Math 12 Independent Study Unit Graphing Technology for Math 12

Math 12 – Independent Study Unit – Graphing Technology for Math 12

1. Assignment Component (4 points) Due: December 1st, 2010

While you will have some class time to work on your independent study, this unit is intended to be completed outside of regular class time. You are expected to:  read the objectives and notes for meaning. These are important ideas – think about what you are reading.  look carefully at the examples. They will help you to understand the concepts and what is expected as you complete the written component.  complete all exercises using a graphing calculator.  make use of the available class time and extra help periods to use school graphing calculators; if needed, download the Virtual TI from class website to use on home computer.  include required aspects of a solution using technology, such as neat and accurate sketches of graphs and other (pay close attention to examples).  check answers to selected questions that will be provided.  follow the directions that accompany each question. Graphs must be neatly and accurately drawn. Clearly indicate question numbers and keep your work organized. If you do not submit the assignment before corrected assignments are returned to other students, you will get a zero on this component.

2. Formal Assessment Component (6 points) Date: December 1st, 2010

 The assessment will be based on the concepts and skills developed in the unit. The questions will address the objectives of each section.  If your assignment is completely done (and submitted by the due date) and you score 80% or higher on the quiz, you will automatically be given a perfect score for the Assignment Component.  If you score less than 80% on the quiz, your assignment will be graded according to the marking scheme. Math 12 ISU Page 2

General TI-83+ Tips

Reset: It is a good idea to reset your graphing calculator to clear lists, functions, and prior graphing settings. To reset: 2nd + (“Memory”) … 7: Reset … 1: All RAM … 2: Reset

Contrast: Sometimes the screen appears very dark or dim. To adjust the screen contrast: 2nd … hold up or down arrow as needed

Diagnostic On: 2 2 When using the regression feature, it is useful to see the r value. The closer r is to one, the 2 nd better the equation fits the data. If r  1 , it is a perfect fit. To turn diagnostic on: 2 0 (“Catalog”) … scroll down to find “Diagnostic On” … ENTER to select … then ENTER again to execute

PART A: Graphing Functions

Objectives:  identifying types of functions; entering equations into TI-83+  adjusting window appropriately

Notes: There are three main types of functions that you will need to be able to recognize: Linear Functions, Exponential Functions, and Quadratic Functions. Below are examples for each type.

x 2 y  2x  3 y  2(3)  2 y  x  3x  4

You should know how to identify these functions from the equations and their graphs.

Before you can use the calculator to examine a function, you must remember to enter the equation into Y=. Once you enter it, you can access information about the function from the graph or table.

When graphing functions, be sure the window shows the characteristic shape for that function. In addition, you should always show the origin to help give some scale to your graph. Because every Math 12 ISU Page 3 graphing solution you give needs to have a sketch of the graph, setting an appropriate window is extremely important. You can adjust the window manually, but this can be time consuming and sometimes very difficult.

The calculator has built in zoom functions to quickly change the window to suit the graph. The following zoom tools are useful and can be found using ZOOM:  1: ZBox – allows you to zoom in to a user-defined region of the screen by enabling you to create a rectangular box around a selected area of your graph; very useful for making key points much clearer  3: Zoom Out – must click ENTER on the graph to activate; widens the scale significantly to see more of the function; very useful  6: ZStandard – sets the window back to default settings; very important to do for every new function  9: ZoomStat- sets the window to fit plotted points from the lists; very useful when entering data instead of an equation  0: ZoomFit – automatically adjusts the window so most of the graph is visible; very useful, but window will often need further adjustment

Tips for setting best graph window:  always start with 6: ZStandard; if you still can’t see enough of the function …  0: ZoomFit; if you still can’t see enough of the function …  3: Zoom Out to see more OR 1: ZBox to see less; if you still can’t see enough of the function …  you may need to Zoom Out and/or ZBox multiple times  this will always work … if you’re not seeing the graph start back at ZStandard  if necessary, adjust window manually using WINDOW to show the origin

Examples: Try the following. You should go through the steps to be sure you can view the graph as shown.

  x 1. y 3x 40 2. y  18(0.99)  8

(Zoom 6, Zoom 0, Zoom 3) (Zoom 6, Zoom 0, Zoom 3-three times, Zoom 1) Math 12 ISU Page 4

2 2 3. y  57.5x  2  6 4. y  0.02x  4.8x  15

(Zoom 6, Zoom 1) (Zoom 6, Zoom 0, Zoom3-twice, Zoom 1)

► Exercise A: Graph each of the following, then draw an neat and accurate sketch of what you see on your screen. Be sure to zoom appropriately: the key shape of the graph needs to be clear and the origin must be visible. 2 x 2 1. y  2x  53x  250 2. y  38(0.85)  15 3. y  28x  135x  9 x y   253 x  26 2 4. y  15(1.5)  8 5. 8 6. y  1.05(x  15)  23.5

PART B: Determining Key Points on Graphs

Objective:  locating axes intercepts and identifying maximum or minimum points using “Calc” and “Table”

Notes: Once you’ve entered your function into Y=, you can view points on the graph. While TRACE can give points, it is unlikely to give key points like intercepts. We will use two methods: “Table” and “Calc”.

