MAT1033C - 2014-2015 - Instructor

7/17/2019 MAT1033C - 2014-2015 - Instructor

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Activity Topic Page

Graphing Calculator Scavenger Hunt!!! Calculator 1–3

Equation = Graph pictures Graphing/Solving Linear Equations 5–6

Pan Balance – Linear Expressions and Equations Linear Equations 7–8

Graph Your Motion Modeling Linear Functions 9–11

Linear Modeling with Polygons Linear Functions 13–14

Finding a Special Pattern with Functions! Functions 15–

17The Mysterious Bone Linear Functions 19–21

Where can I get the best car rental deal? Comparing Linear Equations 23–24

Yes, we do mix things in real life!!! Mixture Problems with two Variables 25–29

Pan Balance – Linear Inequalities Linear Inequalities 31–32

Can you give me the solution in 5 formats? Writing Linear Inequality Solutions 33–36

I remember it well!!! Exponents 37–38

Find the infamous polynomial!!! Factoring and Polynomials 39–40

Do you want me to Solve or find x – intercepts? Factoring and Polynomials 41–42

Polynomial Factoring BINGO Factoring and Polynomials 43

Discovering Quadratic Graph Secrets!!! Calculator 45–46

Different words – Same meaning Quadratic Transformations 47–49

Get the Ball Rolling Modeling Quadratic Functions 51–55

Could you pass this Test? Solving Quadratic Equations 55–57

How high and far can you hit the ball? Real Life Quadratic Functions 59–61

Pan Balance – Quadratics Quadratic Equations 63–64

Stop Stealing My Blueberries! Exponential Functions 65–66

The more you compound my account the better!!! Compound Interest 67–

69Better than Average!!! Rational Functions 71–72

Cost–Benefit Rational Functions 73–74

From a Distance Modeling Rational Functions 75–77

What makes you think this is Rational? Rational Expression Review 79–80

Distance to the Horizon Radical Functions 81–83

The Cycle of i !! Imaginary Numbers 85–87

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Instructor’s Key:

Graphing Calculator Basics (TI 84)

Graphing Calculator Scavenger Hunt

Turn the calculator

1. Calculate 7 5 2 by pressing:

a. What is the result?17

b. Why is the result NOT 24?Due to the order of operations;multiplication occurs before division

c. How could you change the expression toforce the calculator to give a result of 24? (7 5) 2

2. Calculate 229 by pressing:

(or alternatively )

a. What is the result?

841

b. What is 2

34 ?

1156

i. Is it equivalent to 234 ?

No;−

1156

ii. Why or why not?

Exponents must be simplified beforemultiplying the negative

c. What would you press to calculate 93 ?

What is the result?

3^9; 19683

3. Calculate 105625 by pressing:

[For newer operating system, press ]


b. Determine 1.72265625

1.3125

c. Now press

What happened?

Solution changed to fraction form (21/16)

4. Calculate the cube root of 17576, 3 17576 , bypressing:

[For newer operating system, press ]

a.

What is the result?26

b. Determine 3 1.728 . Write the result as adecimal number and as a fraction.

1.2,6

5

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The button that reads can be used to introducevariable expressions.

5. Press: to store 3 as thevalue for X . (the calculator should show 3 in the next line)

Press:


b. What happened?The calculator replaced x with 3, &simplified the expression

6. Now store 5 for x by pressing:

. Then, press:

a. What happened each time you pressed the

combination: ?

Brings back the previous inputted entries

Press again.

b. What is the result?

21

c. What happened?The calculator replaced x with 5, &simplified the expression

Note: Your calculator cannot simplify variable

expressions. If you type in 3 7 2 4 x x (which

simplifies as 5 3 x ), the calculator will recall thestored value for x and evaluate the expression.

7. Create a table of values for 7 3 y x startingwith 0 x and increasing by 1.

Steps:

Go to the Y= screen by pressing

Remove any previous equations by pressing .

Input the 7 3 x

using appropriate key presses. Go to the TBLSET screen by pressing

Make sure the TblStart = line reads 0 and the ∆Tbl = reads 1. Also, the Indpnt: line should read Auto andthe Depend: line should read Auto

Press . You should now see a table ofvalues for the equation.

a. What is the y value when x = 4? 25

b.

What happens each time the x valueincreases by 1? The y value increases by 7

8. Now, use similar techniques to create a table ofvalues for 4 3 y x a. What is the y value when x = 10?

−37

b. What happens each time the x value

increases by 1? The y value decreases by 4

9. Looking back for the PATTERN! a. What pattern do you see between the

equation and what happens each time weincrease the x value by 1?

The solution changes by the value of thecoefficient

b. Consider the equation:1

52

y x . Based

on your answer to 9a), what would youexpect to see on the table each time weincrease the x value by 1? The y value increases by

1

2

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10. Create a graph for 7 3 y x Steps:

Go to the Y= screen by pressing

Input the 7 3 x using appropriate key presses.

Go to the Window screen by pressing

Input the following settings: Xmin=−10, Xmax=10,Xscl=1, Ymin=−10, Ymax=10, Yscl=1

Press . You should see a figure similar to this:

a. Graph1

52

y x using the same window

settings. Sketch your graph on the provided

grid.

b. Graph2

4

4

x y

x

using the same window.

If you graph it correctly, your graph shouldappear to be in three “pieces.” If yourgraph does not appear to have three“pieces,” you did not use parenthesesappropriately.Sketch your graph on the provided grid:

11. Mixed practice; using your calculator:

a. Evaluate: 4

139876 87 2 45

455

b. If 5 x , evaluate 3 24 7 2 9 x x x 694

c.

Create a table of values for 4(2)

x

y

,starting at 0 and increasing by 1.

i. What is the value for y when 0 x ? 4

ii. What happens each time the x −valueincreases by 1? The y value doubles

d. The graphs of

214 and 2

3 y x y x will

intersect twice. In which two quadrants willthey intersect? III, IV

Sketch your graph.

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Instructor’s Key:

Graphing / Solving basic linear equations

Equation values = Graph picturesFill in the blanks.

1.

A __ __ __ __ __ on a graph represents the x and y value in an equation that creates a truestatement.

2. Where do you start counting from to find a graphpoint? __ __ __ __ __ __

3. (5, −3) The 5 of this ordered pair refers to whaton a graph? __ − coordinate

4. (5, −3) The −3 of this ordered pair refers to what

on a graph? __ − coordinate

5. The point (4, 6) is located in which quadrant? _____

6. The point (−5, 7) is located in which quadrant? _____

7. The ___ −coordinate of a point is usuallyassociated with the horizontal axis.

8. The ___ −coordinate of a point is usuallyassociated with the vertical axis.

9. Define slope using words:Rise over run

Define slope using symbols: y

x

10. Write the Slope –Intercept form of a linearequation:

y mx b

11. Write the Point –Slope form of a linear equation:

1 1( ) y y m x x

12.

Plot the following points on the graph paperand label as indicated. Then connect eachpoint to the next one in alphabetical order (alsoconnect H to A). (use line segments to connect thepoints) Graph paper is on the next page!

A: (5, 8)B: (9, 4)C: (9, −2)D: (5, −6)E: (−3, −6)

F: (−7, −2)G: (−7, 4)H: (−3, 8)

13. Plot the following points on the graph paperand label as indicated. Then connect eachpoint to the next one in alphabetical order. (useline segments to connect the points) Graph paper is on the next page!

J: (−5, −1)K: (−2, −3)

L: (4, −3)M: (7, −1)

14. What is the slope of JK? 2 / 3 Equation of JK? 2 3 13 3 y x

15. What is the slope of LM? 2 3

Equation of LM? 2 3 17 3 y x

16. What is the slope of KL? 0 Equation of KL? 3 y

17. What is the slope of AB? −1 Equation of AB? 13 y x

18. What is the slope of BC? undefined Equation of BC? 9 x

p o i n t

o r i g i n

x

y

I

II

x

y

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Solve the following linear equations:

19. 2 4 10 x 3 x

25. 1 46

x

18 x

20. 5 1 29 x 6 x

26. 2 8

3 9 x

4

3 x

21. 2

63

x

9 x 27. 3 2

25

x 4 x

22. 3 1

24 2

x 2 x 28. 4 7 9 5 x x

2

5 x

23. 2 5 4 x x 9 x

29. 2

5 2 33

x x

1

3 x

24. 4 12 2 x x 2 x

30. 1 2 1 1

6 3 2 3 x x 5 x

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Instructor’s Key:

Pan Balance – Linear Expressions & Equations

Using your favorite internet browser, go to http://illuminations.nctm.org/Activity.aspx?id=3529 . Alternatively,perform a search for the following terms: illuminations pan balance expressions . It should be the first option.

Instructions

Place an algebraic expression in each of the red and blue pans. These expressions may or may not include thevariable x . Enter a value for x , or adjust the value of x by moving the slider.As the value of x changes, the results will be graphed. Use the Zoom In and Zoom Out buttons, or adjust thevalues for the x and y axes with the sliders, to change the portion of the graph that is displayed. The ResetBalance button removes the expressions from the pans and clears the graph.

Try it out:

Use the Pan Balance to see the relationshipbetween 2 10 x and 2 x

1. Type 2 10 x in the left pan (red)

2. Type 2 x in the right pan (blue)

3. You should see two lines

4.

Let 5 x (type 5 in for the x value). What isthe corresponding output for each expression?Which pan is heavier?

0;3

5. Let 1 x (type 1 in for the x value). What isthe corresponding output for each expression?Which pan is heavier?

12;3

6. By using the slider, or trial and error, find thevalue of x which balances the pans. Record thisvalue.

4 x

7. Look at the graph, you should see either twolines intersecting at a single point, two parallellines, or a single line.

8. Using symbolic methods(addition/subtraction/multiplication/division toisolate the variable), solve the equation2 10 2 x x . Record your work and thesolution.

9. Compare your answers for exercise 6 and 7.Anything interesting?

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Using a similar process, solve each of the following equations. You should record on separate paper:

a.

The expressions you used for the left and right pans.b.

The value(s) of x that balances the pan, estimating if necessary. (You may need to adjust the settings on thegraph to find some of the values.

c.

The graphical situation that you see: two lines intersecting at a single point, two parallel lines, or a single line.What does that tell you about the type of solution to the equation?

d.

The exact solution that you find using symbolic methods. Show your work for this part!

Appropriate work for the sample exercise:

A. 8 5 3 x

1 x

B. 3 2 5 x x 7

2 x

C.

13 2 22 3 x x

2 x

D. 6(2 8) 4(3 6) x x

No solution

E. 2 1 17

27 2 2

x x x

7 x

F. 3(6 4 ) 2( 6 9) x x

Infinite solutions

G.

1 5 47 ( 2)

2 6 3 x x x

No solution

H. 0.3( 15) 0.4( 25) 25 x x

15 x

I. 1.3 2(0.6 4) 3 0.5( 16) x x x x

Infinite solutions

J. 1 (3 1) 5 x x 5

2 x

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Instructor’s Key:

Graphing Your Motion (©2000 Vernier Software & Technology/©1997 Texas Instruments Inc.)

