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.

[pic]

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.

ORDER OF OPERATIONS

You must understand and know the order of operations in order to evaluate an expression correctly using your calculator. To evaluate the expression [pic], you must 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 [pic] |[pic] |

| | |

|) [pic] ( 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 |[pic] |

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

GRAPHING A LINE

In order to graph a line, you must solve it for [pic]. To graph the line [pic], you must first rewrite it as [pic] before you enter it into the calculator.

| | |

|Y= ( (-) 3 [pic] 4 ) | |

|x,T,[pic],n |[pic] |

| | |

|+ 2 | |

| | |

|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 [pic] you would see:

|[pic] | [pic] |[pic] |

| | | |

| |Go to WINDOW to change what you see on the screen. |Press GRAPH . Now you can see the graph and both of|

| |Since you know that the y-intercept is 27, you need |its intercepts. |

| |only to adjust the Ymax entry to a number larger than| |

| |27. | |

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 [pic] from the previous example,

let’s find the x-intercept.

|[pic] |[pic] |[pic] |

| | | |

|Start by pressing 2ND |The calculator will ask you for a left and right |0 ENTER |

| |bound. This simply means that you should enter a | |

|TRACE 2 . Note that a “zero” is |number you think is to the left of the x-intercept and| |

|equivalent to an “x-intercept” in mathematics. |one that is to the right. Just type in these numbers | |

| |as you are given the prompts. Here, you simply press | |

| | | |

| |( - ) 5 ENTER . | |

|[pic] |[pic] | |

| | | |

|When the calculator asks you to guess, simply take|Finally, press ENTER to view the answer | |

|your |displayed at the bottom of the screen. The | |

|cursor close to the point of intersection. |x-intercept is at x = -1.8. | |

SCATTER 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:

|[pic] |1 |2 |

|[pic] | [pic] |NOTE: You may need to adjust the viewing |

| | |window to see a better picture of your scatter |

|Use the cursor keys to navigate through the |Finally, press GRAPH . |plot. Also, if there are other graphs on your |

|different selections on the screen. Press | |viewing screen, press Y= and clear any|

|ENTER to select an option. | |equations you may have. However, if you need |

| | |the equations in the Y= screen for a later |

| | |date, you can simply de-activate them by |

| | |placing the cursor on the equal sign and |

| | |pressing ENTER . This allows you to keep the |

| | |equations, but they won’t be graphed. |

|[pic] | [pic] |[pic] |

| | | |

|To find the equation of the line of best fit, press |Then press 2ND 1 |Notice that the values for a and b are given. |

|STAT and move the cursor over to CALC and select |, 2ND 2 , |The value of r is the correlation coefficient. |

|option 4. | |The closer the value is to 1, the closer the |

| |VARS and move the cursor over to the Y-VARS option |plotted points are to the line of best fit. If|

| |and select option 1. Finally, select option 1 again |you do not see the value for r, do the |

| |for Y1. What this allows you to do is to get the |following: |

| |line of best fit and have it pasted to your Y= | |

| |screen. Press ENTER to get the line of best fit. |2ND 0 [pic] and scroll down|

| |(If you press Y= , you will notice that the |until you find DiagnosticON. Press ENTER |

| |regression equation you just calculated is in Y1.) |twice and repeat the steps in the previous |

| | |column. |

|[pic] | | |

| | |NOTE: If you return to the Y= screen, you will notice that PLOT1 is highlighted at|

|Press the GRAPH key and you will see the scatter | |the very top. It is important to turn all plots off in order to be able to graph |

|plot graphed along with the line of best fit. | |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. This can also be |

| | |done by pressing 2ND Y= and going into each of the plots to turn them off|

| | |in the same way you turned them on. If you don’t turn off the plots, 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 order to find the point of intersection for the linear system[pic], you would do the following:

|[pic] |[pic] |[pic] |

| | | |

|Enter the equations into the calculator. |Graph it on the standard viewing window. Since |Notice that the point of intersection is in |