Using “Table” (2nd GRAPH) allows you to scroll through the table of values at a set x-interval. You may be able to identify the y-intercept (where x=0) and any x-intercepts (where y=0). If you cannot see the values you need, you can adjust the table settings in “TblSet” (2nd WINDOW). If you use the table to identify a key point, your solution should include the part of table where you found your point: show at least one point above and one point below. Remember, not all key points will appear on the table. You may need to use “Calc”.

Using “Calc” (2nd TRACE) will have the calculator determine key points for you. The ones that are most often used are:  1: value – allows you to sub in any x-value and it gives the corresponding y-value  2: zero – gives x-intercepts  3: minimum – gives the lowest point (vertex) of a parabola  4: maximum – gives the highest point (vertex)of a parabola  5: intersect – shows where two functions cross (this will be used more later) Math 12 ISU Page 5

Examples: Try the following on your calculator to ensure you can follow the steps to get the same results.

2 1. Use “Table” to find the x-intercepts for y  x  x  20 .  2nd GRAPH ... Find points where y=0  Draw the parts of the table that you used ... this is part of your solution.

So the x-intercepts are x=-4 and x=5. They can also be written as coordinates: (-4,0) and (5,0)

2 2. Use “Calc” to find the x-intercepts for y  x  x  20 .  Adjust the zoom if necessary.  2nd TRACE ... 2: zero  “left bound?” means you needs to use the arrow keys to move the cursor until it is left of the x-intercept you want to find first ... press ENTER  “right bound?” means you needs to use the arrow keys to move the cursor until it is right of the x-intercept you want to find first ... press ENTER  “guess?” means you can move the cursor to the approximate location of the x-intercept ... not necessary though ... press ENTER  Repeat this process for the other x-intercept.  Sketch your graph with the x-intercepts clearly marked.

So the x-intercepts are x=-4 and x=5. Math 12 ISU Page 6

2 3. Use “Calc” to find the vertex for y  3.5x  28x  12 .  Adjust the zoom if necessary.  Since the parabola has a high point at the vertex, use “maximum”  2nd TRACE ... 4: maximum  “left bound?” means you needs to use the arrow keys to move the cursor until it is left of the vertex ... press ENTER  “right bound?” means you needs to use the arrow keys to move the cursor until it is right of the vertex ... press ENTER  “guess?” means you can move the cursor to the approximate location of the vertex ... not necessary though ... press ENTER  Sketch your graph with the vertex clearly marked.

So the vertex is (4, 44).

x 4. Use “Calc” to find the value of y  1250(0.75) 9 when x=22.87.  Adjust the zoom if necessary.  2nd TRACE ... 1: value  Enter the x-value ... 22.87 ... press ENTER  Sketch your graph with the key point clearly marked.

When x=22.87, y=601.76

► Exercise B: Use the graphing calculator for each of the following. You can choose to use “Calc” or “Table”. If you use “Calc”, you must show your graph with the point(s) clearly identified. If you use “Table”, you must show the part(s) of the table that shows the point(s). State your final answer very clearly. 1. Find the x-intercept for each function: 2x 2 c) y  0.5x  766 x a) y  6.5(2)  8 b) y  x  15x  63 d) y  25(0.964) 2  12 Math 12 ISU Page 7

2. Find the y-intercept for each function: x y   14 x  75 c) y  3(x  7)(x  24) a) y  160(5)  3.2 b) 55

3. Find the vertex for each quadratic function: 2 2     a) y  x  22 b) y  5.5x  8x  144 c) y 1.5(x 2)(x 50)

PART C: Using Regression to Determine Equations of Functions

Objective:  finding the equation of a function from a table of values or a set of points using the regression functions on the graphing calculator

Notes: If you have data that represents a function and you want the equation, you can input the data and the calculator will determine the equation for you.

ENTERING DATA You must start by entering your data (x- and y-values) into L1 and L2; x-values go into L1 and y-values go into L2. Use STAT ... 1: Edit to access the lists. Your lists must be the same length (i.e. you can’t have a different number of x- and y-values). To delete a single number in a list, use DEL; to clear an entire list, go to the top heading on the list and press CLEAR ... ENTER.

Before getting the equation, you need to know what kind of function the points represent. We will be looking at linear, quadratic, and exponential functions only.

VIEWING POINTS Once your data is entered, turn on the plotting feature using “Stat Plot” (2nd Y=) ... then ENTER (to access plot #1) ... and ENTER again to turn this plot “ON”. Now, to view the points, you need to adjust the zoom. There is a zoom for just this purpose: ZOOM ... 9: ZoomStat will adjust the window perfectly to show the points you’ve entered.

