An Algebra II Student S Guide

An Algebra II Student’s Guide

to the

TI-84+ Graphing Calculator

prepared by: Mrs. P. Marques Seneca High School Welcome to Algebra II! This guide is intended to assist you as you use your graphing calculator throughout the course. Your teacher will require you to be able to do many problems both with and without the graphing calculator.

CONTENTS

1. Getting Started ...... 1

2. Order of operations ...... 2

3. Graphing a line ...... 3

4. Adjusting the viewing window...... 3

5. Finding x-intercepts of a linear function ...... 4

6. Scatter plots and lines of best fit ...... 5

7. Solving a system of linear equations ...... 7

8. Entering matrices ...... 8

9. Finding x-intercepts of a quadratic function ...... 9

10. Finding the vertex of a quadratic function ...... 9

11. Solving polynomial equations ...... 10

12. Permutations and combinations ...... 11

13. Evaluating rational exponents ...... 12

14. Finding nth roots ...... 12

15. Absolute value ...... 12

16. Technical Support ...... 13 GETTING STARTED

Whether you are using a TI-84+ or a similar graphing calculator, keep in mind that every function is color-coded. Below is an image of the TI-84+ taken from the following website: 21ctf.fi.ncsu.edu/msms/ti84.html.

In order to use the blue functions listed above the keys, you must first press the blue “2nd” key. In order to access the light green letters and functions listed above the keys, you must first press the light green “ALPHA” key.

-1- ORDER OF OPERATIONS

You must understand and know the order of operations in order to evaluate an 1 2  32 expression correctly using your calculator. To evaluate the expression , you must 2(8) remember that the division bar (vinculum) groups the numerator and denominator, so the equation must be entered with parentheses grouping the numerator and parentheses grouping the denominator in the following way:

( 1 - 2 + 3 x2

)  ( 2 ( 8 )

) ENTER

If you want the answer to be in the form of a fraction, you can access this function by pressing the MATH key at the end of your expression. You can also avoid retyping an expression by recalling it using the 2ND ENTER keys.

2ND ENTER MATH ENTER ENTER

Keep in mind that the TI-84+ family of graphing calculators will only display reduced fractions in either proper or improper form.

-1- GRAPHING A LINE

In order to graph a line, you must solve it for y . To graph the line 3x  4y  8 , you 3 must first rewrite it as y   x  2 before you enter it into the calculator. 4

Y= ( (-) 3  4 ) x,T, ,n

In order to graph the line in a standard viewing window where the x-axis runs from -10 to 10, and the y-axis runs from -10 to 10, press the ZOOM 6 keys after entering the above key strokes.

NOTE: You can always return to the home screen at any time by pressing 2ND MODE .

ADJUSTING THE VIEWING WINDOW

Let’s say that when you graph the line you cannot see its x- or y-intercepts on the screen. In that case, you must press the WINDOW key in order to adjust the window. For example, if you want to graph the line y  15x  27 you would see:

Go to WINDOW to change Press GRAPH . Now you can what you see on the screen. see the graph and both of its Since you know that the y- intercepts. intercept is 27, you need only to adjust the Ymax entry to a number larger than 27.

-1- There is no specific method to follow in order to adjust the viewing screen window. Sometimes you just have to make educated guesses when changing the maximum and minimum values.

FINDING X-INTERCEPTS OF A LINEAR FUNCTION

In order to find the x-intercepts of a line on the graphing calculator, you must first graph the line as outlined above. Using the line y  15x  27 from the previous example, let’s find the x-intercept.

Start by pressing 2ND The calculator will ask you for a 0 ENTER left and right bound. This simply TRACE 2 . Note that a means that you should enter a “zero” is equivalent to an “x- number you think is to the left of intercept” in mathematics. the x-intercept and one that is to the right. Just type in these numbers as you are given the prompts. Here, you simply press ( - ) 5 ENTER .

When the calculator asks Finally, press ENTER to you to guess, simply take your view the answer displayed at cursor close to the point of the bottom of the screen. The intersection. x-intercept is at x = -1.8.

-1- S CATTER PLOTS AND LINES OF BEST FIT

When you are given a list of data, you can graph it on you calculator. Let’s say you were given the following information:

x 1 2 3 4 5 6 7 8 9 10 11 y -2 2 5 1 6 10 3 5 5 12 9 You must enter this information into your calculator by creating lists for your information.

NOTE: You may need to adjust the viewing window to see a better picture of your scatter plot. Also, if there are other graphs on your viewing screen, press Y= and clear any equations you may have.

Press the STAT key and You will enter the x values in However,Once your if you lists need are the entered,equations press in the Y=2ND screen Y= for a select Use optionthe cursor 1. keys to L1 and Finally, the y pressvalues GRAPH in L2. . This is done by taking your cursor to the tolater access date, the you STAT can simply PLOT de- navigate To find throughthe equation the different of the Then press 2ND 1 Notice that the values for a top of each list and typing in the function.activate them Press by ENTERplacing theto lineselections of best on fit, the press screen. STAT Press cursorand b areon the given. equal The sign value and number , followed 2ND by 2 the , select Plot 1. andENTER move to selectthe cursor an option. over to pressingof r is the ENTER correlation . This allows you ENTER key. You will do this for CALC and select option 4. tocoefficient. keep the equations, The closer but thethey VARS and move the cursor over each number entered. To move won’tvalue isbe to graphed. 1, the closer the to the Y-VARS option and select onto the next list simply move plotted points are to the line option 1. Finally, select option 1 your cursor over to that column of best fit. If you do not again for Y1. What this allows you and repeat the steps. see the value for r, do the to do is to get the line of best fit following: and have it pasted to your Y=

screen. Press ENTER to get the 1 2ND 0 x and line of best fit. scroll down until you find (If you press Y= , you will DiagnosticON. Press ENTER notice that the regression twice and repeat the steps equation you just calculated is in in the previous column. Y1.)

