Keystrokes 2nd 3 x 2 + 4 x 2) ENTER 2+3 x -1 ENTER 10 x 5 ENTER LOG 200 ENTER 2.34 2nd
Display 5 2.333333333 2.301029996 234000

log 200 2.34

Additional mathematical operations and functions are available from the MATH menu. Press MATH to see the various options (Figure II.5). You will learn in your mathematics textbook how to apply many of them. As an example, calculate 7 by pressing MATH and then either 4 ( or ENTER; finally press 7 ENTER to see 1.912931183. To leave the MATH menu and take no other action, press 2nd QUIT or just CLEAR.
Figure II.5: MATH menu The factorial of a nonnegative integer is the product of all the integers from 1 up to the given integer. The symbol for factorial is the exclamation point. So 4! (pronounced four factorial ) is 24. You will learn more about applications of factorials in your textbook, but for now use the TI-83 to calculate 4! The factorial command is located in the MATH menus PRB sub-menu. To compute 4!, press these keystrokes: 4 MATH 4 ENTER or 4 MATH ENTER ENTER. Note that you can select a sub-menu from the MATH menu by pressing either once than to press three times to get to the PRB sub-menu. or. It is easier to press
Figure II.6: Complex number calculations On the TI-83 it is possible to do calculations with complex numbers. To enter the imaginary number i, press 2nd i. For example, to divide 2 3i by 4 2i, press ( 2 + 3 2nd i ) ( 2nd i ) ENTER. The result is 0.1 0.8i (Figure II.6). To find the complex conjugate of 4 5i press MATH ENTER 4 + 5 2nd i ENTER (Figure II.6).

Functions and Graphs

II.2.1 Evaluating Functions: Suppose you received a monthly salary of $1975 plus a commission of 10% of sales. Let x your sales in dollars; then your wages W in dollars are given by the equation W 1975 .10x. If your January sales were $2230 and your February sales were $1865, what was your income during those months? Heres one method to use your TI-83 to perform this task. Press the Y= key at the top of the calculator to display the function editing screen (Figure II.7). You may enter as many as ten different functions for the TI-83 to use at one time. If there is already a function Y1, press or as many times as necessary to move the cursor to Y1 and then press CLEAR to delete whatever was there. Then enter the expression 1975 .10x by pressing these keys: 1975 +.10 X,T,,n. (The X,T,,n key lets you enter the variable X easily without having to use the ALPHA key.) Now press 2nd QUIT to return to the main calculations screen.

Figure II.7: Y= screen

Figure II.8: Evaluating a function
Assign the value 2230 to the variable x by using these keystrokes (see Figure II.8): 2230 STO X,T,,n. Then press ALPHA : to allow another expression to be entered on the same command line. Next press the following keystrokes to evaluate Y1 and find Januarys wages: VARS 1 [Function] 1 [Y1] ENTER. It is not necessary to repeat all these steps to find the February wages. Simply press 2nd ENTRY to recall the entire previous line, change 2230 to 1865, and press ENTER. Each time the TI-83 evaluates the function Y1, it uses the current value of x. Like your textbook, the TI-83 uses standard function notation. So, to evaluate Y12230 when Y1x 1975 .10x, press VARS (2230) ENTER (see Figure II.9). Then to evaluate Y11865), press 2nd ENTRY to recall the last line, change 2230 to 1865, and press ENTER.
Figure II.9: Function notation You may also have the TI-83 make a table of values for the function. Press 2nd TBLSET to set up the table (Figure II.10). Move the blinking cursor onto Ask beside Indpnt:, then press ENTER. This configuration permits you to input values for x one at a time. Now press 2nd TABLE, enter 2230 in the X column, and press ENTER (see Figure II.11). Continue to enter additional values for X and the calculator automatically completes the table with corresponding values of Y1. Press 2nd QUIT to leave the TABLE screen.
Figure II.10: TBLSET screen
Figure II.11: Table of values
For a table containing values for x 1, 2, 3, 4, 5, and so on, set TblStart 1 to start at x 1, Tbl 1 to increment in steps of 1, and both Indpnt and Depend to Auto.
Technology Tip: The TI-83 does not require multiplication to be expressed between variables, so xxx means x3. It is often easier to press two or three xs together than to search for the square key or the powers key. Of course, expressed multiplication is also not required between a constant and variable. So, to enter 2x3 3x2 4x 5 in the TI-83, you might save keystrokes and press just these keys: 2 X,T,, n X,T,, n X,T,,n + 3 X,T,,n X,T,,n 4 X,T,,n + 5. II.2.2 Functions in a Graph Window: Once you have entered a function in the Y= screen of the TI-83, just press GRAPH to see its graph. The ability to draw a graph contributes substantially to our ability to solve problems. For example, here is how to graph y x3 4x. First press Y= and delete anything that may be there by moving with the arrow keys to Y1 or to any of the other lines and pressing CLEAR wherever necessary. Then, with the cursor on the top line Y1, press (-) X,T,,n MATH 3 + 4 X,T,,n to enter the function (as in Figure II.12). Now press GRAPH and the TI-83 changes to a window with the graph of y x3 4x (Figure II.13). While the TI-83 is calculating coordinates for a plot, it displays a busy indicator at the top right of the graph window. Technology Tip: If you would like to see a function in the Y= menu and its graph in a graph window, both at the same time, open the MODE menu, move the cursor down to the last line, and select Horiz screen. Your TI-83s screen is now divided horizontally (see Figure II.12), with an upper graph window and a lower window that can display the home screen or an editing screen. The split screen is also useful when you need to do some calculations as you trace along a graph. For now, restore the TI-83 to Full screen.

