X-values are restricted to [Xmin,Xmax].

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1 A. TRACE Working With A Graph TRACE is a very useful tool in graph analyses. Even when a graph is not visible, you can use TRACE to find Y-values. When using TRACE, the X-values are restricted to the interval [Xmin, Xmax]. For values outside of that interval you will have to change the window or use TABLE. However, the Y-values are not restricted to [Ymin,Ymax]. Locate TRACE. X-values are restricted to [Xmin,Xmax]. You will get an error if you enter an X beyond the set Window.

2 You can use the left and right cursors to trace along a curve or you can enter a value of x by hitting the TRACE key and then a numerical value. If you use the cursors to trace along the curve, the graph will be redrawn as you move beyond the set window and the window will change accordingly. Example 1: Enter the function y=x(x+1)(x-2) and graph using ZOOM4. Hit TRACE. Cursor is at (0,0). Use the cursor to move to x=.5 Enter the value x=1. Press Enter. For x=4 the Y-value is not shown on the graph. For x=-6, there is an error message.

3 Example 2: In this example, the y-values are so small that the graph does not show on the screen. However, the TRACE key can still be used to find Y-values. Enter the function. Use ZOOM4 Enter TRACE and then 1. ZOOM4 is called a friendly window because TRACE moves in increments of x=0.1. For other windows, the TRACE will most likely not move to a value you want, so you can just enter the x-value you desire.

4 B. TABLE The TABLE function is more versatile than TRACE in evaluating a function because you are not restricted to any range of values. You can use a full screen for TABLE or you can use a split screen by changing the MODE.

5 Example 1: Graph the function shown below and then use TABLE in full screen and split screen to evaluate the function. Enter the function. Graph with ZOOM4. Evaluate with the TABLE. Go to MODE and change. Graph the function. Use TRACE and notice the change in the TABLE. One valuable use of TABLE is to examine the behavior of a function right around certain values or to look at extremely large values of x. This is useful in both algebra and calculus when analyzing the behavior of functions around vertical and horizontal asymptotes.

6 Example 2: Look at values of the function very close to x=2 and then look at values of x as it increases in value. Enter the function. Graph with ZOOM4. Access the TABLE. Evaluate for large values. Use a table to evaluate a function at values that are beyond the range of [Xmin, Xmax].

7 C. CALC The CALC key menu has seven applications that can be used with a graph. The last two are used in calculus problems. Access the CALC MENU This is the screen.

8 1: value This key can be used to evaluate the function at any value within the range [Xmin, Xmax]. It performs like TRACE key when you enter a value. You can repeat the process by entering other values. If you enter a value that is outside the range, you will get an error message. Example 1: Enter the function shown and evaluate it at x=3 and x=-2. Enter the function. Use ZOOM6 Access value. When you see this screen, enter a value. Press ENTER and a value will be shown. You can repeat for other values.

9 2: zero Use zero to find the x-value where a function's graph crosses the x-axis. This is an extremely useful tool in solving equations. Example 2: Find a zero of the function shown in the first window. When you enter the left and right bounds, try to enter values that are relatively close to the zero so that other variations in the graph do not impede the process. After you enter a left bound and a right bound, the cursors shown should have the zero between them. Enter. Set a window. Graph. Access 2:zero. Enter a Left Bound of x=-2. Enter a Right Bound of x=-1.5. Ignore "Guess?" and press enter. This is the zero.

10 Note: When you are entering a Left Bound and a Right Bound, you will see two cursors. These should "box in" the area.

11 3: minimum This will find the minimum value of the function over an interval that you designate with a left bound and a right bound. Example 3: Using the function in the example above, find the minimum value of the function shown on the screen. Graph the function. Access "minimum". Enter the left bound. Enter the right bound. Ignore "Guess" and press enter. Minimum is at the point (-1,-4).

12 4: maximum This key works exactly like the minimum key. It will find a maximum value over a given interval. Example 4: Find the local maximum of the function shown above. Graph the function. Access "maximum". Enter the left bound. Enter the right bound. Ignore "Guess" and press enter. Maximum is at the point (0,-3). Note: The minimum in the screen above shows an x value of -2.77E-6. This is the number which is rounded

13 5: intersect Finding where two graphs intersect can be useful in many applications, solving an equation, for example. However, if you have more than two graphs on the screen at once, it is suggested that you turn one of them off before using the intersect option. To turn off a function, move the cursor over the = sign and press enter.