You should be able to predict what kind of data you are dealing with.

DETERMINING THE EQUATION To access the regression functions, go to STAT ... “Calc”. The three types of regression that we will be using are:  4: LinReg (for linear data)  5: QuadReg (for quadratic data)  0: ExpReg (for exponential data) Once you select the type of regression, press ENTER to show the results. For most data that we use, your values will typically be integers or reasonable decimal/fraction numbers. Math 12 ISU Page 8

If you have turned on the diagnostics (see the tips at the very beginning of the unit notes), you will 2 likely see r  1 which means the equation fits the data perfectly. If it is less than one, you should 2 try a different type of regression to get a better fit. With “real” data, you may never see r  1 , but you would aim to have this value as close to one as possible.

If you like, enter your new equation into Y=, then press GRAPH. You should see the line/curve passing through all of your data points.

** NOTE: The only work you need to show is the type of regression and then your equation. If you use your equations to get other information afterwards, you must also show your graph (or table) with points indicated.

Determine the equation that best fits each data set: 1. STEP #1: View Points x y -2 30 -1 16 0 8 1 6 2 10 * Data appears to be quadratic STEP #2: STAT ... “Calc” ... 5: QuadReg Check by entering in equation ...

2 The equation is y  3x  5x  8 2 STEP #3: Work shown: QuadReg → y  3x  5x  8

Math 12 ISU Page 9

2. STEP #1: View Points (-3, 11), (1, 1), (7, -14), (9, -19)

* Data appears to be linear STEP #2: STAT ... “Calc” ... 4: LinReg Check by entering in equation ...

The equation is y  2.5x  3.5 STEP #3: Work shown: LinReg → y  2.5x  3.5

3. STEP #1: View Points x y -2 2.56 0 16 1 40 3 250 5 1562.5 * Data appears to be exponential STEP #2: STAT ... “Calc” ... 0: ExpReg Check by entering in equation ...

x The equation is y  16(2.5) x STEP #3: Work shown: ExpReg → y  16(2.5)

Once you have the equation you can use “Table” or “Calc” to determine key points. If you are using the graph to determine points, you must draw the graph and show the key points. Math 12 ISU Page 10

► Exercise C: Use the graphing calculator for each of the following. Show your work properly.

Find the equation that best fits each data set, then determine any x- and y-intercepts. If the function is quadratic, also determine the coordinates of the vertex. (you will need to show these key points on a graph or table) 1. (2, 241), (5, 220), (10, 185) 2. 3. x y x y -2 6.25 -7 460 -1 2.5 -1 850 0 1 3 910 1 0.4 4 900 2 0.16 6 850 4. 5. (-8, 98304), (-4, 6144), 6. x y (-3, 3072), (-1, 768) x y -15 491 1 -150 -11 2735.16 5 -490 -7 4978.04 6 -575 -5 6099 8 -745

PART D: Solving Equations

Objectives:  solving single-variable equations using points of intersection  solving single-variable equations using zeros

Notes: You can solve any type of equation using graphs on the TI-83+. You must remember that this method is not an “algebraic” method. If you are asked to solve algebraically or if you need to get exact answers (non-rounded) you need to show your solving steps. Otherwise, solving by graphing is a perfectly acceptable method any time including on tests and your exam as long as you show proper work.

SOLVING USING POINTS OF INTERSECTION If you are given an equation to solve, the idea is to find the value of the variable that makes both sides of the equation equal. If you graphed each side of the equation as a separate function, you could solve the equation by looking at where the curves/lines crossed each other.

You need to enter each side of the equation into Y=, then adjust the zoom so you can clearly see the general shape of the curves, the origin, and where the two functions intersect. Use “Calc” (2nd TRACE) ... 5:Intersect to determine the point of intersection (i.e. where the functions are equal). You will be asked for the “First curve?” (your cursor will automatically go to your first function) press ENTER. Then you will be asked for the “Second curve?” (your cursor will automatically go to your second function) press ENTER. Lastly you will be asked to “Guess?” – move your cursor close to the Math 12 ISU Page 11 point you want (this is really only necessary if there are multiple points of intersection), then press ENTER.

Since the unknown from your original equation is the “x” on your graph, the x-value of the point of intersection is the solution to the equation. Clearly mark the point of intersection on your graph and indicate the x-value to solve the equation.

Solve: x5  1  Enter each side of the equation into Y=; you must also write down these 1. 22x1     8  equations as part of your solution.

y  22x1   1 x 5 y     8 

Adjust the zoom, then use “Calc” ... 5:Intersect. Sketch your graph with the intersection clearly marked.