Figure II.17: Custom window
Figure II.18: Graph of y x3 4x
Sometimes you may wish to display grid points corresponding to tick marks on the axes. This and other graph format options may be changed by pressing 2nd FORMAT to display the FORMAT menu (Figure II.19). Use arrow keys to move the blinking cursor to GridOn; press ENTER and then GRAPH to redraw the graph. Figure II.20 shows the same graph as in Figure II.18 but with the grid turned on. In general, youll want the grid turned off, so do that now by pressing 2nd FORMAT, use the arrow keys to move the blinking cursor to GridOff, and press ENTER and CLEAR.
Figure II.19: FORMAT menu
Figure II.20: Grid turned on for y x3 4x
Technology Tip: On the TI-83, the style of your graph can be changed by changing the icon to the left of Y1 on the Y= screen. To change the icon press Y= and then ENTER repeatedly to scroll through the different styles available. II.2.3 Graphing Step and Piecewise-Defined Functions: The greatest integer function, written x, gives the greatest integer less than or equal to a number x. On the TI-83, the greatest integer function is called int and is located under the NUM sub-menu of the MATH menu (see Figure II.5). So, calculate 6.by pressing MATH 5 [int(] 6.78 ENTER. To graph y x, go into the Y= menu, move beside Y1, and press CLEAR MATH 5 [int(] X,T,,n GRAPH. Figure II.21 shows this graph in a viewing window from 5 to 5 in both directions. The true graph of the greatest integer function is a step graph, like the one in Figure II.22. For the graph of y x, a segment should not be drawn between every pair of successive points. You can change from Connected line to Dot graph on the TI-83 by opening the MODE menu. Move the cursor down to the fifth line; select whichever graph type you require; press ENTER to put it into effect, and GRAPH to see the result.
Figure II.21: Connected graph of y x
Figure II.22: Dot graph of y x
Make sure to change your TI-83 back to Connected line, because most of the functions that you will be graphing should be viewed this way. The TI-83 can graph piecewise-defined functions by using the options in the TEST menu (Figure II.23) that is displayed by pressing 2nd TEST. Each TEST function returns the value 1 if the statement is true, and the value 0 if the statement is false.