14 Example 5: Solve the equation by using the intersect option. Find the intersection for the two functions shown. Enter. Use ZOOM4. CALC 5:intersect Move cursor close to point. When you see this just press ENTER. When you see this press ENTER again. This is the point of intersection. Other point of intersection.

15 D. Style Functions can be turned on and off by moving the cursor over the equal sign. When you hit, the function will change from on to off or from off to on.

16 Style options affect the appearance of the graph. Unlike other options on the calculator, these are accessed directly on the window. To access these options, move the cursor all the way to the left and you will see the option blinking. There are seven different choices. Hitting the ENTER button will scroll through the options.

17 1. Line: This is the default setting and graphs will be a thin solid line. Enter the function. Graph is solid thin line.

18 2. Thick: The graph will appear as a thick line. If you have two graphs, this option applied to one of the graphs will allow you to distinguish between them easily. Put it in thick mode. Graph will be thick.

19 3. Shade above: This will shade the area above the graph. To change the shading style, go to the DRAW menu. Access the shade above mode. Graph will be shaded above.

20 4. Shade below: This will shade the area below the graph. To change the shading style, go to the DRAW menu. Access the shade below mode. Graph will be shaded below.

21 5. Path: This will trace the path as it is graphed. This can be useful when graphing in parametric mode. This will trace the path as it is graphed. The path will be traced out.

22 6. Animate: This will trace the path of the graph but will not show the graph. Access the animate mode. Graph path will be traced only.

23 7. Dot: When in this mode, the points are not connected. This mode is useful when you are graphing functions that have vertical asymptotes or piece-wise defined functions. Access dot mode. Points will not be connected.

24 E. FORMAT This menu gives you some control over the axes and coordinates that are shown when TRACE is used. To move through the menu, use the cursors to select the format you want and press enter.

25 RectGC and PolarGC: This determines the type of coordinates that will be displayed when TRACE is used. RectGC stands for rectangular graphing coordinates and gives the coordinates in (X,Y) form. PolarGC stands for polar graphing coordinates and will give the coordinates in polar form. PolarGC is usually used when you are graphing in Polar Mode. The other choices are displayed below.

26 RectGC PolarGC CoordOn with TRACE CoordOff with TRACE GridOff GridOn AxesOn AxesOff LabelOff LabelOn ExprOn - Shows y1 ExprOff

27 F. ZOOM and MEMORY The ZOOM menu is a very useful tool when working with graphs. It not only offers preset windows, but other ways of looking at a graph. The MEMORY menu offers you a quick way to return to a window and to change the Zoom In and Zoom Out factors.

28 1: ZBox allows you to box in a part of the graph. This is useful when you need to look at a small part of the graph. ZPrevious will restore the previous window. Example: Graph the function using ZOOM4 and then use ZBox to examine the graph close to the origin.

29 2: Zoom In will zoom in on the graph with the position of the cursor serving as the new center. The default for Zoom In is a factor of 4, but this can be changed by accessing the MEMORY menu and using 4: SetFactors. Example: Use the function from the previous example to access the Zoom In function and then use ZPrevious to restore the graph.

30 3: Zoom Out works exactly like Zoom In. 4: ZDecimal is a "friendly window" which means that TRACE moves in increments of 0.1. This is a useful window to use when the graph will fit because you can move to particular X-values easily. The use of this window is demonstrated in the polynomial examples in the Setting Windows section. 5: ZSquare will redraw the graph by adjusting [Xmin,Xmax] so that the tick marks are an equal distance apart on each axes. This is particularly useful when you need to see an accurate shape, such as a circle. ZDecimal is also a square window. Example: Graph the circle by entering the functions and using ZStandard and ZSquare.

31 6: ZStandard is a preset window of [-10,10] by [-10,10] with tick marks set at integers. The use of this window is demonstrated in the polynomial examples in the Setting Windows section. 7: ZTrig is a trigonometry preset window with [Xmin,Xmax]=. This window is covered in the trigonometry examples in the Setting Windows section. 8: ZInteger will redraw the graph and reset the window to Xscl=10 and Yscl=10. The TRACE key will move in increments of 1.

32 9: ZoomStat will change the graph window so that statistical points will all be shown. This window is generally not used in algebra, trigonometry or calculus. 10: ZoomFit will redraw the graph by resetting [YMin, YMax] to include the maximum and minimum values of the function over the interval [Xmin, Xmax]. This window can produce some rather distorted graphs and should be used with care.