«Appendix A – Using a Graphing Calculator Section 2: Tables and Graphs There are lots of great features available on your graphing calculator. ...»
Appendix A – Using a Graphing Calculator
Section 2: Tables and Graphs
There are lots of great features available on your graphing calculator. First,
though, you’ll need to be able to graph a function, or a collection of functions,
using your graphing calculator. This is a three step process:
• Entering the function(s) into the calculator
• Setting the viewing window
• Graphing the function
As you learn to do this, you will also need to be able to use the TABLE feature of the calculator. All of these topics are covered in this section.
Entering Functions in Y= To enter a function in the Y= screen, press Y= and select a line for the equation.
Press CLEAR to delete an equation that is no longer needed. Use Y1 for most problems. If you are entering more than one equation, you’ll use more than one line on the Y= screen.
Use the numbers on the keypad to enter coefficients and the X key to enter variables. Even if the problem uses a different variable, use the X key when entering variables in the calculator. For fractional coefficients, use parentheses, so enter (1/2) for.
Use the operation symbols for addition, subtraction, multiplication and division that appear in the problem. Use (-) for a leading negative rather than subtraction.
To raise a variable to the second power, type X and then press x 2. To raise a variable to another power, such as the kth power, type X and then press ^k. For a square root, press 2nd x 2 and then enter the number or expression. For a different root, such as the kth root, press ^(1/k). Note that the when working with radicals, the calculator automatically opens parentheses. Take care to close parentheses when needed.
Math 1314 Page 1 of 14 Appendix A, Section 2 Example 1: Enter f ( x) = x − 5 x 2 − x + 9 on the Y1 line on the Y= screen.
Solution: Note the use of parentheses and exponents when enter this function.
*** x+7 Example 2: Enter f ( x) = − x + 4 − on the Y2 line on the Y= screen.
x−3 Solution: Note that we want to retain the function entered in Example 1 and enter a second function. Use the arrow keys to scroll down to the Y2 line; then enter this function. Note the use of parentheses.
*** Example 3: Enter f ( x) = 3 2 x 2 − 5 x + 4 on the Y3 line on the Y= screen.
Solution: First rewrite the function as f ( x) = ( 2 x − 5 x + 4 ). Then scroll down to the Y3 line and enter the function.
*** Math 1314 Page 2 of 14 Appendix A, Section 2 Tables The calculator has two different formats for viewing tables of values. In the first format, you set the starting number and the increment value for x – the difference between two values of x that will be used in the table. The calculator automatically generates a table of values.
To set up this type of table, press 2nd WINDOW. This is the resulting screen.
In setting up the table, you select the start point (TblStart) and the increment ∆ Tbl. Use the up and down arrow keys until the character to the right of TblStart= is highlighted. Enter the starting number. Arrow down to the next line and enter the desired increment. An increment of one is often useful, but at times a larger or smaller increment may work better. To finish TABLE SETUP for an automatic table of values, make sure the word Auto is highlighted on both the Indpnt: and Depend: lines.
To view the table of values, press 2nd GRAPH.
Example 4: Using the functions that were entered in Examples 1 - 3, create a table of values starting at x = 0 with an increment of 1.
This is the TABLE SETUP:
To view the values for the function entered in Y3, use the right arrow and cursor over until the Y3 column comes into view.
Notice that the table can display a limited number of decimal places. To view an entry to more decimal places, move the cursor to the desired value and view the number to 12 or 13 significant digits at the bottom of the screen. Note also that the calculator will display ERROR if the function is not defined at a listed value for x.
In the screen shot displayed above, the function entered in Y2 is not defined when x = 3.
The table currently only displays the y values associated with x values from 0 to 6.
However, by scrolling up and down the list of x values, other y values will be displayed in the table. In the table below, y values associated with x values from 26 to 32 are displayed.
Press ENTER. Now press 2nd GRAPH to view the table. The table that was there before has disappeared. The calculator is waiting for x values to be entered.
Now, enter the desired values for x in the X column, and corresponding values for Y1, Y2 and Y3 will appear in those columns.
Example 5: Use the functions that were entered in the Examples 1 - 3, create a table of values that gives only the function values when x = 1, x = -10 and x = 12.5.
Follow the steps given above to change to an “Ask” table.
Solution: Put the cursor in the top of the X column. Press 1, ENTER, -10 ENTER, 12.5 ENTER. This is the result.
Once a function is entered in Y=, graph the function by pressing GRAPH.