Figure II.30: Free-moving cursor Remove the free-moving cursor and its coordinates from the window by pressing GRAPH, CLEAR, or ENTER. Press an arrow key again and the free-moving cursor will reappear at the same point you left it. Press TRACE to enable the left and right arrow keys to move the cursor from point to point along the graph of the function. The cursor is no longer free-moving, but is now constrained to the function. The coordinates that are displayed belong to points on the functions graph, so the y-coordinate is the calculated value of the function at the corresponding x-coordinate (Figure II.31). The TI-83 displays the function that is being traced in the upper left of the screen while the TRACE feature is being used.
Figure II.31: TRACE Now plot a second function, y .25x, along with y x3 4x. Press Y=, move the cursor to the Y2 line, and enter.25x, then press GRAPH to see both functions. Note in Figure II.32 the equal signs next to Y1 and Y2 are both highlighted. This means both functions will be graphed as shown in Figure II.33. In the Y= screen, move the cursor directly on top of the equal sign next to Y1 and press ENTER. This equal sign should no longer be highlighted (see Figure II.34). Now press GRAPH and see that only Y2 is plotted (Figure II.35).
Figure II.32: Two functions
Figure II.33: y x3 4x and y .25x
Figure II.34: Y= screen with only Y2 active
Figure II.35: Graph of y .25x
Many different functions can be stored in the Y= list and any combination of them may be graphed simultaneously. You can make a function active or inactive for graphing by pressing ENTER on its equal sign to highlight (activate) or remove the highlight (deactivate). Now go back to the Y= screen and do what is needed in order to graph Y1 but not Y2. Now activate both functions so that both graphs are plotted. Press TRACE and the cursor appears first on the graph of y x3 4x because it is higher up in the Y= list. You know that the cursor is on this function, Y1, because this function is displayed in the upper left corner of the screen (see Figure II.31). Press the up or down arrow key to move the cursor vertically to the graph of y .25x. Now the function Y2 -.25x is displayed in the upper left corner of the screen. Next press the left and right arrow keys to trace along the graph of y .25x. When more than one function is plotted, you can move the trace cursor vertically from one graph to another with the and keys. Technology Tip: Trace along the graph of y .25x and press and hold either or. Eventually you will reach the left or right edge of the window. Keep pressing the arrow key and the TI-83 will allow you to continue the trace by panning the viewing window. Check the WINDOW screen to see that Xmin and Xmax are automatically updated. If you trace along the graph of y x3 4x, the cursor will eventually move above or below the viewing window. The cursors coordinates on the graph will still be displayed, though the cursor itself can no longer be seen. When you are tracing along a graph, press ENTER and the window will quickly pan over so that the cursors position on the function is centered in a new viewing window. This feature is especially helpful when you trace near or beyond the edge of the current viewing window.

The TI-83s display has 95 horizontal columns of pixels and 63 vertical rows. So when you trace a curve across a graph window, you are actually moving from Xmin to Xmax in 94 equal jumps, each called x. You Xmax Xmin would calculate the size of each jump to be x . Sometimes you may want the jumps to 94 be friendly numbers like 0.1 or 0.25 so that, when you trace along the curve, the x-coordinates will be incremented by such a convenient amount. Just set your viewing window for a particular increment x by making Xmax = Xmin 94 x. For example, if you want Xmin 5 and x 0.3, set Xmax 0.3 23.2. Likewise, set Ymax = Ymin 62 y if you want the vertical increment to be some special y. To center your window around a particular point, h, k), and also have a certain x, set Xmin = h 47 x and Xmax h 47 x. Likewise, make Ymin k 31 y and Ymax k 31 x. For example, to center a window around the origin 0, 0), with both horizontal and vertical increments of 0.25, set the range so that Xmin 0.25 11.75, Xmax 0.25 11.75, Ymin 0.25 7.75, and Ymax 0.25 7.75. See the benefit by first graphing y x2 2x 1 in a standard viewing window. Trace near its y-intercept, which is 0, 1, and move towards its x-intercept, which is 1, 0. Then press ZOOM 4 [ZDecimal] and trace again near the intercepts. II.2.6 ZOOM: Plot again the two graphs for y x3 4x and for y .25x. There appears to be an intersection near x 2. The TI-83 provides several ways to enlarge the view around this point. You can change the viewing window directly by pressing WINDOW and editing the values of Xmin, Xmax, Ymin, and Ymax. Figure II.37 shows a new viewing window for the range displayed in Figure II.36. The cursor has been moved near the point of intersection; move your cursor closer to get the best approximation possible for the coordinates in the intersection.