Graphs made using a CBR 2™motion detector can be used to study motion. In this experiment, you will use a CBR2™motion detector to make graphs of your own motion.

ObjectivesIn this experiment, you will:

Use a motion detector to measure distance and velocity Produce graphs of your motion Analyze the graphs you produce

Data collection: Distance vs. Time Graphs

1. Place a CBR 2™motion detector to a tabletop facing an area free of furniture and other objects. TheCBR 2™motion detector should be at a height of about 15 centimeters above your waist level.

walk back and forth infront of the CBR 2™

motion detector

2. Use short strips of masking tape on thefloor to mark the 1 m, 2 m, 3 m, 4 mdistances from the CBR 2™motiondetector.

3. Connect the CBR 2™motion detector tothe calculator using an appropriate cable.

4.

On the calculator, press

A and selectEasyData to launch the app. (Note: EasyDatawill launch automatically if the CBR 2™motiondetector is connected to a TI 84 Plus.)

5. To set up the calculator for data collection:

a. Select Setup (press @ ) to openthe Setup menu.

b. Press 2 to select 2: Time Graph to openthe Time Graph Settings screen.

c. Select Edit (press #) to open theSample Interval dialog window.

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d. Enter 0.1 to set the time betweensamples to 1/10 second.

e. Select Next (press #) to advanceto the Number of Samples dialogwindow.

f. Enter 50 to set the number of samplesto collect. The experiment length willbe 5 seconds (number of samplesmultiplied by the sample interval).

g. Select Next (press #) to display asummary of the new settings.

h. Select OK (press %) to return to the

main screen.

6. Explore making distance vs. time graphs.

a. Stand at the 0.5m mark, facing away fromthe CBR 2™ motion detector.

b. Signal your group member to select Start(press @)

c. Slowly walk to the 4.0m mark and stop.

d. When data collection ends, a graph plot isdisplayed.

e. Sketch your graph on the empty graphprovided. (label the axes and scalinginformation on the graph)

f. Use > and < to move along the graph.Select two points on the graph anddetermine the slope from the x and y

coordinates.

1 x 0 1

y 0.954

2 x 2.101 2

y 1.798

Calculate the slope: m = 0.402

g. The y−intercept is the y−value where the x−value is equal to zero. Use the arrow keysto move to the y−intercept of this line andrecord the value below.

y−intercept: b = 0.954

h. Use the slope and the y−intercept to writethe equation of the line using the slopeintercept form. ( y mx b )

0.402 0.954 y x

7. Check to see if this equation matches the datacollected by the CBR 2™motion detector. Press$

then % to exit out of the EasyData

app. Then press !. Enter your equation fromthe previous step in one of the functionregisters. Press%. Does your equationmatch with the data? If not, check the values ofthe slope and the y intercept. If necessary,make adjustments and record your new equationbelow.Answers will vary

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8. What does the slope and the y−interceptrepresent in the graph? (Hint: Look at the units.)

The slope represents the speed or velocity. They intercept represents the starting position.

9. Write the equation for a person who starts 1meter from CBR 2™and walks away at a speed

of 1 meter per second for 5 seconds. Sketch thegraph of this motion, Include scale markers onyour axes. Be sure to correctly label the y

intercept and use the correct slope for thiswalker.

y = 1 x

10. How would your motion graphs differ if thewalker moved towards the CBR 2™with a

constant speed?

The graph would show a negative slope.

11. Repeat the above activity for a person walkingtowards the CBR 2™at a constant speed. Thestarting point should be at least 4 meters fromthe CBR 2™. Record your graph in the spaceprovided. Answers will vary (#11−14)

12. Select two points on the graph which are notclose together. Record the values below.

1 x 1.05 1

y 3.096

2 x 2.25 2

y 2.191

13. Calculate the slope, y intercept and use them to

write the equation of the line in slope interceptform. ( y mx b )

0.754 3.888 y x

14. Check to see if this equation matches the datacollected by the CBR 2™. (If you needinstructions, repeat those in question 7.) Doesyour equation match with the data? If not,check the values of the slope and the

y−intercept. If necessary, make adjustmentsand record your new equation below.

15. Describe the characteristics of any equation ofmotion for a person moving at a constant speedaway from the CBR 2™.Any equation of a person moving away from theCBR™at a constant speed is linear with positiveslope.

How does this equation differ if the person ismoving at a constant speed towards the CBR2™?The equation for the motion of a person walkingtowards the CBR™at a constant rate is linearwith a negative slope.

16. Velocity differs from speed in that it indicatesdirection. If the slope of the Distance Time

graph is positive, the velocity is positive. If theslope (as in the above example) is negative, thevelocity is negative. What does it mean whenthe CBR 2™indicates that a person is movingwith a negative velocity?

A person moving with negative velocity ismoving towards the CBR™.

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Instructor’s Key:

Linear Modeling with Polygons

Using your favorite internet browser, go to http://illuminations.nctm.org/ActivityDetail.aspx?ID=9 Alternatively,

perform a search for the following terms: illuminations angle sums . The site should be the first option.

You should be brought to a page displaying atriangle. Using your mouse, click and drag one ofthe vertices of the triangle. You should note somevalues changing on the right hand side of thescreen. You should also note something thatremains constant while you drag one of thevertices.

1. What changes?

The angle measure changes.

2. What remains constant?

The sum of the angles = 180°

Click on the quadrilateral box near the bottom of thescreen and repeat the experiment that you did for thetriangle.

3. What changes?

The angle measure changes.

4. What remains constant?

The sum of the angles = 180°

5. What happens if you drag a vertex over anotherside?

The angle/measure box disappears.

Continue this procedure with the remaining availableshapes (pentagon, hexagon, heptagon, octagon).

6.

What remains constant for each of the shapes?

The sum of the angle measures.

Is There A Pattern? Let’s go back and record some important information.For each of the six shapes, note, in the table, thenumber of sides (n) and the corresponding data for thepart that remains constant (S).

n S

3 180

4 360

5 540

6 720

7 900

8 1080

7. The table should describe a linear pattern.Explain why this is true. Use a completesentence.

For every increase in the number of sides by 1,there is an 180° increase for the sum of theangle measures.

8.

Determine the slope of the linear pattern.

180m

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9. Interpret the slope as a rate of change. Write acomplete sentence. Be sure to include unitsand the word “per.”

The sum of the angle measures increase by180° per increase in the number of sides by 1.

10. Determine the slope intercept form of a linearfunction that models this linear pattern. Call it

S n .

( ) 180 360S n n

11.

If you have answered part 9 correctly, youshould be able to find that 9 1260S .

Verify.

(9) 180(9) 360

1620 360

1260

S

12.

A dodecagon is a polygon with 12 sides.Evaluate 12S

(12) 1800S

13. What does your answer to exercise 12 tell

you? Write a complete sentence.

The sum of the angle measure for a dodecagonis 1800°

14. What is a reasonable domain for S n ?

Explain!

[3,4,5,6,....)

In order to create a polygon, you would need tostart with at least 3 sides increasing to aninfinite number; using only natural numbers.

15. Evaluate 17.5S . Is your result sensible?

Why or why not?

(17.5) 2790S

While the sum of the angle measures is

sensible, it would be difficult to create a

polygon with 17.5 sides.

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Instructor’s Key:

Functions

Finding a Special Pattern with Functions

Each exercise has three parts. Parts 1 & 2 of each exercise ask you to evaluate a function following the given

rule. The rule may be stated verbally, symbolically, graphically or numerically. Part 3 of each exercise asks youto use the answers you found in parts 1 & 2 to determine some other number. Locate your answer to part 3 in thenumber search and highlight (or circle) the numbers. After you have completed all the exercises, the numbersremaining in the number search will form a special pattern.

1. 3( ) 2 5 A x x

Part 1.: Evaluate (4) A 133

Part 2.: Evaluate (2) A 21

Part 3.: Determine (4) (2) A A 2793

2.

( ) B x is the function that multiples the input by 100 and then subtracts 5.

Part 1.: Evaluate (9) B 895

Part 2.: Evaluate (10.34) B 1029

Part 3.: Determine (9) (10.34) B B 1924

3. C x is the function defined by the following graph. The graph window is: [ 10,10,1] by [ 10,10,1]

Part 1.: Evaluate (2)C 3

Part 2.: Evaluate ( 1)C 6

Part 3.: Determine 4 2

2 1C C

2916

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4. ( ) D x is the function defined by the following table of values: x −3 −2 −1 0 1 2 3 4 5( ) D x −1045 2032 −2045 5204 −9612 6457 −4576 3578 −7412

Part 1.: Evaluate (0) D 5204

Part 2.: Evaluate (5) D 7412

Part 3: Determine (0) (5) D D 12616

5. ( ) 4 33 E x x

Part 1: Evaluate (22) E 11

Part 1: Evaluate (82) E 19

Part 3: Determine E(22) 8(82) E

6859

6. ( )F x is the function that determines the absolute value of the cube of the input

Part 1.: Evaluate (5)F 125

Part 2.: Evaluate ( 7)F 343

Part 3: Determine (5) ( 7)F F 42875

7. G x is the function defined by the following graph. The graph window is: [ 3,5,1] by [ 60,80,10]

Part 1.: Evaluate ( 1)G 20

Part 2.: Evaluate (2)G 10

Part 3.: Determine

3 2

1 2 4 1

10

G G G

798

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8. ( ) H x is the function defined by the following table of values:

x −30 −25 −20 −15 −10 −5 0 5 10( ) H x 3 6 12 24 48 96 192 384 768

Part 1.: Evaluate ( 5) H

96

Part 1.: Evaluate (10) H

768 Part 3: Determine 7 10 3 5 H H

5664

9. Now, find your answers to the “Part 3” exercises in the number search. They may be horizontally,vertically or diagonally, and may be backwards. Highlight or circle the numbers.

4 2 8 7 5 3

8 9 7 1 1 2

6 8 5 9 4 9

1 5 2 9 3 1

2 4 6 6 5 6

6 1 2 6 1 6

Once you have found all the numbers, list the remaining digits as they are listed, from left to right, top tobottom. 31415926

10. There is something special about the remaining digits. What is it? [Hint: put a decimal point in after thefirst listed digit!]

3.1415926

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Instructor’s Information Sheet:

Linear equations

The Mysterious BoneStory

A bone has been found in the woods near Valencia College East Campus. It is a humerus bone, which is the bonethat goes from the elbow to the shoulder. Investigators can use the length of the humerus bone, which is 12inches, to determine the height of the person, but they’ve forgotten the formula they need to use. They recallonly that it was a linear equation, ( ) H x mx b , where x is the length of the humerus bone and H ( x) is the

height, with both measurements in inches. Can you help them?

Part 1

Find a linear equation that models a person’s height in inches, based on the length of their humerus bone also ininches, using data from the class to estimate the best line.

