﻿ Graphing

## Using the TI-83 to Explore Functions

In Calculus problems, it is often helpful to sketch a graph of the function involved. Doing that by hand is tedious and error-prone. Using Excel, one can easily generate nice-looking graphs. However, probably the fastest way to get a sketch of the graph is to use your TI-83 calculator. You can also use your calculator to evaluate functions at specific values of x.

### Entering the Function

1. To enter the function you want to graph, begin by pressing the button. You can enter up to six functions, but most frequently, you'll want only one. If necessary, any functions appearing in the calculator window. Suppose we want to graph the function y = x2 - 3x + 2. Using the up and down cursor keys, make sure that you are on the line that starts out "Y1 =". Enter the function using the button for the variable x:

2 3 2

You'll see this on the calculator screen:

`Y1=X^2-3*X+2`

2. For what it's worth, the multiply operator is not actually needed. You can enter the function as follows:

2 2

If you do, you'll see:

`Y1=X^2-3X+2`

While the multiply operator can often be omitted on the TI-83, it can never be omitted when entering expressions in an Excel worksheet.

3. Again, for what it's worth, there is one other way we could enter this function:

2

If you do, you'll see:

`Y1=X2-3X+2`

### Setting up the Coordinate System

Before graphing the function, it is usually a good idea to set up the coordinate system. Begin by pressing the button. This brings up a screen that allows you to set up 7 properties of the graph window (which corresponds to the coordinate system on which the graph will be drawn):

 Property Purpose Xmin The minimum value of x; the left edge of the window Xmax The maximum value of x; the right edge of the window Xscl The scale to use on the x-axis; the distance between tick marks. Ymin The minimum value of y; the bottom of the window Ymax The maximum value of y; the top of the window Yscl The scale to use on the y-axis; the distance between tick marks. Xres The x-axis resolution resolution (1 - 8)

The Xres setting needs a little explanation. If Xres = 1, the function value is calculated for every pixel coordinate from the left side of the screen to the right.  For complicated functions that can be time-consuming. If the Xres property is set to 2, the function value is calculated only for every other pixel location. This cuts the graphing time approximately in half. The Xres value is the distance between pixels for which the function value is calculated. The maximum is Xres = 8 which would calculate the value of the function at every 8th pixel location. In general, I recommend that you leave the Xres setting at 1 (the default value).

```Xmin = -10 (that's 10 rather than 10)
Xmax = 10
Xscl = 1
Ymin = -10
Ymax = 10
Xres = 1```

### Displaying the Graph

To display the graph, just press the button.

### Calculating Function Values

#### Calculating Function Values Using the "Calculation" Menu

To evaluate the value of your function at a specific value of x, use the calculate feature: . Select the first option in the "Calculate" menu (the "value" option). The graph of your function will be displayed and at the lower-left corner you will be prompted to enter a value for x. When you enter a value and tap the button, the corresponding value of y will be displayed at the bottom of the screen.

Assuming your function is still Y1=X2-3X+2, enter a value of X = 0. The corresponding value of Y is 2. Now, type in the value 3 and the corresponding value of y is 2. Once you have entered the calculate mode, you can continue to type in new values for X and the calculator will display the corresponding values of Y. The only restriction is that the value of X must be within the window. If you set the window to go from x = -10 to x = 10, then the value you enter for x must be between -10 and 10.

#### Calculating Function Values Using the "Vars" Menu

There is another way to calculate the value of a function. Press the button, select the "Y-VARS" option, select the "1: FUNCTION..." option, and then select the "1: Y1" option. Enter the value for which you wish to calculate Y1 in parentheses. When you tap the button, the corresponding function value will be displayed. For example, if your function is still Y1=X2-3X+2, then Y1(0) should give you 2 and Y1(4) should give you 6.

The advantage of this approach is that you can calculate the functional values for a whole list of x-values all at once. All you need to do is enclose the comma-delimited list of x-values in braces within the parentheses of the function argument. For example, if you enter Y1({0,1,2,3,4,5}) the list of corresponding y-values, {2 0 0 2 6 12}, will be displayed in the window.

### Calculating Zeroes of a Function

The zeroes of a function are the values of x for which f(x) = 0. Make sure your function is still Y1=X2-3X+2 and that the window goes from -10 to 10 in both directions. Draw the graph and note that the parabola appears to cross the x-axis near x = 2. Modify the window to get a better view. Set x to run from 0 to 4 and set y to run from -2 to 2. I suggest the scale values be 0.25 in both directions. Redraw the graph and you can see that the parabola crosses the x-axis near 1 and again near 2. Now let's use the calculator to find the specific values of x.

To find the zero of a function use the calculate feature, , and choose the "zero" option in the "Calculate" menu. The calculator needs a left bound, a right bound, and an estimate in order to find a zero of the function. To set the left bound, use the left and right cursor keys to move to a point on the curve corresponding to an x-value a little to the left of the first point at which the function has a value of zero. To set the left bound, tap the button.

To set the right bound, use the left and right cursor keys to move to a point a little to the right of the first zero and tap the button. To enter your guess, use the cursor keys to move to a point as close to the zero as you can and tap the button. The calculator will tell you that the function crosses the x-axis at the point (1,0). That is, x = 1 is a zero of the function.

Repeat this procedure to find the second zero of the function. Use the calculate feature, , and choose the "zero" option in the "Calculate" menu. Use the cursor keys to move a little to the left of the second zero and tap the button. Use the cursor keys to move a little to the right of the second zero and tap the button. Finally, use the cursor keys to move as close to the zero as possible and tap the button. The calculator will tell you that the function crosses the x-axis at (2,0). That is, x = 2 is also a zero of this function.

### Calculating the Derivative at a Point

You calculator can also calculate the approximate derivative of a function at a given value of x. The process is basically the same as for calculating function values (as described in the previous section) except you choose option "6: dy/dx" on the "Calculate" menu. Using our example function, dy/dx at X=10 is 17 and dy/dx at X=20 is 37. Unfortunately, for each new value of X, you must go through the "Calculate" menu to find the corresponding value of dy/dx.

### Working with a Function and Its Derivative

It is easy to calculate values for both a function and its derivative. Assuming that Y1 is still X2-3X+2, enter Y2=2X-3. Notice that Y2 is the derivative of Y1. Go to the "Calculate" menu using the combination and choose the "1: value" option. The graphs for both Y1 and Y2 will be displayed and you'll be prompted for X in the lower-left corner of the window. Enter a value of 2 and you'll find that the corresponding value of Y1 is zero. Tap the down cursor key and the function indicated at the top of the window will change to Y2 and the corresponding value of Y2, which is 1, will be displayed at the bottom of the window. Tapping the up cursor key will take you back to Y1. You can enter additional values of X and use the up and down cursor keys to view the corresponding values of Y1 and Y2.

There is a significant difference between finding the derivative this way and using the technique in the previous section. In this section, you are entering the equation of the derivative and using it to calculate the value. In the previous section (using option "6: dy/dx" on the "Calculate" menu), the calculator uses numerical techniques to approximate the value of the derivative. The numerical technique basically calculates the slopes of secant lines using smaller and smaller values of h until the difference between successive slope calculations is less than 0.001.