Figure II.36: New WINDOW

Figure II.37: Closer view
A more efficient method for enlarging the view is to draw a new viewing window with the cursor. Start again with a graph of the two functions y x3 4x and y .25x in a standard viewing window (press ZOOM 6 for the standard window). Now imagine a small rectangular box around the intersection point, near x 2. Press ZOOM 1 [ZBox] (Figure II.38) to draw a box to define this new viewing window. Use the arrow keys to move the cursor, whose coordinates are displayed at the bottom of the window, to one corner of the new viewing window you imagine. Press ENTER to fix the corner where you have moved the cursor; it changes shape and becomes a blinking square (Figure II.39). Use the arrow keys again to move the cursor to the diagonally opposite corner of the new rectangle (Figure II.40), then press ENTER. The rectangular area you have enclosed will now enlarge to fill the graph window (Figure II.41).

Figure II.38: ZOOM menu

Figure II.39: One corner selected
You may cancel the zoom any time before you press this last ENTER. Press ZOOM once more and start over. Press CLEAR or GRAPH to cancel the zoom, or press 2nd QUIT to cancel the zoom and return to the home screen.

10 0.65

Figure II.40: Box drawn
Figure II.41: New viewing window
You can also quickly magnify a graph around the cursors location. Return once more to the standard viewing window for the graph of the two functions y x3 4x and y .25x. Press ZOOM 2 [Zoom In] and then press arrow keys to move the cursor as close as you can to the point of intersection near x 2 (see Figure II.42). Then press ENTER and the calculator draws a magnified graph, centered at the cursors position (Figure II.43). The range variables are changed to reflect this new viewing window. Look in the WINDOW menu to verify this.

10 1.85

Figure II.42: Before a zoom in
Figure II.43: After a zoom in
As you see in the ZOOM menu (Figure II.38), the TI-83 can Zoom In (press ZOOM 2) or Zoom Out (press ZOOM 3). Zoom out to see a larger view of the graph, centered at the cursor position. You can change the horizontal and vertical scale of the magnification by pressing ZOOM 4 [SetFactors.] (see Figure II.44) and editing XFact and YFact, the horizontal and vertical magnification factors (see Figure II.45).
Figure II.44: ZOOM MEMORY menu
Figure II.45: ZOOM MEMORY SetFactors.
The default zoom factor is 4 in both directions. It is not necessary for XFact and YFact to be equal. Sometimes, you may prefer to zoom in one direction only, so the other factor should be set to 1. As usual, press 2nd QUIT to leave the ZOOM menu. Technology Tip: The TI-83 remembers the window it displayed before a zoom. So, if you should zoom in too much and lose the curve, press ZOOM 1 [ZPrevious] to go back to the window before. If you want to execute a series of zooms but then return to a particular window, press ZOOM 2 [ZoomSto] to store the current windows dimensions. Later, press ZOOM 3 [ZoomRcl ] to recall the stored window. II.2.7 Value: Graph y x 3 4x in the standard viewing window (Figure II.13). The TI-83 can calculate the value of this function for any given x (between the Xmin and Xmax values). Press 2nd CALC to display the CALCULATE menu (see Figure II.46), then press 1 [value]. The graph of the function is displayed and you are prompted to enter a value for x. Press 1 ENTER. The x-value you entered and its corresponding y-value are shown at the bottom of the screen and the cursor is located at the point 1, 3 on the graph (see Figure II.47).