1. You will be put in groups and given tape measures (1 for a group of three, two for a group of 4).

2. Take turns measuring the length of your partners’ humerus bone (from the elbow to the shoulder ) and theirheight (if unknown). Both measurements are to be done in inches – to the nearest half inch. Record yourmeasurements in the table below.

Name Length of Humerus (inches) Height (inches) Ordered Pairs

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3. Create ordered pairs with the humerus bone as the independent variable and the height as the dependentvariable, (humerus, height).

4. Share your data with the other groups. You should have 10 or more ordered pairs to get started. Be sure toget data from students with a variety of heights. You need to have a mix of short, medium, and tall people.

5. Create a scatterplot on the graph below, with the horizontal axis as the length of the humerus bone and thevertical axis as the height. If you need help, your lab instructor can give you guidance on the best scales to

use based on the span of the data. Otherwise you may have your data points piled up in one area.

6. Next, use a straight edge to draw a line that connects two data points such that the line runs through allthe data points fairly well.

7. Create the equation for the line that passes through these two points.

Part II

Use the equation found in Part I to estimate the height of the person whose bone was found. Remember thehumerus bone was found to be 12 inches long.

1. Estimate the height.

2. State your answer in a complete sentence, in context of the problem. In other words, how will you passyour solution on to the investigators so that they understand what you are telling them?

x

y

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Instructor’s Key:

Comparing linear equations

Where can I get the best car rental deal?

Upon landing at the airport in Anaconda, Montana I needed to rent a car so I could visit all the wonderful sites in

the area, especially the snake monument that stood 24 feet high. After finding the car rental desk I had tochoose between the only 2 options available. Option #1 : A fee of $10 plus a charge of 5¢ per mile. Option #2: A charge of 17¢ per mile.

1. Fill in the following table:

Number of milestraveled Cost of Option #1 Cost of Option #2 Best choice option

10 $10.50 $1.70 #2

25 $11.25 $4.25 #2

43 $12.15 $7.31 #2

68 $13.40 $11.56 #2

94 $14.70 $15.98 #1

133 $16.65 $22.61 #1

2. When did you find Option #1 to be the bestchoice?At 94 miles

3. When did you find Option #2 to be the bestchoice?Prior to traveling 94 miles

4. If you were to travel m miles, write an equationthat would show the cost of the car rental usingoption #1 (C1):

1 0.05 10C m

5. If you were to travel m miles, write an equationthat would show the cost of the car rental usingoption #2 (C2):

2 0.17C m

6. Put these 2 equations into your calculator andset your table to start at 0 and change (Δ) by 5.Also put both the Independent and Dependenton Auto. Notice the changes as you scroll downyour table. Write a few sentences to explainwhat the numbers are telling you about thecosts of Option #1 compared to Option #2.

For every 5 miles traveled, the cost ofoption #1 increases by 25¢, while the costof options #2 increases by 85¢.

7. Draw graphs of your 2 equations. Put the axesand scaling information on the graph.

8. x axis represents: miles

9. y axis represents: cost

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10. Using the graphs you have drawn, estimate the point of intersection.

85; 14 x y

11. Using the graphing features of your calculator, determine the point of intersection of the two graphs. Wasyour estimate reasonable?

83.33; 14.17 x y Yes

12. Explain, in the context of the situation, what the point of intersection tells you.

After traveling around 83.33 miles, the cost of both optionsis same at about $14.17

13. Take the expressions you wrote for exercises #4 and #5, and set them equal to each other. Solve theresulting equation symbolically.

0.17 0.05 10

1083.33

0.12

m m

m

14. Your solution to the equation from exercise #13 should be familiar. Explain why.

The solution represents the point of intersection of the two lines/options.

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Instructor’s Key:

Mixture problems using two variables

Yes, we do mix things in real life!

The ability to picture the mixture story will give you the best opportunity to understand what is happening andachieve the correct results. Using a chart can additionally help organize the information from the problem.

1. The left bank is offering CD’s at 5%. The right bank is offeringa money market account at 4%. If I have a total of $10,000 toinvest in these accounts and want a return (accrued interest)on my money of $436 from both accounts at the end of oneyear , how much should I put into each account?

Let C represent $$ you put into CD. Let M represent the $$ you put into the money market.Complete the chart below using information from the problem:

Mixture problem Principal($ invested)

Rate(% as decimal) Time (yrs.)

Interest($ earned oninvestment)

CD account C 0.05 0.05CMoney market M 0.04 0.04M

Totals for the twobank accounts $10,000

Cannot be found byadding values

above1 $436

The first column in the chart should contain information necessary to write one equation containing the twovariables and the last column should show information to write another equation containing the two variables.Fill in the boxes below with the information from those columns to set up the system of equations:

+ =

+ =

Solve algebraically. State your answer in complete sentences. $3600

$6400

C

M

$$ put into CD accountC

$$ put into money marketM

Total $$ invested$10,000

CD interest (at 5%)0.05C

Money market interest (at 4%)0.04M

Total interest$436

Principal • Rate • Time = Interest (return)

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2. The Daytona 500 started in 1959 with an average speed of 135 mph butbefore they raced on the big speedway they would race up and down onthe beach. In 1954 Lee Petty raced on the beach for a total 160 miles. Asthe 2 hour race developed Lee averaged 90 mph for the first part of therace but only 60 mph for last part. How long did he race at 90 mph andhow long at 60 mph?

Let F represent the hours he raced at 90 mph. Let L represent the hours he raced at 60 mph.Complete the chart below using information from the problem.

Use the formula to work across and complete any empty spaces in the last column:

Fill in the boxes below with the information from the chart to set up the system of equations:

+ =

+ =

Solve algebraically. State your answer in complete sentences.

4. @ 90

3

2. @ 60

3

F hr mph

L hr mph

Mixture problem Rate (mph) Time (hours) Distance (miles)First part of race 90 F 90F

Last part of race 60 L 60L

Totals for entire raceCannot be found by

adding the twoaverages above!

2 160

Rate (mph) • Time (hours) = Distance (miles)

Distance for first part90F

Distance for last part60L

Distance for whole race160

Time for first partF

Time for last partL

Time for whole race2

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3. Luis was hired to make some jewelry for a special occasion.He needs to make the jewelry out of an alloy that is 54%silver and he requires 10 ounces. He has a large amount of45% silver and also some that is 75% silver. How manyounces of the 45% silver and 75% silver does he need?

Let X represent the amount of 45% silver. Let Y represent the amount of 75% silver.Complete the chart below using information from the problem .

Use the formula to work across and complete any empty spaces in the last column:

Again use the chart’s information to fill in the boxes below and set up the system of equations:

+ =

+ =

Are you starting to see a pattern with these mixture problems?


7 45%

3 75%

x ounces of silver

y ounces of silver

Mixture problem Quantity (ounces) Percent (% as decimal) PURE silver totals45% alloy X 0.45 0.45X

75% alloy Y 0.75 0.75Y

Totals for the 54%

mixture 10 This information must be given

0.54(10)

Quantity • Percent = Amount of PURE Material

Amount of 45% silver x

Amount of 75% silver y

Total amount of 54% silver10

Pure silver in 45%

0.45 x

Pure silver in 75%

0.75 y

Total amount of PURE silver

0.54(10)

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4. Connie has been collecting all her extra nickels and dimesinto a vase this past year. The vase is full so she decided tocount them and see if she has enough money to buy herselfa new dress. She was surprised to find that she had 3400coins worth $290. How many nickels (N) and how manydimes (D) did she have in the vase?

Fill in the words and the value (numeric or algebraic) that accompany each piece of information requested in thechart below. Then, as above, use the chart to fill in the boxes and complete the system of equations.

+ =

+ =


1000

2400

N

D

Mixture problem Quantity (_______) Value (______) Totals of __________Nickels N 0.05 0.05NDimes D 0.10 0.10D

Totals for3400

This information normally given$290

Quantity of_________ N

Quantity of __________D

Value of ________0.05N

Value of_____________0.10D

Total _____________$290

Total _____________3400

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Now, can you solve some problems without the charts given to you?

5. Joel sells cotton candy at the Magic games for

$4 per bag. He also sells peanuts at the gamesfor $2.50 per bag. One day he sold 160 bags andcollected $460. How many of each item did he

sell?

160

4 2.5 460

40

120

C P

C P

C

P

(cotton candy)

(peanuts)

6. For a chemistry experiment, Agatha needs 50milliliters of 20% acetic acid solution. Thelaboratory only has 10% and 50% acetic acidsolutions. Agatha plans to make the 20% aceticacid solution by mixing together appropriateamounts of the two solutions. Determine theamounts of 10% and 50% acetic acid solutionthat should be mixed to make 50 milliliters of20% acetic acid solution.

50

0.10 0.50 0.2(50)

37.5 10%

12.5 50%

x y

x y

x ml of acetic acid

y ml of acetic acid

7. A long distance runner trains for 3 hours. During

the first part of the training, he keeps a pace of7 miles per hour (mph). For the second part, heslows down to 5 mph. At the end of the run, he

has traveled a total of 19.5 miles. How long didhe spend at each speed?

3

7 5 19.5

2.25 . @ 7

0.75 . @ 5

F S

F S

F hr mph

S hr mph

8. Jody plans to invest a total of $7000 betweentwo different accounts. She has done someresearch and found a savings account that willpay 2.5% interest per year and a money marketaccount that pays 6% per year. If she wants toearn $385 after one year, how much should sheinvest in each account?

70000.025 0.06 385

$1000

$6000

S M S M

S in a savings account

M in a money market acco

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28 M AT1033C

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Instructor’s Key:

Pan Balance – Linear Expressions & Inequalities

Using your favorite internet browser, go to http://illuminations.nctm.org/Activity.aspx?id=3529.Alternatively, perform a search for the following terms: illuminations pan balance expressions . It should be thefirst option.

Instructions:

Place an algebraic expression in each of the red and blue pans. These expressions may or may not include thevariable x. Enter a value for x, or adjust the value of x by moving the slider.

As the value of x changes, the results will be graphed. Use the Zoom In and Zoom Out buttons, or adjust thevalues for the x and y axes with the sliders, to change the portion of the graph that is displayed.

The Reset Balance button removes the expressions from the pans and clears the graph.

Try it out:

Use the Pan Balance to see the relationship between 2 10 x and 2 x and solve 2 10 2 x x

1. Type 2 10 x in the left pan (red)


3. You should see two lines

4. By using the slider, or trial and error, find the value of x which balances the pans. Record this value. Thisvalue is the solution of the related equation 2 10 2 x x and is the boundary for the solutions of the

inequality.

5. We want 2 10 x to be greater than 2 x . So, in terms of the Pan Balance, we want the left pan to be“heavier.” Move the slider to the side of the value you found in part d that makes the left pan heavier. [Youshould also see that the point associated with the left expression is above the point associated with the rightexpression!] In which direction did you move the slider? (left or right)

6. Write the solution of the inequality. Remember, if you moved the slider to the left in part e, your new x valueis less than the value from part d. If you moved the slider to the right in part e, your new x value is greaterthan the value from part d. The solution should take the form: x number or x number .