| |you can’t see the point of intersection, adjust |quadrant IV. Use this fact to help you choose |

| |the window. |the max/min values and graph the lines again. |

|[pic] |[pic] |[pic] |

| | | |

|Now that you can see the point of intersection, | |The calculator will ask you which curves (lines) |

|press | |you want to find the intersection of. The top of |

| | |the screen displays one of the equations we want. |

|2ND TRACE 5 |ENTER |Press ENTER . |

|[pic] |[pic] |[pic] |

| | | |

|The second line we want is displayed on the top |Here, you may just want to move your cursor to |You can now see that the point of intersection of|

|so press |the point of intersection although this is not |the linear system is approximately at |

|ENTER . |necessary. Press ENTER . |(3.5, -12.7) . |

ENTERING MATRICES

In order to enter the matrix [pic]into the calculator, follow the steps below:

|[pic] |[pic] |

| | |

|Press 2ND [pic] 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.

|[pic] |[pic] |[pic] |

| | | |

|Enter the equation into the calculator by going |Press 2ND TRACE 2 |One x-intercept is at about |

|to the Y= screen. Graph the equation in the |and follow the prompts as you did for finding the|x = -1.9. To find the other x-intercept, follow |

|standard viewing window and adjust the window as |x-intercept of a line. |the same steps as before. When the calculator |

|necessary. | |asks for a guess, simply move the cursor closer |

| | |to the other x-intercept you want to find. |

|[pic] | | |

| | | |

|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 .

|[pic] |[pic] |[pic] |

| | | |

|Once the graph appears on the screen, follow the |Because the vertex appears to be at about (0, |x = 2 appears to be to the right of the vertex so|

|left/right bound prompts as you did before. |-7), the choice of x = -2 for a left bound seems |it is a good choice for the right bound. |

| |appropriate. | |

|[pic] | | |

| | | |

|When the “Guess” prompt appears, simply press | |NOTE: In order to find the vertex of a parabola which is a maximum, follow the |

|ENTER . The vertex is at (-0.3, -7.3). | |previous steps, but select option 4 (maximum) in the CALCULATE screen. |

SOLVING POLYNOMIAL EQUATIONS

Suppose you are asked to solve the polynomial equation [pic]. 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:

|[pic] |[pic] |[pic] |

| | | |

|Enter the left side of the equation into Y1 of |Graph the equations. It may be necessary to | |

|the Y= screen. Enter the right side as |adjust the viewing window. |Press 2ND TRACE 5 . |

|Y2. | | |

|[pic] |[pic] |[pic] |

| | | |

|Answer the prompts as you did when solving a |The second point of intersection is at about |The last point of intersection is at about (0.9, |

|system of linear equations. Remember to move the|(-0.4, 10) |10) |

|cursor to the desired point of intersection each | | |

|time the prompt asks you to “guess.” The first | | |

|point of intersection is at about (-2.8, 10). | |From these three points of intersection, we can |

| | |determine that the solutions to the equation |

| | |[pic] 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).

|[pic] | | To compute 5P3, for example, press 5 |

| | |MATH and arrow over to PRB. Then press |

| | |2 to select option 2. Finally, press 3 |

| | |ENTER . |

| | |[pic] |

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

EVALUATING RATIONAL EXPONENTS

| To evaluate [pic]simply press 2 | | |

|4 ^ | | |

| | |[pic] |

|( 3 / 5 | | |

| | | |

|) ENTER . | | |

| | | |

FINDING nth ROOTS

| To evaluate [pic]simply press 5 | | |

|MATH 5 |[pic] |[pic] |

| | | |

|8 6 ENTER . | | |

| | | |

ABSOLUTE VALUE

| To evaluate [pic] using the graphing | | |

|calculator, press MATH and arrow over to NUM. |[pic] |[pic] |

|Select option 1 by either pressing | | |

|1 or ENTER . Type in the rest of the | | |

|quantity remembering to end with | | |

|) . Press ENTER . | | |

| | | |

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.

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