Figure II.46: CALCULATE menu
Figure II.47: Finding a value
Note that if you have more than one graph on the screen, the upper left corner of the TI-83 screen will display the equation of the function whose value is being calculated. Press the up or down arrow key to move the cursor vertically between functions at the entered x-value. II.2.8 Relative Minimums and Maximums: Graph y x3 4x once again in the standard viewing window. This function appears to have a relative minimum near x 1 and a relative maximum near x 1. You may zoom and trace to approximate these extreme values.
First trace along the curve near the relative minimum. Notice by how much the x-values and y-values change as you move from point to point. Trace along the curve until the y-coordinate is as small as you can get it, so that you are as close as possible to the relative minimum, and zoom in (press ZOOM 2 ENTER or use a zoom box). Now trace again along the curve and, as you move from point to point, see that the coordinates change by smaller amounts than before. Keep zooming and tracing until you find the coordinates of the relative minimum point as accurately as you need them, approximately 1.15, 3.08. Follow a similar procedure to find the relative maximum. Trace along the curve until the y-coordinate is as great as you can get it, so that you are as close as possible to the relative maximum, and zoom in. The relative maximum point on the graph of y x3 4x is approximately 1.15, 3.08. The TI-83 can automatically find the relative minimum and relative maximum points. Press 2nd CALC to display the CALCULATE menu (Figure II.46). Choose 3 [minimum] to calculate the minimum value of the function and 4 [maximum] for the maximum. You will be prompted to trace the cursor along the graph first to a point left of the minimum/maximum (press ENTER to set this left bound). Then move to a point right of the minimum/maximum and set an right bound and press ENTER. Note the two arrows at the top of the display marking the left and right bounds (as in Figure II.48). Next move the cursor along the graph between the two bounds and as close to the minimum/maximum as you can; this serves as a guess for the TI-83 to start its search. Good choices for the left bound, right bound, and guess can help the calculator work more efficiently and quickly. Press ENTER and the coordinates of the relative minimum/maximum point will be displayed (see Figure II.49).
Figure II.48: Finding a minimum
Figure II.49: Relative minimum on y x3 4x
Note that if you have more than one graph on the screen, the upper left corner of the TI-83 screen will display the equation of the function whose minimum/maximum is being calculated. II.2.9 Inverse Functions: The TI-83 draws the inverse function of a one-to-one function. Graph y xas Y1 in the standard viewing window (see Figure II.50). Next, press 2nd DRAW to display the DRAW menu. Use to move down and then choose 8 to draw the inverse function (see Figure II.51). Press VARS ENTER (see Figure II.52). These keystrokes instruct the TI-83 to draw the inverse function of Y1. The original function and its inverse function will be displayed (see Figure II.53). Note that the calculator must be in function mode in order to use DrawInv.

Figure II.59: A zero of y x3 8x TRACE and ZOOM are especially important for locating the intersection points of two graphs, say the graphs of y x3 4x and y .25x. Trace along one of the graphs until you arrive close to an intersection point. Then press or to jump to the other graph. Notice that the x-coordinate does not change, but the y-coordinate is likely to be different (see Figures II.60 and II.61).

3.2 3.2

Figure II.60: Trace on y
Figure II.61: Trace on y .25x
When the two y-coordinates are as close as they can get, you have come as close as you now can to the point of intersection. So zoom in around the intersection point, then trace again until the two y-coordinates are as close as possible. Continue this process until you have located the point of intersection with as much accuracy as necessary. The points of intersection are approximately 2.062, 0.515, 0, 0, and 2.062, 0.515. You can also find the point of intersection of two graphs by pressing 2nd CALC 5 [intersect]. Trace with the cursor first along one graph near the intersection and press ENTER; then trace with the cursor along the other graph and press ENTER. Marks are placed on the graphs at these points. Finally, move the cursor near the point of intersection and press ENTER again. Coordinates of the intersection will be displayed at the bottom of the window. More will be said about the intersect feature in Section II.3.3. II.3.2 Solving Equations by Graphing: Suppose you need to solve the equation 24x3 36x 17 0. First graph y 24x3 36x 17 in a window large enough to exhibit all its x-intercepts, corresponding to all the equations real zeros (roots). Then use zoom and trace, or the TI-83s zero finder, to locate each one. In fact, this equation has just one real solution, x 1.414. Remember that when an equation has more than one x-intercept, it may be necessary to change the viewing window a few times to locate all of them. Technology Tip: To solve an equation like 24x36x, you may first rewrite it in general form, 24x3 36x 17 0, and proceed as above. However, you may also graph the two functions y 24xand y 36x, then zoom and trace to locate their point of intersection.
II.3.3 Solving Systems by Graphing: The solutions to a system of equations correspond to the points of intersection of their graphs (Figure II.62). For example, to solve the system y 2x 5 and y 2x 1, first graph them together. Then use zoom and trace, or use the intersect option in the CALCULATE menu, to locate their point of intersection, which is 1, 3 (see Figure II.63). The solutions of the system of two equations y 2x 5 and y 2x 1 correspond to the solutions of the single equation 2x 5 2x 1, which simplifies to 4x 4 0. So you may also graph y 4x 4 and find its x-intercept to solve the system.
Figure II.62: Solving a system of equations
Figure II.63: The point of intersection is 1, 3). 3x x 4. To solve it with your 2

II.3.4 Solving Inequalities by Graphing: Consider the inequality 1 TI-83, graph the two functions y 1
3x and y x 4 (Figure II.64). First locate their point of 2 3x intersection, at x 2. The inequality is true when the graph of y 1 lies above the graph of 2 y x 4, and that occurs for x < 2. So the solution is x 2, or , 2.