7.

Using symbolic methods (addition/subtraction/multiplication/division to isolate the variable), solve theinequality 2 10 2 x x . Record your work and the solution.

8. Compare your answers for parts f) and g). Anything interesting?

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Using a similar process, solve each of the following equations. You should record on separate paper :a. The expression you used for the left pan. (How did you type it in to the website?)b. The expression you used for the right pan. (How did you type it in to the website?)c. The value of x that balances the pan, estimating if necessary, and the direction you would move the

slider to make the appropriate side heavier. (You may need to adjust the settings on the graph tofind some of the values.)

d. The exact solution that you find using symbolic methods.

Appropriate“

work”

for the sample exercise:

A.

8 5 3 x 1 x

B.

3 7 2 11 x x 4 x

C. 6 3 2 4 4 x x x 1 x

D. 5 3 6 3 2 1 4 x x x

16

7 x

E. 5 4 2 2 3 0 x x

10

9

x

F. 3 1

15 2

x x

15

16 x

G. 5 4 3 5

6 4

x x

7 x

H. 3.1 3 2.9 x x

1.55 x

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Instructor’s Key:

How to write a linear inequality solution

Can you give me the solution in 5 formats?

Solve: 2 3 11 x

Solution written in English: All real numbers less than or equal to 7.

Solution using algebraic notation: 7 x

Solution using interval notation: ( ,7

Solution using set builder notation: | 7 x x

Solution using a number line:

| | | | | | | | | | | | | | | | | | | | | | | | | | |-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16

Circle any of the following values that are included with this solution.2 1

, 0, 4 , 453, 2.4, 5.678, 11, 7, 13 3

1. Solve: Symbolically

3 5 7 x

4 x

A. English solution:

All real numbers less than 4.

B. Algebraic solution:

4 x

C. Number line solution:

D. Interval notation solution:

, 4

E. Set builder notation solution:

| 4 x x

2. Solve: Symbolically 4 5 13 x

2 x

A. English solution:

All real numbers greater than or equal to −2

B. Algebraic solution:

2 x

C. Number line solution:

D. Interval notation solution:

2,

E. Set builder notation solution:

| 2 x x

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2 1 y y

1 y

2 y

3. Solve Graphically 2 4 1 x x

A. On the graph 1 2 4 y x and 2 1 y x

B. Find when 1 2 y y (Hint: x coordinate of the point of intersection– thisis the boundary for the solutions of the inequality)

1 x

C. We now need to decide what ‘ x’ value(s)are represented when 1 2 y y ? Looking at

the graph what ‘ x’ value(s) did you find?(Hint: where is the Y 1 line lower than the Y 2 line?)

1 x

D. Solutions:

a.

English:All real numbers less than −1

b. Algebraic:

1 x

c. Number line:

d. Interval notation:

, 1

e. Set builder notation:

| 2 x x

4. Solve: Graphically 1 2 y y

A. Find when 1 2 y y

2 x

B. Where is 1 2 y y ?

2 x

C. Solutions:a. English:

All real numbers less than 2

b. Algebraic:

2 x

c. Number line:


, 2


| 2 x x

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5. Solve: Numerically 1

2 32

x x

A. On the table 1

1

2 y x and 2 2 3 y x

B. Where does the chart show that 1 2 y y ?(Hint: this is the boundary for the solutions of theinequality)

2 x

C.

We now need to decide what ‘ x’ value(s)are represented when 1 2 y y ? Looking at

the chart what ‘ x’ value(s) did you find?2 x

D. Solutions:a. English:

All real numbers less than 2

b. Algebraic:

2 x

c. Number line:


, 2


| 2 x x

6. Solve: Numerically 1 2 y y

A. Where does the chart show that 1 2 y y ?(Hint: this is the boundary for the solutions of theinequality)

1 x

B. We now need to decide what ‘ x’ value(s)are represented when 1 2 y y ? Looking at

the chart what ‘ x’ value(s) did you find?1 x

C. Solutions:a. English:

All real numbers less than −1

b. Algebraic:

1 x

c. Number line:


, 1


| 1 x x

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7. AND – OR sentences:A. If you are in the Math Center wearing a hat

OR over 25 years old then you will get a giftcertificate for $10. If you are in the MathCenter wearing a hat AND over 25 years oldthen you get a gift certificate for $25.

B. What do you think is the difference when

comparing the 2 statements in part A?In the statement, you need to meet at leastone criteria to get the $10 certificate, whileyou need to meet both criteria to get the$25 gift certificate.

C. If you were giving out gift certificates today

would you anticipate giving out more of the$10 or $25 certificates? Explain youranswer.

$10. It is more likely that a person will meetone criteria than both.

D. AND creates which of the followingpatterns (Circle the correct answer) :

Union of the groups (∪) combine

Intersection of the groups (∩) overlap

E. OR creates which of the following patterns:

Union of the groups (∪) combine

Intersection of the groups (∩) overlap

8. Write solution in 5 formats: 1 4 x or x

A. Number line:

B. English:

All real numbers greater than or equal to −4

C. Algebraic:

4 x

D. Interval notation:

4,

E. Set builder notation:

| 4 x x

9.

Write the solution in 5 formats:

2 5 x and x

A. Number line:

B. English:

All real numbers greater than or equal to

−5, but less than 2.

C. Algebraic:

5 2 x

D. Interval notation:

5,2

E. Set builder notation:

| 5 2 x x

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Instructor’s Key:

Exponents

I remember it well!

I remember my instructor telling me the definition of an ___________________ is a little number in the top

right hand corner of a number or variable that TELLS me how many times to multiply the number or variable byitself.

We are not changing that definition in the following problems but are just skipping steps to simplifying theseexpressions (otherwise called a shortcut).

1. 4

5 A

x x In this equation the value of A = 20

2. 7 2 B

x x x In this equation the value of B = 9

3.

7

3

C x

x

x

In this equation the value of C = 4

4. 1 D

x In this equation the value of D = 0

5.

110

100

E

In this equation the value of E = −2

6. 4

1 F

x

x

In this equation the value of F = −4

7. 3 4

4 3

G

In this equation the value of G = −1

Does A + B + C + D + E + F + G = 26? If not,

discuss it with your group and instructor

before going on to the next problem.

8. These questions represent what most studentsconsider a “trick question” by the instructor.Will they “trick” you?

a. 2 2

3 3

True or False False

b. 22

3 3 True or False False

c. 2 2

3 3 True or False False

If your answers to a, b, and c were not the same,

then you need to discuss it with your group!

9. All the following are incorrectly simplified.Explain what’s wrong and simplify theexpression correctly.

a. 2

4 8

3 6 x x 2 8

3 9; 9 x

b. 0

4 0 x 0

1; 4 x

c.

2215

5 x

x

2

2

55 x

x

d. 1

82

4

x

x

1

282

4

x x

x

exponent

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Simplify, write all final responses with positive exponents:

1. 2 4 655 y y y 2. 23

x

x x

3. 3 5

5 3

2

2

315

25 5

r

r

d d

d r 4.

4 82( )m m

5.

5 3 8

w w w 6.

3 3

4 4( ) 6r r

7. 3 6 12 24( 2166 )r x x r 8. 5 3 112( )r r r

9. 4 2 6(7 )(5 5) 3r r r 10.

32

5 9

6

121

4

5

8

0 x

x

x

11. 5 7

2

6

4

3

3

g p y

g p y

g p

y 12. 0 1r

13. 3 02 (4 ) 2r r r 14. 06 6w

15. 0 3 93(2 ) 8 f t f 16. 0

5 5

7

4 21

1 x

x x

17.

05

8

10

241

m

m

18. 28

1

64

19. 2(3 2

6 )3

20. 35 2 y y y

21. 4 3( 3 ) ( 33) 22. 3

3 1m

m

23. 2 3

55

5m m

m

24. 15 5

cc

25. 3 2 1

52 ( 2 )

1

gg g

26. 4( 5) 1

625

27. 4 34

3

4 4

ph

h p

28. 7

4

3v

vv

29. 5

2 3

1

y y

y

30. 5

3

2

x

x x

31. 1

2 3

6 3

2

p

p p

32.

4 3

3 85

7k r

k r w

k

r w

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Instructor’s Key:

Factoring / Polynomials

Find the infamous polynomial!Following are 17 polynomials that need to be factored. Find the factors of each polynomial andthen cross them off on the chart below. When completed you will have 2 blocks left on the chartthat are not crossed off.

1. 23 4 7 x x

(3 7)( 1) x x 2.

26 5 4 x x (3 4)(2 1) x x

3. 210 59 22 x x

(5 2)(2 11) x x

4. 25 32 35 x x

(5 7)( 5) x x 5.

224 50 9 x (6 1)(4 9) x x

6. 26 7 10 x x

(6 5)( 2) x x

7.

2

10 37 7 x x

(5 1)(2 7) x x 8.

2

8 30 27 x x

(2 9)(4 3) x x 9.

2

30 31 44 x x

(5 4)(6 11) x x

10.

230 103 51 x x (5 3)(6 17) x x

11.

22 17 30 x x (2 5)( 6) x x

12.

212 29 14 x x (4 7)(3 2) x x

13. 218 15 7 x x

(3 1)(6 7) x x 14.

212 47 40 x x (3 8)(4 5) x x

15. 24 17 4 x x

(4 1)( 4) x x

16.

212 8 39 x x (2 3)(6 13) x x

17.

24 23 33 x x (4 11)( 3) x x

(6 5) x

(2 1) x

( 3) x

(5 7) x

(3 4) x

(4 1) x

(3 8) x

(5 3) x

(2 7) x

(4 3) x

(6 13) x

( 6) x

(5 6) x

(3 2) x

(4 9) x

(2 3) x

( 4) x

(6 7) x

(4 7) x

( 1) x

(6 11) x

(3 7) x

(5 2) x

(2 11) x

( 2) x

(4 11) x

(3 5) x

(6 1) x

(2 9) x

(5 4) x

(2 5) x

(6 17) x

(5 1) x

( 5) x

(4 5) x

(3 1) x

18. The 2 factors that were not crossed off: ________________________ 19. Multiply (FOIL) the 2 factors from question #18: _______________________

NOTE : If you got – 43x as the middle term in your trinomial for #19 then continue, if not recheck your work.

(5 6)(3 5) x x

215 43 30 x x

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1. 23 4 7 x x 2.

26 5 4 x x 3. 210 59 22 x x

4.

25 32 35 x x 5.

224 50 9 x x 6.

26 7 10 x x

7. 210 37 7 x x 8.

28 30 27 x x 9. 230 31 44 x x

10.

230 103 51 x x 11.

22 17 30 x x 12.

212 29 14 x x

13. 218 15 7 x x 14.

212 47 40 x x 15. 24 17 4 x x

16.