Finish by giving the solution to the original equation: x  2.8

2 2 2. 3p  5p  p  9 Enter each side of the equation into Y=; you must also write down these equations as part of your solution.

2 y  3x  5x 2 y  x  9

Adjust the zoom, then use “Calc” ... 5:Intersect. Sketch your graph with the intersection clearly marked. Since there are two points of intersection, you will need to go through these steps twice. You don’t need to draw two graphs, but you do need to indicate both points on the graph. Math 12 ISU Page 12

Finish by giving the solutions to the original equation: p  1 ; p  2.25

SOLVING USING ZEROS If you are given an equation to solve, you could bring all terms to one side so the other side equals zero. If you then replace the “0” with “y” you can graph the function, then you can solve the equation by looking at where the curve/line crosses the x-axis since this is where y=0.

You need to bring all terms to one side of the equals sign (you don’t need to simplify) and enter into Y=. Adjust the zoom so you can clearly see the function (especially any x-intercepts). ** NOTE: It is possible that your graph looks very unfamiliar because you may be combining functions. Use “Calc” (2nd TRACE) ... 2:Zero to determine all x-intercepts (i.e. where the function equals zero). Follow the process outlined in Part B.

Clearly mark the x-intercept(s) on your graph and indicate the x-value to solve the equation.

Solve: x5 x5  1   1  1. 22x1    Rearrange equation: 22x1     0  8   8  Enter the equation into Y=; you must also write this down as part of your solution.

x5  1  y  22x1     8 

Adjust the zoom, then use “Calc” ... 2:Zero. Sketch your graph with the x-intercept clearly marked.

Finish by giving the solution to the original equation: x  2.8 Math 12 ISU Page 13

2 2 2 2 2. 3p  5p  p  9 Rearrange equation: 3p  5p  p  9  0 Enter the equation into Y=; you must also write this down as part of your solution. 2 2 y  3x  5x  x  9 OR 2 y  4x  5x  9

Adjust the zoom, then use “Calc” ... 2:Zero. Sketch your graph with the x-intercepts clearly marked. Since there are two points of intersection, you will need to go through these steps twice. You don’t need to draw two graphs, but you do need to indicate both points on the graph.

Finish by giving the solutions to the original equation: p  1 ; p  2.25

** NOTE: Every time you carry out a “Calc” operation, the x and y values are saved as “X” and “Y”. You can access them on the main calculator screen and convert answers to fractions if you like. 

► Exercise D: Use the graphing calculator for each of the following. Show your work properly.

Solve for the unknown in each: 2x 2  1  5 2.  15n  60  10.25n  16 3. x  2x  48  0 1.    2  8 

2 (x  5)  2  4 (3x  1) 2 b2 b 4. 3 7 5.  3x  4x  15  0 6. 3  7

2 1   7. 3  2c  16c  2c 2 9. log x 2log3 log 4 8. x  81 8

2 x 4 2 3 2 1 x  7  5  3  x   x  a  8a  14 (a  4)  32 10. 2 11. 5   45   3  0 12. 3    

x x 3 x 1 2 h 2 13. 2  2  2  14 14. 84  3h  2  1 15. log 3x  x  5x  12 2 Math 12 ISU Page 14

► Review Exercise: Use the graphing calculator for each of the following. Show your work properly.

1. Label all key points (intercepts and vertex) on the graph of each function: 2    2 a) y  0.02x  38x  800 b) y 13x 255 c) y  x  16x  55

x 2 x d)     1  f) y  3000.87  28.5 y 5 18 e) y 4x 2 x 16

y  8 (x  44)  80 x 4 2 g) 29 h) y  1.752  12 i) y  15x  260x  500

2. Determine the equation of the function that best fits the given data set: a) b) c) x y x y x y -2 -18 -4 187500 -4 74 -1 -15 -3 37500 -2 38 0 -12 -2 7500 0 18 1 -9 -1 1500 2 14 2 -6 0 300 d) (-2, 0.0243), (-1, 0.081), e) (3, 13), (5, 44), (7, 75) f) (14, 218), (15, 261), (16, 308) (0, 0.27), (1, 0.9)

3. Determine the vertex of the parabola that passes through the given points: a) (4, 16), (6, 42), (8, 80) b) (-6, -192), (-5, -165), (4, -102) c) (12, 724), (13, 749), (14, 776) d) (9, 66), (10, 35), (11, -2) e) (-3, 76.6), (-2, 36.8), (0, 1.6) f) (5, -375), (6, -460), (7, -565)

4. Solve each equation: 2 x 6 a)  3x  9  4x  16 b) 2e  13e  7  0 c) 2  64

2 2 2(x  5)  2 (x  14)  0 3p 2 d) 3t  8  t  4t e) 3 f) 0.2  p  4p

    m1 m 3 2 g) 3(2x 1) 6 7(6 x) h) 2  3  85 i) q  6q  q  5q  2