Figure II.64: Solving 1

3x x4 2
The TI-83 is capable of shading the region above or below a graph or between two graphs. For example, to graph y x2 1, first graph the function y xas Y1. Then press 2nd DRAW 7 [Shade(] VARS , 10 ) ENTER (see Figure II.65). These keystrokes instruct the TI-83 to shade the region above y xand below y 10 (chosen because this is the greatest y-value in the graph window) using the default shading option of vertical lines. The result is shown in Figure II.66.
To clear the shading, press 2nd DRAW 1.
Figure II.65: DRAW Shade Now use shading to solve the previous inequality, 1
Figure II.66: Graph of y x2 1
3x x 4. The function whose graph forms the 2 lower boundary is named first in the SHADE command (see Figure II.67). To enter this in your TI-83, press these keys: 2nd DRAW 7 X,T,,n 4 , X,T,,n 2 ) ENTER (Figure II.68). The shading extends 3x left from x 2, so the solution to 1 x 4 is x 2, or , 2 (see Figure II.68). 2
Figure II.67: DRAW Shade command

Figure II.68: Graph of 1

More information about the DRAW menu is in the TI-83 manual.

Trigonometry

II.4.1 Degrees and Radians: The trigonometric functions can be applied to angles measured either in radians or degrees, but you should take care that the TI-83 is configured for whichever measure you need. Press MODE to see the current settings. Press twice and move down to the third line of the mode menu where angle measure is selected. Then press or to move between the displayed options. When the blinking cursor is on the measure you want, press ENTER to select it. Then press CLEAR or 2nd QUIT to leave the mode menu. Its a good idea to check the angle measure setting before executing a calculation that depends on a particular measure. You may change a mode setting at any time and not interfere with pending calculations. Try the following keystrokes to see this in action.
Expression sin 45 sin sin sin 45 sin
Keystrokes MODE SIN 45 ENTER ENTER CLEAR
Display.7071067812.0548036651 0.8509035245.5
SIN 2nd ENTER MODE ENTER CLEAR SIN 2ND ENTER SIN 45 ENTER SIN 2nd 6 ) ENTER
The first line of keystrokes sets the TI-83 in degree mode and calculates the sine of 45 degrees. While the calculator is still in degree mode, the second line of keystrokes calculates the sine of degrees, approximately 3.1415. The third line changes to radian mode just before calculating the sine of radians. The fourth line calculates the sine of 45 radians. Finally, the fifth line calculates the sine of radians (the 6 calculator remains in radian mode). The TI-83 makes it possible to mix degrees and radians in a calculation. Execute these keystrokes to calculate tan 45 sin as shown in Figure II.69. TAN 45 2nd ANGLE 1 [ ] ) + SIN 2nd 6 ) 2nd ANGLE [ r ] ENTER. Do you get 1.5 whether your calculator is set either in degree mode or in radian mode?

Figure II.69: Angle measure Technology Tip: The automatic left parenthesis that the TI-83 places after functions such as sine, cosine, and tangent (as noted in Section II.1.3) can affect the outcome of calculations. In the previous example, the degree sign must be inside of the parentheses so that when the TI-83 is in radian mode, it calculates the tangent of 45 degrees, rather than converting the tangent of 45 radians into an equivalent number of degrees. Also, the parentheses around the fraction are required so that when the TI-83 is in radian mode, it converts 6 into radians, rather than converting merely the 6 to radians. Experiment with the placement of parentheses 6 to see how they affect the result of the computation. II.4.2 Graphs of Trigonometric Functions: When you graph a trigonometric function, you need to pay careful attention to the viewing window and to your angle measure configuration. For example, graph sin 30x in the standard viewing window in radian mode. Trace along the curve to see where it is. Zoom y 30 in to a better window, or use the period and amplitude to establish better WINDOW values. Technology Tip: Because 3.1, when in radian mode, set Xmin 0 and Xmax 6.3 to cover the interval from 0 to 2. Graphing Technology Guide
Next graph y tan x in the standard window first, then press ZOOM 7 [ZTrig] to change to a special window for trigonometric functions in which the Xscl is 1.5708 or 90 and the vertical range is from to 4. The TI-83 plots consecutive points and then connects them with a segment, so the graph is not exactly what you should expect. You may wish to change from Connected line to Dot graph (see Section II.2.3) when you plot the tangent function.

