Functions


INTRODUCTION TO GRAPHING CALCULATORS: by the Moravian College Mathematics Department.

A powerful feature of the TI-83 is the ease of working with functions.

Entering Functions:

To enter a function on the TI-83, first press the [Y=] key.  You may enter as many as 10 functions; each function is labeled using the variable Y with a subscript.  To enter the function "the square root of x + 1", or sqrt(x+1), press the following sequence of keys:
[Y=] [2nd] [x2]  [X T q n] [+] [1] [ ) ].
Notice that this function was entered into Y1; to enter a second function into Y2, simply press [ENTER] or the down cursor.

Now enter the function y = 3(x-2) into Y2.

Note that the equals sign = is highlighted on both functions you have entered. This means that they are selected. If you press [GRAPH], both functions will be graphed, first Y1, and then Y2.  If you have several functions entered and do not wish to graph all of them, you can deselect the functions you do not want graphed as follows:

To return to the home screen, press [2nd] [MODE] (which is QUIT).
Once functions have been entered, they can be evaluated for different numerical values of the variable x.  There are several ways to evaluate a function on the TI-83.

Evaluating Graphically with Trace:

If you graph the selected function(s), then you can ask for the value of each graphed function for any x that is in the x-range of the window you have chosen for the graphs.  Let's first carry this out with just one function.

Press [Y=] and deselect the function in Y2, so the only selected function is that in Y1.  Press [WINDOW] and set Xmin = -3, Xmax = 10, Xscl = 1, Ymin = -1, Ymax = 4, Yscl = 1, and Xres = 1.  Now press [GRAPH].

To find the value of the graphed function sqrt(x+1) when x = 1, press  [2nd][TRACE] (which gives the CALCULATE menu), then [ENTER].  On your screen is X=, with a blinking cursor.  Type 1, and press [ENTER]. You will see the y-value Y1(1) at the lower right of the screen and the formula for the function Y1 at the upper left of your screen. You will also see a point highlighted on the graph at (1, Y1(1) ). After displaying a graph, you can also simply press [TRACE] and then the value of X you would like displayed, followed by [ENTER]. Thus, to see the value of Y1(6), display the graph, press [TRACE] 6 [ENTER].

Exercise: Repeat the process just described (beginning with pressing  [2nd] [TRACE] or [TRACE] ) to find the y-values of Y1 for each of the following values of x:

x = 3,  x = -1, x = -2, x = 7.2, x = 11.

For one of these values of x, no y-value will appear, and for another one of these values of x, you will get an error message.  Explain what is wrong in each case.

Now press [Y=], highlight Y2 in addition to Y1, and then press [GRAPH]. Both graphs should appear.  Follow the previous directions to evaluate both functions at x = 1. When you type the number 1 to ask for the value of a function at x = 1, you will see exactly what you saw before: the function Y1 and its y-value for x = 1.  Now press the "up" (or "down") cursor to toggle to the function Y2.  The formula for Y2 will replace that of Y1 in the upper left corner of the screen and you will see the y-value of Y2 at the lower right of your screen.

Exercise: Find the y-values of the function Y2 for x = 3, x = -1, x = -2, x = 7.2, x = 11.
 

Home Screen Evaluation With Functional Notation:

Another way to evaluate a function that has been entered into the [Y=] list (without graphing it) is as follows:
Go to the home screen (press [2nd] [MODE] if necessary). Press the [VARS] key, then the right cursor to highlight Y-VARS at the top of the menu. Then press [ENTER], and from the list of names of functions that is displayed, use the cursor key to select the one you want and press [ENTER]. The function will be displayed on the home screen, and you can type in parentheses and the value of x at which you want to evaluate it.  Here is how to find the value of Y1(2):
Press [VARS] [right-arrow] [ENTER] (select Y1) [ENTER] [( ] [2] [ )] [ENTER].

Exercise: Use this method to find the following values: Y2(2), Y1(15), Y2(-3.4).

The TI-83 can find several values of a function with one procedure. Various ways to do this are described next.

Several Values Using Lists:

You can make a list of x-values at which you want to evaluate the function, store the list, and use the functional notation procedure above to evaluate your function for all values in the list.  A list is defined with curly brackets, with the values in the list separated by commas. Here is the procedure to evaluate Y1 at the following list of x-values: {-1, 2.2, 3, 5.9}:
 


The output is a list of y-values that have as many as 10 decimal digits each, so the list extends beyond the visible window (indicated by the ...). To see all the values, use the right cursor key to scroll the list of values.

Exercise: Use the list procedure above to evaluate the function Y2 at the list of x-values: {-3.1, 0, 2.9, 5}.
 

Using Tables for Evaluation:

You can use the Table function of the TI-83 to evaluate one or more selected functions for a specified list of x-values.  The y-values of at most two functions are displayed together. Here is the procedure to evaluate Y1 and Y2 at the list of x-values: -1, 2.2, 3, 5.9.
Press [2nd] [WINDOW] (which is TABLE SETUP).  Ignore the first two lines and use the down cursor to move to Indpnt, then use the right cursor to highlight Ask, and press [ENTER]. Move to the next line, highlight Auto and press [ENTER]. Now press [2nd] [GRAPH] (which is the TABLE).  You will see three columns, labeled X, Y1, and Y2.  If there are numerical values in the columns, you will want to clear them. Enter each of the numbers in the list of x-values: -1 [ENTER] 2.2 [ENTER] 3 [ENTER] 5.9 [ENTER], and you will see the corresponding y-values of Y1 and Y2 appear in the center and right columns of the table. To see up to 12 decimal places of a y-value, move the cursor to highlight the y-value in the table and the fuller value appears at the bottom of the screen.

If you have more than two functions that you want to evaluate at a list of x-values using Table, first check your [Y=] list to make sure that all the functions you want evaluated are selected. Then press [2nd] [GRAPH] to get to the Table. To see the values of Y3, etc. in the table, use the right cursor key one or more times to scroll to the next selected functions--you will see their y-values displayed for the given values in the X-column.

Exercise:  Use the procedure above to evaluate the functions Y1 and Y2 at the list of x-values: -3.1, 0, 2.9, 5.
 

Automatic Table Values:

You can use the Table function to have the calculator generate as many evenly spaced values of the function(s) as you wish.  You specify the first x-value, and the size of the space between successive x-values (the increment).  The calculator does the rest.  Here is how to generate a list of values of Y1 and Y2, beginning with x = -1, at intervals of Dx = 0.1:
  In the X-column is the list of values beginning with x = -1 (which is highlighted), increasing by increments of 0.1, and the Y1 and Y2 columns display the corresponding y-values.  Use the down cursor to scroll in the X-column; note that you can extend the lists of values as long as you wish.  Now use the up cursor to scroll the X-column in the opposite direction.  "Error" appears in the Y1 column (but not in the Y2 column) for many values of X.  Explain why.

Exercise:  Clear the table, and then follow the procedure above to generate values for Y1 and Y2 beginning with x = 0 and increments of DTbl = .05.