If more than one equation is entered in Y=, the calculator will graph all of them, unless you turn off the functions that should not be graphed. To disable or “turn off” an equation, move the cursor so that it is on the equal sign of the graph to turn off and press ENTER. The equal sign will not be highlighted. To enable the function again, move the cursor onto the equal sign and press ENTER. The equal sign will be highlighted again, and the function will show up when graphing.
Example 6: Graph f ( x) = x − 5x2 − x + 9.
Solution: This is the same function that we say in Example 1. We have already entered this function as Y1. Of the three functions listed on the screen shot below, only Y1 will appear on the graph, since the other two functions have been disabled.
Press GRAPH to view the graph of the function.
*** Without modification, the calculator will use a standard 10 unit by 10 unit viewing window. This means that the minimum x value is -10 and the maximum x value is
The graph above shows that the standard viewing window is sometimes not sufficient to view all of the features of the graph of the function.
In this case, the turning point of the graph in the fourth quadrant is not shown in this viewing window. To change the viewing window, press WINDOW and change the appropriate values.
To revert to the standard viewing window, press ZOOM, 6.
Example 7: Select an appropriate viewing window for f ( x) = x − 5x2 − x + 9.
Solution: From Example 6, we found the graph shown in this screen shot.
As we noted, the graph cuts off part of the graph in Quadrant 4. The graph seems to show enough graph in the x direction, and there seems to be enough graph in the positive y direction. We’ll need to change the y minimum to get a better view of the function.
To find an appropriate Ymin value, use trial and error, or scroll down the TABLE to see the smallest y value in the list between x values of 2 and 7. The table below shows a y value of −36.42, so a Ymin of −50 will display the whole graph and will allow you to see the information at the bottom of the screen without covering the graph.
More detail along the negative x axis is needed, and more of the quadrant 4 portion of the graph should be displayed. Unfortunately, it may not be possible to get a good view of the entire graph on one window. Change the viewing window to [-20, 12, -8, 5] and check the result. Change Xscl to 4 and Yscl to 5.
Here’s the result:
Math 1314 Page 9 of 14 Appendix A, Section 2 This does not provide a good view of the graph and shows one of the limitations of relying exclusively on a graphing utility. To view the portion in quadrants 2 and 3 in greater detail, change the window to [ −5, 2, −1,6] with scales of 1 on both axes.
It seems as though the function is not defined for values of x that are less than −4.
You can use the table to confirm this. Use an Auto table with start value 0 and increment 1, and then scroll up until you see what happens to the y values.
To view the quadrant 4 portion in greater detail, change the window to [0,12, −12, −4] with scales of 1 in both directions.
The original graph gives an overview of the graph of the function. The last two provide the details of the graph of the function.
Solution: This function is already entered as Y3. Disable the functions in Y1 and Y2 and enable Y3.
Then press GRAPH.
The modified window was still in place, so the display shows no part of the graph of the function entered in Y3. Revert to the standard viewing window by pressing ZOOM, 6.
To enter a function on a limited domain, put parentheses around the function; then press /; then enter the limitation on the domain and put parentheses around the domain.
The interval [ −1,5] is the same as the inequality −1 ≤ x ≤ 5. We can split this into two statements, −1 ≤ x and x ≤ 5. This is how we’ll enter the interval.
First, press / and then use this sequence of key strokes to enter the interval: (, (-)1, 2nd MATH, 6, X, 2nd MATH, LOGIC, and, X, 2nd MATH, 6, 5).
Press GRAPH to view the graph. Note that it may be necessary to adjust the viewing window to see an appropriate display of the graph of the function. The window used in the screen shot below is [-2, 8, -40, 10].
A piecewise defined function is a function that is defined using two or more functions, each of which has its own domain. We can use a graphing calculator to graph these.
Example 11: Graph the piecewise defined function:
Solution: Enter the function on the Y= screen as Y1= (2, X, +, 3) * (X, 2nd MATH, 6, 1) + ( X, x 2, -, 2) * ( X, 2nd MATH, 3, 1).
Using a standard viewing window, the graph of the function will look like this:
*** An alternate means of entering a piecewise defined function on a TI84 calculator is to enter as Y1 the top line of the function (in parentheses) followed by its domain (in parentheses) with no operation symbol separating the parentheses and then to enter as Y2 the second line of the function and its domain.
Either method generates the same graph.
Some older TI calculators can require different operation symbols for graphing piecewise defined functions. If the multiplication symbol between the function and its domain does not work, try the division symbol.