212 8 39 x x 17.

24 23 33 x x

Subtraction of polynomials: In this section you are to find 2 of the polynomials (#1 #17) thatif subtracted would equal the given polynomial.

Example:

_____________ − _______________ =

2 39 45 x x

Looking at the list of polynomials we need to find 2 that when subtracted the coefficient of the 2 x term

would be −1. Once you find 2 that meet this criteria then subtract the rest of the polynomial to check theother coefficients.

2 2 2

5 3 2 3 5 6 7 1 0 3 9 4 5 x x x x x x

Therefore:

#4 – #6 = 2 39 45 x x

Polynomial from list – Polynomial from list = Difference of the 2 polynomials

20. #__________ – #_________ _ =27 15 5 x x

21. #__________ – #_________ _ =26 19 35 x x

22. #__________ – #_________ _ =2 15 31 x x

23. #__________ – #_________ _ = 21 53 x

24. #__________ – #_________ _ =214 9 31 x x

Write the equation of the following graphs in factored form:

25. ____________________ 26. ________________ 27. __________________

4

9

15

12

3

14

5

4

16

5

( 2)( 3) x x

( 3)( 6) x x

( 8)( 1) x x

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Instructor’s Key:

Polynomials – Connecting factors, solutions, and x intercepts

Do you want me to Solve or find x intercepts?

Completely Factor the

expression

Solve the equation Determine the x intercepts

of the graph 1. Example:

29 12 x x

3 3 4 x x

2.

Example: 29 12 0 x x

3 3 4 0 x x

3 0 3 4 0 x or x 4

03

x or x

3. Example:29 12 y x x

4

0,0 ,03

and

4.

22 7 x x

(2 7) x x

5.

22 7 0 x x

70,

2 x

6.

22 7 y x x

7(0,0) ,0

2

7.

2 12 35 x x

( 5)( 7) x x

8.

2 12 35 0 x x

7, 5 x

9. 2 12 35 y x x

( 7,0) ( 5,0)

10.

2 6 72 x x

( 6)( 12) x x

11.

2 6 72 0 x x

12,6 x

12. 2 6 72 y x x

( 12,0) (6,0)

13. 24 12 8 x x

4( 1)( 2) x x

14. 24 12 8 0 x x

2, 1 x

15. 24 12 8 y x x

( 2,0) ( 1,0)

16. 24 4 35 x x

(2 5)(2 7) x x

17. 24 4 35 0 x x

5 7,

2 2 x

18. 24 4 35 y x x

5 7, 0 , 0

2 2

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Use a graphing calculator to solve #19 #22. Include a sketch that supports your conclusion.

19. 20.5 1.75 0 x x 3.5,0

(0,0) ( 3.5,0)

x

20. 4 3 20.5 1.5 5 12 0 x x x x (Note: This equation is difficult to solve withfactoring. Use an appropriate graph)

3,0,2,4

3,0 , 0,0 , 2,0 , 4,0

x

21. 2 6 4 0 x x (Note: This equation cannot be solved byfactoring. Use an appropriate graph. Round yoursolutions to the nearest tenth)

0.6,6.6

0.6,0 , 6.6,0

x

22. 3 23 6 8 0 x x x

2,1, 4

2,0 , 1,0 , 4,0

x

23. Using your reasoning powers

derived from this lab draw a linefrom the equation to the graphthat it matches.

2

4 3 2

3 2

6 16 6

2 31 26 24

3 3 36

2 3

x x

x x x x

x x x

x

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Instructor’s Information Sheet:

Polynomials/Factoring

Polynomial Factoring BINGO

Student Instructions

Directions: 1. Select a BINGO game card.2. Start factoring the polynomials as they are written on

the board.3. Cover up those factors that appear on your BINGO card.

(Some students might have both factors, some might just have one, depends on the factors present on theirBINGO card)

4. Continue down the list of polynomials until you have 4

in a row, vertically, horizontally, or diagonally.5. Yell out “BINGO” and have the instructor verify youranswers.

Workspace (use the space below and the back of this page to show your work)

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Instructor’s Key:

Looking for graph patterns from an equation on a calculator.

Discovering Quadratic Graph Secrets

This lab will require the use of your graphing

calculator. Push on your calculator and

any equations you might have in it already.

1. Graph the following 5 equations:

2

1 y x Set your window to the following

settings:2

2 2 y x

X min = −4X max = 4X scl = 1

Y min = −1Y max = 10Y scl = 1

2

3 3 y x

2

4 5 y x

2

5 10 y x Push GRAPH after entering all of the

equations.

2. What conclusion(s) can you draw after seeingthe 5 different parabolas?Note: Use the TRACE and then the ↑↓ keys to help

you identify which equation is which.

As the coefficient increases, the graphstretches.

Push on your calculator and allequations from question 1.


2


settings:

2

2 1 2 y x



2

3 1 3 y x

2

4 1 5 y x 2

5 1 10 y x Push GRAPH after entering all of

the equations.

4. What conclusion(s) can you draw after seeing

the 5 different parabolas?Note: Use the TRACE and then the ↑↓ keys to help


As the coefficient falls below 1, the graph iscompressed.



2


settings:

2

2 1 y x

X min = −10X max = 10

X scl = 1

Y min = −1Y max = 10

Y scl = 1

2

3 2 y x

2

4 5 y x

2

5 8 y x

Push GRAPH after entering all of theequations.

6. What conclusion(s) can you draw after seeingthe 5 different parabolas?Note: Use the TRACE and then the ↑↓ keys to help


The graph shifts to the left.

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2


settings:

2

2 1 y x



2

3 2 y x

2

4 5 y x

2

5 8 y x

Push GRAPH after entering all of theequations.

8. What conclusion(s) can you draw after seeingthe 5 different parabolas?

The graph shifts to the right.

9. What logical reason can you find for therelationship between the signs in question 5and 7 with your conclusions in questions 6 and8?

Finding the zeros.



2


settings:2

2 3 y x



2

3 5 y x

2

4 3 y x

2

5 5 y x Push GRAPH after entering all of the

equations.


Graphs shift up with a positive constant term,and shift down with a negative constant term.



2

1 3 y x Set your window to the following

settings:

2

2 3 y x



2

3 6 y x

2

4 6 y x

25 6 y x Push GRAPH after entering all of

the equations. 2

6 6 y x


Graphs shift up with a positive constant term,and shift down with a negative constant term;are reflected about the x axis with a negativecoefficient term.

14. Using your conclusions from the work abovedescribe how the following parabolas will lookwhen graphed compared to the standardparabola of 2

y x

a. 22 5 y x

graph stretches by a factor of 2; shiftsdown 5 units

b. 2

3 2 y x

shifts up 2 units and 3 units to the right

c. 21 y x

graph shifts down 1 unit; reflects on x axis

d. 2

4 3 y x

graph shifts down 3 units and 4 units to theleft

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Instructor’s Key:

Transformations of quadratic equations

Different words – Same meaning!One instructor said, “Write x squared on your paper” . A second instructor said, “Write x to the second power on

your paper” . Hopefully all the students would know that both instructors wanted you to write the same thing.

Often different words will mean the same thing as we will see in this activity.This is the graph of

2( ) f x x where each tic mark represents 1 unit. This graph will

be used in this activity as the “bench mark” for the comparisons that we will be makingin all the following questions.

1. Below are graphs that are different from our “bench mark” . In addition to the letter on the graph find all theletters with info that matches up with the graph.

Note: Not all the words or equations will be used and some may be used more than once.

U. E. Wider O.

P. Narrower

Z. “a” is a proper fraction

T. “a” is greater than 1

S. Compressed

I. Stretched

A. Shifted downward

M. Shifted upward

C. Reflected

G. Shifted left

W. Shifted right

N. 2( ) 5 f x x

R. 21( )

5 f x x

B. 2( ) ( 5) f x x

F. 2( ) ( 5) f x x

D. 2( ) 5 f x x

K. 2( ) 5 f x x

2. Unscramble all the letters belonging to each graph to fill in the blanks below.

The independent variable is the __________. ________ are also x intercepts.INPUT ZEROS

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3. Below are graphs that are different from our “bench mark” . In addition to the letter on the graph find all theletters with info that matches up with the graph.

Note: Not all the words or equations will be used and some may be used more than once.

“Bench Mark ” Graph

N.

A. Wider

U.

G. Narrower

P. “a” is a proper fraction

O. “a” is greater than 1

K. CompressedN. Stretched

B. Shifted downward

I. Shifted upward

E. Reflected

S. Shifted left

C. Shifted right

R. 2( ) 5 f x x

M. 2( ) ( 5) 5 f x x

D. 2( ) ( 5) 5 f x x

H. 2( ) ( 5) 5 f x x

L. 2( ) 5 f x x

P. 2( ) ( ) 5 f x x

4.

Unscramble all the letters attached to eachgraph to fill in the blanks below.

A linear graph is a ___________ . X to thethird power is also called x __________ .

5.

What is the difference when one instructor talksto their class about “shifting a function” andanother instructor is having a class discussionon “transformations”?

6. The 7 different ways we can make our basicfunctions move are:

A.

________________________________B. ________________________________

C. ________________________________D. ________________________________

E. ________________________________

F. ________________________________G.

________________________________

7.

Shifting up means to add the value and shiftingdown means to subtract the value. It seemsreasonable that shifting right you should addthe value, but that is not true. Why?

8 – 13. Note: #8 is an example.

A. In each of the following graphs below noteall transformations involved:

NS = Narrower, Stretched, “a” is greater than 1

WC = Wider, Compressed, “a” is a proper fractionSL = Shift LeftSR = Shift RightSU = Shift Upward

SD = Shift DownwardR = Reflected

B. For each of the graphs below, write aquadratic equation in vertex form.

2( ) ( ) f x a x h k

Use “a” values of ±1, ±5, or ± 1 / 5 accordingly.

LINECUBED

SHIFT UPSHIFT DOWNSHIFT RIGHT

SHIFT LEFT

REFLECTCOMPRESSSTRETCH

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8A. SR, SD

8B. 2

( ) 4.5 2 f x x

9A. _______________________

9B. _______________________

10A. _______________________

10B. _______________________

11A. _______________________

11B. _______________________

12A. _______________________

12B. _______________________

13A. _______________________

13B. _______________________

21( )

5 f x x

2( ) ( 4.5) 12 f x x

2

( ) 5 5 f x x

21

( ) 5 55

f x x

2( ) 5 3 f x x

R, WC

SL, SD

SL, NS

SR, SU, R, WC

SD, NS

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Instructor’s Key:

Get the Ball Rolling

Objective:

Create a mathematical equation that models a ball rolling up a ramp and back down.

Materials needed:

Blocks and 2 meter sticks per set upPlastic Ball2 TI 84 calculators, one being a student calculatorCBR 2™motion detectorUSB cable link for calculator and CBR 2™

Instructions:

1.

Prop the two meter sticks on the blocks to forma ramp at a 15° incline.

