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== 12.3#h(0.6em)Graphing an Equation


    We can graph equations in the variables #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$]. The variable keys are located on the top row of the app 123 keyboard. There are two steps to graphing an equation:


    + Entering the equation
+ Setting the graphing window


    
      
=== A Basic Graph with intercepts


      #examplebox("Example 1")[][
Graph the equation #math.equation(block: false, alt: "x plus y equals 2")[$x + y = 2$].


        In the Algebra View, Press #math.equation(block: false, alt: "x")[$x$]  #math.equation(block: false, alt: "y")[$y$] = #math.equation(block: false, alt: "2")[$2$] .


        The output history shows


        #math.equation(block: true, alt: "eq1 : x plus y equals 2")[$"eq1" : x + y = 2$]
        and the Graphics View shows the line passing through the points #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] and #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]. If we click on either of these two points, GeoGebra will show a label and the coordinates.


        #figure(figph[x+y=2 on Geogebra calculator], alt: "x+y=2 on Geogebra calculator", caption: none)
]


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Special Points on a Graph]


        GeoGebra automatically highlights "Special Points" with  heavy dots on the graph of an equation. GeoGebra will show a label and the coordinates of any Special Point when we clicking on it.


        In particular, any intercepts of a graph are Special Points.
]


      The size and appearance of the displayed graph will vary with the display size and orientation of device we use. We can hide the Algebra View to increase the size of the Graphics View by clicking on the arrow key at the right edge of the border between the Algebra View and the Graphics View. Clicking again on the arrow key brings back the Algebra View.


    
    
      
=== Translate and Zoom


      We can adjust "graphing window", called the "Graphics View" in GeoGebra, without entering numerical values.


      #examplebox("Example 2")[][
Continuing from Example, drag within the Graphics View so that you can see the point on the graph where #math.equation(block: false, alt: "y equals 15")[$y = 15$]. Then zoom out until you can see both that point and the origin.


        Solution: If we click on the arrow key at the right edge of the border between the Algebra View and the Graphics View, the Algebra View disappears to enlarge the Graphing View. But the point on the line where #math.equation(block: false, alt: "y equals 15")[$y = 15$] is still not visible.


        We can tap or click on the Graphics View and drag down and to the right until we can see the point #math.equation(block: false, alt: "open parenthesis minus 13 , 15 close parenthesis")[$( − 13 , 15 )$].


        But the origin is no longer visible. We need to zoom out.


        - To Zoom out on a touch screen or touch pad, we "pinch in".
- To Zoom using a mouse, click on the Graphics View, then hold the Shift key and rotate the mouse wheel forward.


        #figure(figph[x+y=2 translated], alt: "x+y=2 translated", caption: none)
]


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Retrieving the Algebra View]


        To access the Algebra View again, click again on the arrow at the right edge of what had previously been the border between the Graphics View and Algebra View.
]


    
    
      
=== Graphing a Function, Making a Table, and Zooming One Axis


      To graph an equation of the form #math.equation(block: false, alt: "y equals open parenthesis expression in x close parenthesis")[$y = ( "expression in " x )$], we only need enter the expression in #math.equation(block: false, alt: "x")[$x$].


      #examplebox("Example 3")[][
Graph the equation #math.equation(block: false, alt: "y equals x squared minus 5")[$y = x^(2) − 5$] for #math.equation(block: false, alt: "x")[$x$]-values between #math.equation(block: false, alt: "minus 5")[$− 5$] and #math.equation(block: false, alt: "5")[$5$].


        Clear the output history: In the Algebra View (see the Note above if the Algebra View is closed), tap the kebab menu icon (three vertical dots) in any existing history line, and select Delete. (If the menu does not apppear, you may wish to review the Note from the previous section.)


        In the input box, enter #math.equation(block: false, alt: "x")[$x$], then tap the squaring key,  #math.equation(block: false, alt: "5")[$5$] .


        The output history shows


        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared minus 5")[$f ( x ) = x^(2) − 5$]
        and the Graphics View shows the parabola.


        Tap the kebab menu icon (three vertical dots) in the output history line, and select Table of values.


        #figure(figph[Function options on Geogebra calculator], alt: "Function options on Geogebra calculator", caption: none)


        Set the Start value for #math.equation(block: false, alt: "x")[$x$] to #math.equation(block: false, alt: "minus 5")[$− 5$]. Set the End value for #math.equation(block: false, alt: "x")[$x$] to #math.equation(block: false, alt: "5")[$5$]. Set the Step value to #math.equation(block: false, alt: "1")[$1$], then click OK


        Click on the arrow key at the right edge of the border between the Algebra View and the Graphics View to hide the Algebra View and enlarge the Graphing View. Tap on any of the dark dots on the graph to see its coordinates.


        We zoom out just enough so that both #math.equation(block: false, alt: "x equals minus 6")[$x = − 6$] and #math.equation(block: false, alt: "x equals 6")[$x = 6$] are both visible on the #math.equation(block: false, alt: "x")[$x$]-axis. Now we zoom out just the #math.equation(block: false, alt: "y")[$y$]-axis (until #math.equation(block: false, alt: "y equals 20")[$y = 20$] is visible) as follows.


        - On a touch screen, we pinch vertically on the #math.equation(block: false, alt: "y")[$y$]-axis.
- On a computer, we move the cursor above the #math.equation(block: false, alt: "y")[$y$]-axis until the cursor looks like a hand. Then holding down the Shift key, we click (the cursor changes again to an arrow pointing up to a horizontal segment), and drag down.


        The points #math.equation(block: false, alt: "open parenthesis minus 5 , 0 close parenthesis")[$( − 5 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$] are now visible in the Graphics View.


        #figure(figph[y-axis zoomed on Geogebra calculator], alt: "y-axis zoomed on Geogebra calculator", caption: none)
]


      We can also zoom on just the #math.equation(block: false, alt: "x")[$x$]-axis.


      - On a touch screen, we pinch horizontally on the #math.equation(block: false, alt: "x")[$x$]-axis.
- On a computer, we move the cursor above the #math.equation(block: false, alt: "x")[$x$]-axis until the cursor looks like a hand. Then holding down the Shift key, we click (the cursor changes again to an arrow pointing right to a vertical segment), and drag left or right.