Scatter Plots

II.5.1 Entering Data: This table shows total prize money (in millions of dollars) awarded at the Indianapolis 500 race from 1995 to 2003. (Source: Indy Racing League) Year Prize (in millions) 1995 $8.$8.$8.$8.$9.$9.$9.$10.$10.15
Well now use the TI-83 to construct a scatter plot that represents these points and to find a linear model that approximates the given data. The TI-83 holds data in lists. There are six list names in memory L1, L 2, L 3, L 4, L 5, L 6, but you can create as many list names as your TI-83 memory has space to store. Before entering this new data, press STAT 4 [ClrList] 2nd L1, 2nd L2 , 2nd L3 , 2nd L4 , 2nd L5 , 2nd L6 ENTER to clear all data lists. This can also be done from within the list editor by highlighting each list title (L1, etc) and pressing CLEAR ENTER. Now press STAT 1 [Edit] to reach the list editor. Instead of entering the full year, let x 5 represent 1995, x 6 represent 1996, and so on. Here are the keystrokes for the first three years: 5 ENTER 6 ENTER 7 ENTER and so on, then press to move to the first element of the next list and press 8.06 ENTER 8.11 ENTER 8.61 ENTER and so on (see Figure II.70). Press 2nd QUIT when you have finished.

Figure II.74: Correlation coefficient
Figure II.75: Linear regression line
II.5.4 Other Regression Models: After data points have been entered, you can choose from nine different regression models. They are all located in the CALC sub-menu of the STAT menu.
Matrices Note: If you are using a TI-83 Plus or TI-84 Plus, press 2nd MATRX in the keystroke sequences given in this section to access the matrix menu.
II.6.1 Making a Matrix: The TI-83 can display and use 10 different matrices (A through J). Heres how to 4 in your calculator. store this matrix 17
Press MATRX to see the matrix menu (Figure II.76); then press or just to switch to the matrix EDIT menu. Whenever you enter the matrix EDIT menu, the cursor starts at the top matrix. Move to another matrix by repeatedly pressing. For now, press ENTER to edit matrix A. The display will show the dimension of matrix A if the matrix exists; otherwise, it will display 1 1. Change the dimensions of matrix A to by pressing 3 ENTER 4 ENTER. Simply press ENTER or an arrow key to accept an existing dimension. The matrix shown in the window changes in size to reflect a changed dimension.

Figure II.76: MATRX menu

Figure II.77: Editing a matrix
Use the arrow keys or press ENTER repeatedly to move the cursor to a matrix element you want to change. If you press ENTER, you will move right across a row and then back to the first column of the next row. At the right edge of the screen in Figure II.77, there are dashes to indicate more columns than are shown. Go to them by pressing as many times as necessary. The ordered pair at the bottom left of the screen shows the cursors current location withing the matrix. The element in the second row and first column in Figure II.77 is highlighted, so the ordered pair at the bottom of the window is 2 , 1, and the screen shows that elements current value. Continue to enter all the elements of matrix A; press ENTER after inputting each value. When you are finished, leave the matrix editing screen by pressing 2nd QUIT to return to the home screen.