NOTE: If you return to the Y= screen, you will notice that PLOT1 is highlighted at the very top. It is important to turn all plots off in order to be able to graph equations entered in the Y= . You do this by moving your cursor up to the highlighted plot and press ENTER in order to remove the highlight. Press the GRAPH key and This can also be done by pressing 2ND Y= and you will see the scatter plot going into each of the plots to turn them off in the graphed along with the line same way you turned them on. If you don’t turn off of best fit. the plots, -5- your equations will always be graphed together with the scatter plot or you may get an error message. SOLVING A SYSTEM OF LINEAR EQUATIONS

When you solve a system of linear equations, you are finding the point of intersection of that system. First you need to graph the lines and ensure that you can see the point of intersection on the screen by adjusting the viewing window as necessary. In  7x 12  order to find the point of intersection for the linear system 2 , you would do the x 15 3 following:

Enter the equations into the Graph it on the standard Notice that the point of calculator. viewing window. Since you can’t intersection is in quadrant IV. see the point of intersection, Use this fact to help you choose adjust the window. the max/min values and graph the lines again.

Now that you can see the point The calculator will ask you which of intersection, press curves (lines) you want to find the intersection of. The top of the 2ND TRACE 5 ENTER screen displays one of the equations we want. Press ENTER .

-5- The second line we want is Here, you may just want to You can now see that the point displayed on the top so press move your cursor to the point of of intersection of the linear ENTER . intersection although this is not system is approximately at necessary. Press ENTER . (3.5, -12.7) .

ENTERING MATRICES

3  7 12  In order to enter the matrix   into the calculator, follow the steps 0 9  4 below:

Press 2ND x 1 and move the Use the cursor keys to enter the cursor over to EDIT. Select an empty dimensions of the matrix. Then press matrix and press ENTER . ENTER after you type in each entry. To return to the home screen press 2ND MODE .

FINDING X-INTERCEPTS OF QUADRATIC FUNCTIONS

You can find the x-intercepts of a quadratic function in the same way as you did when finding the x-intercept of a line.

-5- Enter the equation into the Press 2ND TRACE 2 One x-intercept is at about calculator by going to the Y= and follow the prompts as you did x = -1.9. To find the other x- screen. Graph the equation in the for finding the x-intercept of a intercept, follow the same steps standard viewing window and line. as before. When the calculator adjust the window as necessary. asks for a guess, simply move the cursor closer to the other x- intercept you want to find.

The other x-intercept is at about x = 1.2.

FINDING THE VERTEX OF A QUADRATIC FUNCTION

The vertex of a parabola is either its maximum or minimum point. The vertex of the above parabola is a minimum and can be found by pressing 2ND TRACE 3 .

Once the graph appears on the Because the vertex appears to x = 2 appears to be to the right screen, follow the left/right be at about (0, -7), the choice of of the vertex so it is a good bound prompts as you did before. x = -2 for a left bound seems choice for the right bound. appropriate.

NOTE: In order to find the vertex of a parabola which is a maximum, follow the previous steps, but select option 4 (maximum) in the CALCULATE screen.

When the “Guess” prompt appears, simply press ENTER . The vertex is at (-0.3, -7.3).

SOLVING POLYNOMIAL EQUATIONS

Suppose you are asked to solve the polynomial equation 3x3  7x2  5x  7  10. If you tried to set the equation to zero and factor, you would find that this would be impossible. It is only possible to solve using the graphing calculator. Follow the steps below:

Enter the left side of the Graph the equations. It may

equation into Y1 of the Y= be necessary to adjust the Press 2ND TRACE 5 . screen. Enter the right side as viewing window.

Answer the prompts as you did The second point of The last point of intersection when solving a system of linear intersection is at about (-0.4, 10) is at about (0.9, 10) equations. Remember to move the cursor to the desired point of intersection each time the From these three points of prompt asks you to “guess.” The intersection, we can determine first point of intersection is at that the solutions to the equation about (-2.8, 10). 3x3  7x2  5x  7  10 are at about x = -2.8, x = -0.4, and x = 0.9.

PERMUTATIONS AND COMBINATIONS

The permutation and combination keys can be accessed by pressing the MATH key. You will need to use your arrow keys to highlight the PRB heading. In this screen, you can now access the permutation function (option 2), the combination function (option 3), and the factorial function (option 4).

To compute 5P3, for example, press 5

MATH and arrow over to PRB. Then press 2 to select option 2. Finally, press 3 ENTER .

Follow the same steps to find a combination selecting option 3 instead.

-5- EVALUATING RATIONAL EXPONENTS

To evaluate 243 5 simply

press 2 4 ^

( 3 / 5

) ENTER .

FINDING n th ROOTS

To evaluate 5 86 simply press 5 MATH 5

8 6 ENTER .

ABSOLUTE VALUE

To evaluate 2(3) 10 using the graphing calculator, press MATH and arrow over to NUM. Select option 1 by either pressing 1 or ENTER . Type in the rest of the quantity remembering to end with ) . Press ENTER .

-5- TECHNICAL SUPPORT

If you are having trouble using your calculator, your teacher is a good source of information. Others in your class may also be able to help you. If you have a technical problem with your graphing calculator call 1-800-TI CARES. Texas Instruments has fantastic technical support and can take you step by step in solving your problems.