2. Connect the CBR to the TI 84 calculator. Turn onthe calculator. The TI 84 will immediately enterthe Easy Data program and begin collectingdata points.

3. Select Setup (press @). Select Dist. for

Distance, select Units (press@) , select(m) meters, and then select OK (press

%). Press Setup (press@). Select

the Time Graph option. Edit (press #):

(use the settings in the table below )

4.

Press ADV (press

@),

select Manual;Next (press #) , then OK (press %).

Select Start (press #) , then OK (press

%) until a blank screen appears.

5. The CBR is now ready to collect data. Positionthe ball at the bottom of the ramp. Position theCBR behind the ball on the floor. When ready,simultaneously press the trigger button on the

CBR, and push the ball up the ramp just hardenough so that the ball stops just short of theend of the ramp and rolls back downward. Youwill hear a “ticking” sound from the CBR as itcollects data points. When the CBR hasfinished collecting data, press APPSA onthe calculator, go to EasyData and select OK(press %) to retrieve the data. At thistime, a graph will appear on the screen. Repeatthe procedure if you do not have a “clean”

graph (no spikes and no flat lines). For“

doovers” , select Main, Start, and OK until you areback to the blank screen.

Sample Interval: (.05 seconds between datareadings)

.05Number of Samples:

100Experiment Length (s): (5 seconds long

5

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Questions:

1. What is the shape of the graph?A downwards parabola

2. What physical property is represented along thex axis?Time

3. What are the units?Seconds (s)

4. What physical property is represented along they axis?

Distance

5. What are the units?Meters (m)

6. What does the highest point on the plotrepresent physically?It represents the top of the ramp.

7. Use the arrows on the calculator to find thefollowing coordinates in (x, y) form:

Vertex: (0.5,1.099)

Sample point: (0.8,1.003)

The vertex form of a parabola is 2( ) y a x h k .

Using the two points above, write the quadratic

equation that models the graph (Refer to the stepsbelow) .

Step 1: Find a by plugging in the vertex coordinatesfor h and k and the sample point for x and y.

1.067a

Step 2: Write the equation for the quadratic in

standard form, 2 y ax bx c .

21.067 1.067 0.832 y x x

Disconnect CBR and calculator. On the calculator,select Main, Quit, and OK.

Enter your equation into the! window and pressGRAPH. The TI 84 will first show the graph from thedata collected from the ball roll and then plot the! graph over the existing graph.

8. How do the graphs compare?

(If the two graphs do not match closely, check yourcalculations in Section 3)

Conclusion:

In your own words, describe how the graph modelsthe motion of the ball.

Why is the ‘a’ negative?

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Repeat the activity beginning with Step 1 in theinstructions, increasing the angle of incline to25°.

How did the graph change?

How did the coefficient of change?

What conclusions can you draw comparing the twooutcomes relating to the incline of the ramp?

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Instructor’s Key:

Different ways to solve the same quadratic equation

Could you pass this Test?

Solve this quadratic equation for f(x) = 0.

24 4 15 y x x

Part A: Solve by Graphing (on your TI−84 graphing calculator):Set the calculator window so that you see thevertex, y−intercept, x−intercept(s), and it fits on thegraph paper below!

What were your Window values? Xmin = −5 Ymin = −20 Xmax = 5 Ymax = 5 Xscl = 1 Yscl = 1

Sketch the graph from your calculator to thegraph above.

Put arrow(s) on the graph to show where youlooked for the solution(s) to the quadraticequation.

1. Looking at your graph what is the solution(s) tothe quadratic equation?

2.5,1.5 x

2. THINK about it : When your instructor asked youto “SOLVE the quadratic equation for f(x) = 0” what information (in words, not numbers) didthe instructor want you to find? Hint: Look at the graphthat you just did!

Find the x−intercepts.

Part B: Solve by using a Table (on your TI−84 graphing calculator):

x Y1 Table SetupTblStart = −3∆Tbl = 1Indpnt: AutoDepend: Auto

Use the Table on yourcalculator to help you fillin the table to the left:

−3 9

−2 −7

−1 −15

0 −15

1 −7

2 9

3 13

3. What should you look for on a table to find the

solutions?

Find the x−values that make the y−values equalzero.

In the table above put an arrow next to the solution or where thesolution should be.

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4. How should you change the Table Setup so thatit will show you the solution(s) to the quadraticequation?

∆Tbl = 0.5

Fill in the chart below after changing the TableSetup to show the solution(s).

x Y1

−3 9

−2.5 0

−2 −7

−1.5 −12

−1 −15

−0.5 −16

0 −15

0.5 −12

1 −7

1.5 0

2 9

5. What is the solution(s):2.5,1.5 x

In Part A your graph information (the pointingarrows) should match your solution(s) from thistable. If not, recheck your work.

6. Did the table support your answer that you putfor question #2?Yes. One solution was between −3 and −2, whilethe other was between 1 and 2.

7. Do you think that using a table to solve otherquadratic equations would be an easy ordifficult method and why do you think so?

Part C: Solve by Factoring:

Show your steps for solving the quadratic equationby factoring.

24 4 15 0 x x

(2 3)(2 5) 0

3 52 2

x x

x x

8. Did you get the same solution(s) as in question#1 and #5? Yes

9. If given the quadratic equation:2

3 6 5 11 x x , what is the first thing youneed to do BEFORE you can consider solving itby factoring?

Set the right hand side to zero by subtracting 11over to the left.

10. What does the “Zero Product Property” mean?

If two real numbers multiply to give zero, then

one or the other (or both) must equal zero.

11. If given the quadratic equation:2

4 9 7 x ,you could solve it by factoring. But it might beeasier to solve by the s q u a r e r o o tmethod.

Hint: Each space in the answer represents a letter of a word.

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Part D: Solve by Quadratic Formula:

12. Write the quadratic formula?2

4

2

b b ac x

a

13. Where do you find the a , b , and c values for thequadratic formula?a is the coefficient of the x2 term, b is thecoefficient of the x term, and c is the constantterm.

14. What is the Standard form of a quadraticequation and why is it important?

20ax bx c

15. What is another method we could use to solve

a quadratic equation?You may want to reference your textbook!

c o m p l e t i n g the s q u a r e


16. Solve the instructor’s original quadraticequation using the quadratic formula. Show

your steps!

24 4 15

4 16

8

3 5;

0

2 2

x

x

x x

x

Part E: Thoughts about solving quadraticequations:

17. Are you able to solve all quadratic equations byfactoring? No

18. Are you able to solve all quadratic equations bythe quadratic formula? Yes

19.

Which method would you try first on a test thatsaid to “solve” a quadratic equation?Factoring

20. If that method does not work what would youtry next?Quadratic Formula

21. Now that you have “SOLVED” the quadratic

equation by 4 different methods what otherinstructions could an instructor have given toget the same solutions?

A. Find the r o o t s of the quadraticequation.

B. Find thex – i n t e r c e p t s of the quadratic equation.

C. Find the z e r o s of the quadratic equation.


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Instructor’s Key:

Real life quadratic functions

How high and far can you hit the ball?

When an athlete hits a ball which of the following would make a difference in the height or

distance that the ball goes?

How hard you hit the ball

The angle of the swing Which way the wind is blowing

How hard the wind is blowing Temperature of the air

Humidity of the air

How high you hit the ball If the ground in front of you is flat

How tall you are

How fast the pitcher throws the ball Your elevation from sea level

If you are male or female

While in real life all of these may affect the ball in some way the only ones we will be

considering in this lab will be:1)

How hard you hit the ball.2) How far off the ground the ball is when you hit it.

General quadratic equation: 2

( )h x ax bx c

“h” relates to the height of the ball after some amount of time

“x” relates to the time (normally in seconds) that the ball has traveled

“a” relates to earth’s gravity with a value of: 16 (if the gravity is greater on anotherplanet then the downward pull would be greater)

“b” relates to the force with which the ball was hit “c” relates to the height above the ground when the ball was hit

Reminder: When the “a” value is negative the parabola will always be facing downwardmaking the vertex the maximum point.

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Part 1 : Hitting a golf ball into a tower?

While on the golf course last weekend Marc hit into the rough landing the ball behind a tall tree.His best option was to get it high enough to get over the tree and hopefully come down in thefairway for his next shot. So with a mighty swing he hit the ball into the air and was surprised tosee the ball hit near the top of a 300 foot tall tower that he had not noticed.

The formula for this shot is: 2( ) 16 120h x x x where h is the height of the ball and x is thenumber of seconds the ball is in the air.

A. How could Marc mathematically try to prove that he hit the ball near the top of the tower?Or is he just making up this story?

Marc could solve for the vertex of the formula.

B.

How high did Marc actually hit the ball?

2

1203.75

2 2( 16)

(3.75) 16(3.75) 120(3.75) 225 .

b x s

a

h ft

Part 2: Hitting the green in one shot!

Later during that same golf outing Marc decided to show off by trying to hit the green in one shot.So with his macho swing he said he hit the green as the ball hung in the air for 15 seconds.

The formula for this shot is: 2( ) 16 200h x x x

A. How can Marc provide proof that his mighty shot actually hung in the air for 15 seconds?Or is this just another one of his lies?

He can provide proof by solving for the x intercept to see if 15 x . Marc is exaggerating.

B. How long did the ball actually hang in the air?

12.5 x s

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Part 3: Hitting the baseball over the lights

Meanwhile at the ballpark Juan was practicing his hitting while talking to the girls standingaround watching. He started telling them how he hit the ball over the 50 foot light toweryesterday. The girls were unsure that he really did that.

The formula for this hit is: 2( ) 16 60 4h x x x where h is the height of the ball and x is thenumber of seconds the ball is in the air.

A. How can Juan provide proof to the girls that he actually hit the ball over the tower?

Juan can provide proof by solving for the vertex of the formula.

B.

How high did Juan actually hit the ball?

Juan hit the ball 60.25ft

Part 4: Hitting the ball into the parking lot

As Juan continued his hitting practice showing off his abilities, one of the balls flew over thecenter field stands and into the parking lot. “Did you see that shot” , he yelled at the girls.“The ball hung in the air for at least 10 seconds” , he exclaimed.

The formula for this hit is: 2( ) 16 83 4h x x x where h is the height of the ball and x is thenumber of seconds the ball is in the air.

A. How can Juan provide proof that the ball actually hung in the air for 10 seconds?

He can provide proof by solving for the x intercepts by using the quadratic formula.

B. How long did the ball hang actually hang in the air?

5.235 x s

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Part 5: What we learned

A. Compare the golf equation of2( ) 16 120h x x x and the baseball equation of

2( ) 16 60 4h x x x . What do you see as a major difference between these 2

formats?

The baseball is hit 4 units above the ground with less force.

B. How does this difference directly relate to hitting a golf ball or a baseball? (Hint: Rereadthe bullets on page 1)

The golf ball is hit from the ground, while the baseball is hit 4 units above the ground.

C. Rewrite the golf equation of2( ) 16 120h x x x to show Marc hitting the ball while

standing on top of a platform that is 10 feet off the ground.

