen. Use the left and right arrow keys to move the bug along the graph, as shown in Figure. Note that the #strong[Trace] feature does not show every point on the graph!


      #figure(figph[GC tracing bug], alt: "GC tracing bug", caption: none)


      #examplebox("Example 2")[][
Use the #strong[Trace] to find the point on the graph with #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$]. Press


        $#h(2em) #h(2em)$ TRACE (-) #math.equation(block: false, alt: "3")[$3$] ENTER


        The bug is off the bottom of the screen, but the coordinates are still shown.
]


    
    
      
=== Multiple Graphs


      You can enter more than one graph at a time. Press ↓ to enter a second equation at #math.equation(block: false, alt: "Y sub 2 equals")[$Y_(2) =$], at #math.equation(block: false, alt: "Y sub 3 equals")[$Y_(3) =$], and so on. When Tracing, press the    and    keys to move from one graph to another.


      To turn off a graph without deleting its equation, press and move the cursor over the #math.equation(block: false, alt: "equals")[$=$] sign in the equation. Press ENTER to deactivate that equation. (When you move the cursor away, the #math.equation(block: false, alt: "equals")[$=$] sign is no longer highlighted.) To reactivate the equation, move the cursor back over the #math.equation(block: false, alt: "equals")[$=$] sign and press ENTER again.


    
    
      
=== Setting the Window


      Of course, the standard window is not suitable for every graph.


      #examplebox("Example 3")[][
Graph #math.equation(block: false, alt: "y equals 0.01 x squared minus 50")[$y = 0.01 x^(2) − 50$] in the window


        #math.equation(block: true, alt: "Xmin, equals minus 100, Xmax equals 100; Ymin, equals minus 60, Ymax equals 50")[$"Xmin" & = − 100 &  & "Xmax" = 100 \ "Ymin" & = − 60 &  & "Ymax" = 50$]
        + Press Y= and enter #math.equation(block: false, alt: "0.01 X squared minus 50")[$0.01 X^(2) − 50$] by keying in
                #math.equation(block: true, alt: "0.01 X , T , θ , n x squared minus 50 Use the X , T , θ , n key to enter X .")[$0.01 " " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display(X "," T "," θ "," n)$] " " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display(x^(2))$] " " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display(−)$] 50 #hide($b l a n k$) "Use the " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display(X "," T "," θ "," n)$] " key to enter" X .$]
+ Press WINDOW and enter the settings as shown in Figure. Use the up and down arrow keys to move from line to line. Then press GRAPH.#figure(figph[GC window settings screen], alt: "GC window settings screen", caption: none)
]


    
    
      
=== Intersect Feature


      We can use the calculator to find the intersection point of two graphs:


      + Enter the equations for the two graphs in the Y= menu.
+ Choose window settings so that the intersection point is visible in the window.
+ Press 2nd TRACE #math.equation(block: false, alt: "5")[$5$] to activate the intersect feacture.
+ Use the left and right arrow keys to position the bug near the intersection point.
+ Respond to each of the calculator’s questions, First curve?, Second curve?, and Guess? by pressing ENTER. The coordinates of the intersection point are then displayed at the bottom of the screen.


      Figure shows one of the intersection points of #math.equation(block: false, alt: "y equals 0.01 x squared minus 50")[$y = 0.01 x^(2) − 50$] and #math.equation(block: false, alt: "y equals minus 0.5 x")[$y = − 0.5 x$].


      #figure(figph[GC intersection of graphs], alt: "GC intersection of graphs", caption: none)


    
    
      
=== Other Windows


      + The #strong[ZDecimal] (Zoom Decimal) window, accessed by pressing ZOOM #math.equation(block: false, alt: "4")[$4$], shows #math.equation(block: false, alt: "x")[$x$]-values from #math.equation(block: false, alt: "minus 4.7")[$− 4.7$] to #math.equation(block: false, alt: "4.7")[$4.7$] only, but the #strong[Trace] feature shows "nice" #math.equation(block: false, alt: "x")[$x$]-values in increments of #math.equation(block: false, alt: "0.1")[$0.1$].
+ The #strong[ZInteger] (Zoom Integer) window shows nice #math.equation(block: false, alt: "x")[$x$]-values in increments of #math.equation(block: false, alt: "1")[$1$] unit. Access the #strong[ZInteger] window as follows: Press ZOOM #math.equation(block: false, alt: "8")[$8$], move the bug with the arrow keys to the center of your new window, and press ENTER.
+ The #strong[ZSquare] window, accessed by pressing ZOOM #math.equation(block: false, alt: "5")[$5$], makes the tick marks on both axes have the same size. In this window, squares look like squares, circles look like circles, and all angles appear true.
+ #strong["Friendly" Windows]: If the difference between Xmin and Xmax is a multiple of #math.equation(block: false, alt: "94")[$94$], the #strong[Trace] feature gives nice values for #math.equation(block: false, alt: "x")[$x$]. A useful example of a friendly window is #math.equation(block: false, alt: "X m i n equals minus 9.4")[$X m i n = − 9.4$], #math.equation(block: false, alt: "X m a x equals 9.4")[$X m a x = 9.4$].


      
        
==== Troubleshooting


        + If the graph is not visible, you may need to adjust your window. Or, the equation may not be activated. Press Y= and check to see if the #math.equation(block: false, alt: "equals")[$=$]sign is highlighted.
+ If you get a range error, ERR: WINDOW RANGE, quit the message and press WINDOW. Alter the window settings so that Xmin is smaller than Xmax and so that Ymin is smaller than Ymax.
+ If you press and get an unfamiliar window, or if the axes are not visible in the ZStandard window, you may need to return the #strong[Mode] or #strong[Format] menus to their default settings. See Troubleshooting in Getting Started.
+ If you get a dimension error, #strong[ERR: INVALID DIM], you may have a StatPlot turned on. Press 2nd Y= #math.equation(block: false, alt: "4")[$4$] ENTER to turn off the StatPlots.
+ If the bug does not move along the curve, TRACE may not be activated. Press TRACE and then the left or right arrow key.
+ If you get the error, #strong[ERR: INVALID], you have probably entered a value of #math.equation(block: false, alt: "x")[$x$] that is outside the window. Adjust the window settings accordingly.
+ If the #math.equation(block: false, alt: "x")[$x$]-axis or #math.equation(block: false, alt: "y")[$y$]-axis is too thick, the tick marks are too close together. Press WINDOW and make Xscl or Yscl larger. Set #math.equation(block: false, alt: "X s c l equals 0")[$X s c l = 0$] or #math.equation(block: false, alt: "Y s c l equals 0")[$Y s c l = 0$] to remove the tick marks.
+ If you get #strong[ERR: NO SIGN CHNG] when using the intersect feature, the calculator did not find any intersection point within the current window. Alter the window settings so that the two curves meet within the window. If the two curves are tangent, the calculator may simply fail to find the point of intersection.