x 2y 3z 9 x 3y 4. 2x 5y 5z 17
First enter this augmented matrix as A in your TI-83: 0 4. Next store this matrix in E (press MATRX 1 STO MATRX 5 ENTER) so you may keep the original in case you need to recall it. Here are the row operations and their associated keystrokes. At each step, the result is stored in E and replaces the previous matrix E. The matrix in row-echelon form is shown in Figure II.82. Row Operation Add row 1 to row 2. Add 2 times row 1 to row 3. Add row 2 to row 3. Multiply row 3 by 1. 2 Keystrokes MATRX ALPHA D MATRX 5 , 1 , 2 ) STO MATRX 5 ENTER MATRX ALPHA F (-) 2 , MATRX 5 , 1 , 3 ) STO MATRX 5 ENTER MATRX ALPHA D MATRX 5 , 2 , 3 ) STO MATRX 5 ENTER MATRX ALPHA E , MATRX 5 , 3 ) STO MATRX 5 ENTER
Figure II.82: Row-echelon form of matrix after row operations So, z 2, y 1, and x 1. Technology Tip: The TI-83 can produce a row-echelon form and the reduced row-echelon form of a matrix. ALPHA A [ref(] MATRX 1 ) The row-echelon form of matrix A is obtained by pressing MATRX ENTER (Figure II.83) and the reduced row-echelon form is obtained by pressing MATRX ALPHA B [rref(] MATRX 1 ) ENTER (Figure II.84). Note that the row-echelon form of a matrix is not unique, so your calculator may not get exactly the same matrix as you do by using row operations. However, the matrix that the TI-83 produces will result in the same solution to the system.
Figure II.83: Row-echelon form
Figure II.84: Reduced row-echelon form
II.6.4 Determinants and Inverses: Enter this square matrix as A: 3 determinant 0 , go to the home screen and press MATRX 5 You should find that the determinant is 2 as shown in Figure II.85.

3 0. To calculate its 5

1 [det(] MATRX 1 ) ENTER.
Because the determinant of the matrix is not zero, it has an inverse, A -1. Press MATRX 1 x -1 ENTER to calculate the inverse of matrix A, also shown in Figure II.85.

Figure II.85: A and A -1

Figure II.86: Solution matrix
x 2y 3z 9 Now lets solve a system of linear equations by matrix inversion. Once more, consider x 3y 4. 2x 5y 5z The coefficient matrix for this system is the matrix 0 , that was entered as matrix A in the previous example. Now enter the matrix 4 as B. Then press MATRX 1 x -1 MATRX ENTER to calculate the solution matrix (Figure II.86). The solution is still x 1, y 1, and z 2.

Figure II.98: Polar Y= menu
Figure II.99: Polar graph of r 4 sin
Figure II.99 shows rectangular coordinates of the cursors location on the graph. You may sometimes wish to trace along the curve and see polar coordinates of the cursors location. The first line of the FORMAT menu (Figure II.19) has options for displaying the cursors position in rectangular (RectGC) or polar (PolarGC) form.
Probability and Statistics
II.9.1 Random Numbers: The command rand generates a number between 0 and 1. You will find this command in the PRB (probability) sub-menu of the MATH menu. Press MATH 1 [rand] ENTER to generate a random number. Press ENTER to generate another number; keep pressing ENTER to generate more of them. If you need a random number between, say, 0 and 10, then press 10 MATH number between 5 and 15, press 5 + 10 MATH 1 ENTER. 1 ENTER. To get a random
II.9.2 Permutations and Combinations: To calculate the number of permutations of 12 objects taken 7 at a time, 12 P7, press 12 MATH 2 [nPr] 7 ENTER. So, 12 P7 3,991,680, as shown in Figure II.100.

Figure II.100:

and 12C [nCr] 7 ENTER.
For the number of combinations of 12 objects taken 7 at a time, 12C7, press 12 MATH So, 12C7 792, as shown in Figure II.100.
II.9.3 Probability of Winning: A state lottery is configured so that each player chooses six different numbers from 1 to 40. If these six numbers match the six numbers drawn by the State Lottery Commission, the player wins the top prize. There are 40C6 ways for the six numbers to be drawn. If you purchase a single lottery ticket, your probability of winning is 1 in 40C6. Press MATH ENTER to calculate your chances, but dont be disappointed. II.9.4 Sum of Data: The following data are a students scores on 8 quizzes and 2 tests throughout an algebra course. 25, 20, 18, 89, 17, 24, 23, 22, 25, 93 To find the total points earned by the student, first enter the data using the TI-83s list editor, as shown in Figure II.101. Then press 2nd LIST 5 2ND L1 ) ENTER. From Figure II.102, the student earned 356 points throughout the algebra course.