2( ) 16 120 10 h x x x

Part 6: Using our quadratic equation knowledge match the following scenarios.

____ 1. Hitting a golf ball on the moon

____ 2. Hitting a golf ball on Earth from the top of a tall building

____ 3. Hitting a baseball on Jupiter

____ 4. Hitting a baseball on Earth by a bionic man

____ 5. Hitting a baseball on Earth by a 10 foot tall person

A. 2( ) 16 500 4h x x x B.

2( ) 50 50 4h x x x

C.2

( ) 8 100h x x x

D.

2( ) 16 75 8h x x x

E.2( ) 16 150 300h x x x

C

E

B

A

D

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Instructor’s Key:

Pan Balance – Quadratic Equations

Using your favorite internet browser, go to http://illuminations.nctm.org/Activity.aspx?id=3529 . Alternatively,perform a search for the following terms: illuminations pan balance expressions . It should be the first option.

Instructions:

Place an algebraic expression in each of the red and blue pans. These expressions may or may not include thevariable x . Enter a value for x , or adjust the value of x by moving the slider.

As the value of x changes, the results will be graphed. Use the Zoom In and Zoom Out buttons, or adjust thevalues for the x and y axes with the sliders, to change the portion of the graph that is displayed.

The Reset Balance button removes the expressions from the pans and clears the graph.

Try it out:

Use the Pan Balance to see the relationship between 25 x x and 2 x and solve 2

5 2 x x x

1. Type 25 x x in the left pan (red)


3. You should see the graph of each.

4. By using the slider, or trial and error, find the value(s) of x which balance the pans. Record the value(s). Thevalue(s) is/are the solution(s) of the equation 2

5 2 x x x .

5. Using symbolic methods (factoring/square root property/quadratic formula), solve the equation2

5 2 x x x exactly. Record your work and the solution(s).

6. Compare your answers for parts d) and e). Anything interesting?

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Using a similar process, solve each of the following equations. You should record on separate paper :a. The expression you used for the left pan. (How did you type it in to the website?)b. The expression you used for the right pan. (How did you type it in to the website?)c. The value(s) of x that balance the pan, estimating if necessary. (You may need to adjust the

settings on the graph to find some of the values.)d. The exact solution(s) that you find using symbolic methods. If the exact solution(s) involve

radicals, use your calculator to approximate the exact values

Appropriate“

work”

for the sample exercise:

A.

22 4 4 x x x 0.637,3.137 x

B.

22 4 5 7 x (What is the high point of the curve on the left hand side???)

1.550 x

C. 216 2 3

2 x x x

0.755,13.245 x

D. 21

6 4 52

x

(What is the low point of the curve on the left hand side???)10.243 x

E. 23 2 4 3 x x x

(Be Careful!) No solution

F. 2

5 7 2 x

(What is the high point of the curve on the left hand side???)7.236 x

G. 214 3 1

4 x x x

2 x

H. 2

3 2 9 12 x (What is the low point of the curve on the left hand side???)

0.646 x

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Instructor’s Key:

Variable exponents

Stop Stealing My Blueberries!!!

A.

Blueberry pickers on a farm were picking berries in such a waythat the number of berries in their bucket doubles every minuteafter beginning with one blueberry. If after just one hour thebucket was full, when was the bucket half full?

B.

On the same farm one picker picks blueberries, but stops after 5 minutes. Another pickerstops after 10 minutes. If the 2 pickers compare their buckets of blueberries, whatconclusion(s) do you think they will come up with?

C.

Another picker is preparing to put blueberries in his bucket, but is surprised to find that italready has 2 blueberries in it before he started. If it normally takes an hour to fill the

bucket, how will these 2 blueberries affect the time it takes to fill the bucket this time?

D.

Does your group all have the same answers as you? If not, can you convince them thatyou are correct?

---------------------------------------------------------------------------------------------------

Biological organisms are made up of cells. These cells divide at a rate which is described by the formula:

( ) 2 x f x where x = number of cellular divisions and f(x) = number of cells

1. How many single cell amoebas would be produced from one cell: after 10 cellular divisions? 1024

after 17 cellular divisions? 131072 after 25 cellular divisions? 33554432

2. If we double the number of cellular divisions, how do you think that will affect the number of cells at theend?It will square the number of cells at the end.

3. Calculate: f(5) = _____________________ f(10) = _______________________

4. How does your thinking on #2 compare with the calculations you found in #3?

5. The calculations you found in #3 should prove the conclusion you came up with in Part B of the blueberrypickers. Looking back, does your logic seem appropriate?

6. If bamboo cells divide every 3 hours then you should expect to have only one (1) cell division at the end of a3 hour period or a total of 2 cells. How many bamboo cells will you have after 9 hours? ___________________ after 21 hours? ___________________

after 2 days? ___________________

32 1024

8

128

65536

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7. A healthy tissue cell divides every 10 hours. After 120 hours, a single tissue cell has produced 256 cells.Would this indicate healthy tissue? _________ Why or why not?

8. Use our function: ( ) 2 x f x to complete the adjacent table.

Reasonable domain of : x 0,7 Number of cell divisions

Reasonable range of ( ) : f x 0,128 Total number of cells

9. Use the domain from your chart tofind the appropriate scaling for thex – axis and then mark the graphaccordingly.

10. Use the range from your chart tofind the appropriate scaling for they – axis and then mark the graphaccordingly.

11.

Plot the points from your table onthe adjacent graph and draw acurve through them smoothly.

---------------------------------------------------------------------------------------------------

Let’s look back to the questions about the blueberry bucket that doubles every minute. This time use the graphand table above to help you find your answers.

Question A: When would the bucket be half full?Hint: Since we do not know how big the bucket is, it could be full at any time.

59 minutes

Question B: How do the 5 minute and 10 minutes picker’s buckets compare? Hint: Remember that the bucket doubles each minute.

Picker two picked 32 times as many blueberries as picker one.

Question C: What affect does starting with 2 blueberries have? Hint: Use the hint from question A to get you started.

It will take one less minute to fill the bucket.

x f x 0 11 22 43 84 165 326 647 128

No

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Instructor’s Key:

Compound interest

The more you compound my account the better!!!Simple interest means the institution gives you interest only on the money you put into your account, butcompound means you additionally get interest on the interest the institution gives you. The formula below isused to determine the amount in your account if compounded annually.

1t

oP r

A = Amount in your account after interest is added.P0 = Initial principal (money you put into your account at the beginning)

r = Annual Percentage Rate (APR)t = Time in years

Note: All work on this lab will use accounts that are compounded annually!

1.

Calculator review:

A. Write the equation and then calculate howmuch you will have in a savings account after 7years if you initially deposited $5000, have anAPR of 4%, and made no further deposits?

75000(1 0.04)

$6579.66

A

A

B. Write the equation and then calculate howmuch a CD would be worth after 4 years if youinitially deposited $9000, have an APR of 6%,and made no further deposits?

49000(1 0.06)

$11362.29

A

A

C.

Write the equation and then calculate howmuch you would have in a stock fund after 10years if you initially deposited $8000, got anAPR of 9%, and made no further deposits?

108000(1 0.09)

$18938.91

A

A

2.

If you put $10,000 into a stock fund with an APRof 10% and I put $10,000 into a savings accountwith an APR of 5%, does it seem reasonable that you will have twice as much money after 20years if we don’t put any more money into ouraccounts? Why or why not?

Yes

Let’s see how these 2 accounts actuallydeveloped over the 20 years:

If using 5%: 20

10000 1 .05 A

If using 10%: 20

10000 1 .10 A

Amount in my account: ______________

Amount in your account: _____________

What conclusion(s) can you draw from your

answers? (Does doubling the APR effectivelydouble the compound interest earned overtime? )

$26532.98

$67274.99

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3. In this question we will look at what happenswhen we vary the amount of time for anaccount.

5000 1 .10t

A Derive the followinginformation from this equation (consider thebasic form listed at the beginning of theactivity):

__________ = Amount that is presently inyour account (y−axis).

__________ = Amount that you put into thebank initially.

__________ = APR

__________ = Years you left your money inthe bank (x−axis).

Put this equation into your calculator under Y1.Enter x for the letter t .

Table setup on calculator:TblStart = 0∆Tbl = 1Indpnt and Depend: Auto

Go to TABLE on you calculator to answer thefollowing questions:

How long would it take to double your money?8 years

How long would it take to have over $20,000?15 years

How long would it take to have at least$50,000? 25 years

Graph the equation on your calculator using thewindow: [0,28,2] [0,70000,5000] Draw your graph BELOW. Include axis andscaling info.

4. In this question we will look at what happenswhen we vary the initial amount.Let’s put different amounts of money into anaccount with a set interest rate over a set periodof time. How will that affect the money in youraccount at the end of the time period?

10

1 .065o

A P Derive the followinginformation from this equation:

__________ = Amount that is presently inyour account (y−axis).

__________ = Amount that you put into thebank initially (x−axis).

__________ = APR

__________ = Years you left your money inthe bank.

With an initial amount of $1 how much will be inyour account: $1.88

With an initial amount of $1000 how much willbe in your account: $1877.14

With an initial amount of $10,000 how muchwill be in your account: $18771.38

Graph the equation on your calculator using thewindow: [0,28000,2000] [0,70000,5000]. Drawyour graph BELOW. Include axis and scalinginfo.

5. What type of graph did you draw for exercise 3?Exponential

6. What type of graph did you draw for exercise 4?Linear

A

5000

10%

t

A

P0

6.5%

10

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7. After doing this activity and looking at the 2graphs from the previous page, whatconclusion(s) can you draw? (Contrast whathappens if you vary the time versus if you varythe initial amount.)

8. Suppose you invest $100 in an account thatearns 6% interest per year. How long will it takefor the investment to double. (Set up a variableequation and table of values answer thisproblem). [Write the equation you used ]

100(1 0.06)

200 100(1 0.06)

11.89 12 .

t

t

A

t yrs

What if the account earned 8% per year? Howlong would it take for the investment to double?

200 100(1 0.08)

9 .

t

t yrs

What if the account earned 9% per year? 12%per year?

200 100(1 0.09)

8 .

t

t yrs

200 100(1 0.12)

6 .

t

t yrs

9. The “Rule of 72” is a quick way to estimate

how long it will take for an investment, earningcompound interest, to double in value.Basically, it states that the amount of time itwill take for the investment to double isapproximately equal to 72 divided by theinterest rate (as a percentage)

Use the Rule of 72 to approximate how long itwill take for the $100 investment earning 6% todouble. Repeat the approximation for the otherinterest rates from exercise 8

How do your approximations using the Rule of

72 compare with your results writing theequations and using the tables of values?

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Instructor’s Key:

Finding an average

Better than Average!

Many situations in business require the use of an “average” function. One example might bethe determination of a function that models the average cost of producing an item. In thisactivity, you will build and use an “average” function.

When the iPhone was brand new, one could buy a 8 gigabyte model for roughly $600. There wasan additional $70 per month service fee to actually use the iPhone as intended. We will assumefor this activity that the monthly service fee does not change.

A. Determine the TOTAL cost of owning an iPhoneafter:

i. 2 months $740

ii. 4 months $880

iii. 6 months $1020

iv. 8 months $1160

B. Write a linear function that represents theTOTAL cost for the iPhone over x months.

( ) 600 70T x x

To determine the average cost per month (or theaverage monthly cost), we take the total cost anddivide by the number of months.

C. Determine the average monthly cost of owningan iPhone for:

i. 2 months $370

ii.

4 months $220

iii. 6 months $170

iv. 8 months $145

D. Write a rational function that represents theAverage Monthly Cost of owning an iPhone for

x months. [Hint: your answer to part (B) will beuseful here.]

( ) 600 70( )

T x x AMC x

x x

E. If your answer from part (D) is correct, youshould find that AMC(15) = 110.

a.

Verify that AMC(15) = 110 symbolicallythrough the use of your function from part(D).

(15) 110 AMC

b. Interpret what AMC(15) = 110 means in thecontext of the situation (using a completesentence).

The average monthly cost of owning aniPhone for 15 months is $110.

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F. Complete the following table of values for the Average Monthly Cost:

G.

Explain, using complete sentences, the result of the table. Is there anything surprising? (In particular, whathappens for large values of x ?)

H. Use your table values, along with your calculator, to sketch the graph of the function

10 20 30 40 50 60 70

50

100

150

200

250

300

350

400

450

500

550

600

650

I. Write and solve (symbolically) an equation that will determine how long it will take for theaverage monthly cost to reach $100. Answerwith a complete sentence. (Mark this point onyour graph)

600 70( ) 100

20

x AMC x

x

x months

J. Write and solve (symbolically) an equation that will determine how long it will take for theaverage monthly cost to reach $78. Answerwith a complete sentence. (Mark this point onyour graph)

600 70( ) 78

75

x AMC x

x

x months

K. Write and solve (symbolically) an equation that will determine how long it will take for theaverage monthly cost to reach $50. What doesthis mean in the context of the situation?

600 70( ) 50

30

x AMC x

x

x months

Since there is a preset $70 monthly fee, it is notpossible for the average monthly cost to reach$50, therefore the answer is invalid.

Months( x)

2 4 6 8 10 12 24 36 48 60 72

AverageMonthly

CostAMC( x)

370 220 170 145 130 120 95 86.67 82.5 80 78.33

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Instructor’s Key:

Rational Functions

Cost Benefit One business situation where mathematics is used is determining whether the cost of taking an action has a

reasonable value or benefit. In this activity we will look at the cost versus benefit of extracting a precious ore.

Suppose that the cost, in millions of dollars, of extracting x percent of a precious ore from a mine is given by the

function 6.3

100

xC x

x

.

A. Verify that 25 2.1C (showing work) and interpret, including units, in the context of the problem.

6.3(25)(25) 2.1

100 25C

B. Determine the cost of extracting 45% of the ore from the mine.

(45) $5.155C million

C. Complete the following table of values for the cost of extracting x percent of the precious ore:

Percent ( x) 0 10 20 30 40 50 60 70 80 90 100

Cost 0 0.7 1.58 2.7 4.2 6.3 9.45 14.7 25.2 56.7 ----

D. Explain, using complete sentences, the result of the table. Is there anything interesting or surprising aboutthe table?

The cost increases as the percentage of extracted ore increases. Once 100% of the ore is extracted, the costfunction becomes undefined.

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E. Complete the following table of values for the Cost of extracting x percent of the precious ore:

Percent ( x) 92 94 95 96 97 98 99 99.5 99.75 99.9 99.99

Cost 72.45 98.7 119.7 151.2 203.7 308.7 623.7 1253.7 2513.7 6293.7 62994

F.

What happens to the cost of extraction as the percentage approaches 100%. Use complete sentences.The cost of extraction increases exponentially.

G. Use your table values, along with your calculator, to sketch the graph of the function. Note any asymptoteswith dotted lines.

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

110

H. Write and solve (symbolically) an equation that will determine the percentage of ore that can be mined for$60 million. (Mark this point on your graph)

6.360

100

90.50%

x

x

x

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Instructor’s Key:

Modeling Light Intensity with a Rational Function

From a Distance

The activity illustrates a natural phenomenon. As you increase the distance from a light source, the perceivedintensity of the illumination will decrease. To investigate the phenomenon, we will:

Use a CBL to help collect realistic data points Create a scatter plot of the data points

Use the data points to determine an algebraic function that models the data Use the algebraic function to answer some questions

1. Use a CBL to help collect data points

Set up the CBL to collect data:o Plug the light sensor in to CH1.o Plug the calculator in to the CBL.o Turn on the calculator.o Set up the CBL application:

Press APPS button. Scroll down to EasyData and press Enter. The calculator should start displaying readings of the light intensity measured by the sensor.

Collect Data:o Place a meter stick next to the light sensor. (Either place both on the floor or put two level desks

together)o Adjust the flashlight so you have a fine beam.o

Move the flashlight so that it is 20cm from the sensor. Adjust the flashlight so that it is pointingdirectly at the sensor.

o The gauge on the calculator should provide a measure of the intensity. Record thatmeasurement, rounding to two decimal places.

o Repeat for the other distances provided on the table.

Distance fromsensor (cm)

20 30 40 50 60 70 80 90 100

IntensityGauge reading

0.31 0.14 0.08 0.05 0.04 0.03 0.02 0.02 0.02

a value(calculate later)

122 125.1 123.2 117.5 126 127.4 140.8 162 150

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2. Create a scatter plot of the data points

Using the axes provided, plot the data points from the table. Consider the Distance from the sensor as theinput (x variable ) and the Intensity as the output ( y variable ). Be sure to indicate the scale that you use oneach axis.

3. Use the data points to determine an algebraic

function that models the data

A. Considering the table of values that youhave recorded, could a linear functionreasonably model the data? Why or whynot? (Do the points show a constant rate ofchange?)

No. (answers may vary)

B. Considering the scatter plot you havedrawn, could a linear function reasonablymodel the data? Why or why not? (If it wasa linear function, what would be true aboutthe points on the scatter plot?)

No. (answers may vary)

C. It is known that this phenomenon can bemodeled by a function of the form

2( )

a

f x x

, where x represents the distancefrom the sensor and ( ) f x represents theIntensity. The a is a parameter that can bedetermined from the data set.Now we will determine the value of theparameter a . We do this by substituting a

data point into the model2

( ( ) )a

f x x

and

solving for a .

See table

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D. Determine the value of a for each of the datapoints that you collected. Round each a value to three decimal places. Write these avalues to the below the data points in thetable of values. Then average all of the a values that you found from the data points.What is your average a value?

132.667a

E. Now that we have the a value, we can reviseour model. Replace the a in the model

2( )

a f x

x with the specific value of a that

you determined in D). Write the revisedmodel!

2

132.667( ) x

f x

F. Using your calculator, graph the functionfrom part 3E, and use this to sketch thegraph along with the scatterplot on theprevious page of this activity. If you havefollowed the directions, you should see thatthe graph reasonably models the points onthe scatterplot. (Note: The graph will notpass through all of the data points, neithershould your sketch!)

See graph

4. Use the algebraic function to answer somequestions. Use the model you wrote in part 3Eto answer these questions. Show your work!

A. Using the function, determine the expected

intensity at a distance of 150cm. Then usethe sensor to measure the intensity. Did thefunction give a reasonable value of theintensity?

0.0059

B. Using the function, how far away from thelight source should you be to observe anintensity of 0.9? (You should write and solvean equation to answer this part.) Then, usethe sensor to find a distance that yields thatintensity. Are the value from the model andthe value from the experiment reasonably

close?

12.141cm

C. Using the function, what happens to theintensity if you double the distance from thelight source? (Consider a certain distanceand use the function to determine theintensity. Then double the distance anddetermine the intensity. Do this for severalpairs of distances. You should see a pattern

develop. Express that pattern. Hint: youshould be able to multiply the first intensityby some number and get the secondintensity.)

If you double the distance from the lightsource, the intensity decreases by a factor of4.

D. Using the function, what happens to theintensity if you triple the distance from the

light source?(Use a similar process as the previous exercise!)

The intensity decreases by a factor of 9.

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Instructor’s Key:

Rational expressions

What makes you think this is Rational?

Rational expressions are fractions with one or more variables in the denominator and possibly oneor more variables in the numerator. These expressions are worked in the same manner asfractions that do not have variable(s) in them.

Applications:

1. When an engineer designs a highway curve, how does he know if it will be safe for the carsthat use it? Formula for the radius (R) of a curve with a banking elevation or slope (m):

1600

( )15 2

R mm

2.

How does your camera know when the object you are trying to take a picture of is in focus?Formula for the distance (D) to the object while knowing the focal length (F) and the

distance (S) from the lens to the film:1 1 1

S F D

3. It normally takes Julius 2 hours to mow the yard, but because he is in a hurry he asks his sonto help him. If just his son was doing the yard work it would take him 3 hours. How long willit take if both are working together? Formula for finding the time it takes to do job together:

1 2

1 1 1both

T T T

4. What real life problem can you think of that would use rational expression(s) or a rationalequation to help you solve it?

1 3

5. A student turned in this work.2

2

6 13 6 13 4

6 8 3

x x x

x

1 4What’s wrong?

Step 1 to simplifying a rational expression: Completely factor all expressions.

Work rational expressions the same way you do fractions:o

Multiplication: Reduce common factor(s) and multiplyo

Division: Find the reciprocal of what follows the division sign, then multiplyo

Addition: Find a common denominator and then add numeratorso Subtraction: Find a common denominator and then subtract numerators

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Match the following rational expressions to their simplified form:

_____ 1.2

3

2 6

8

x x

x

_____ 2.2

2

6 13 6

6 7 3

x x

x x

_____ 3.2

2

12 19 4

8 10 3

x x

x x

_____ 4.2

5 6

20 9

x xy

y x

_____ 5.2 2

2 2

2

9 9

a b a ab b

a b a b

_____ 6.2 2

2 2

6 19 10 2 5 7

10 27 28 2 7 5

x x x x

x x x x

_____ 7.2 2

2 2

6 8 3 6

6 5 1 3 13 4

x x x x

x x x x

_____ 8.2 2

2 2

8 7 1 64 1

4 5 1 (4 1)

x x x

x x x

_____ 9.2 2

2 3

4 4

x x

x x

_____ 10.4 1 7 4

2 3 2 3

x x

x x

_____ 11.4 2 3

2 2 3

x x

x x

_____ 12.2 3 4

4 2 3

x x

x x

_____ 13.2 2

1 2 3

4 2

x x

x x x

_____ 14.

2 2

2 2 2 3

6 13 6 3 11 6

x x

x x x x

_____ 15.2 2

5 1 5 1

6 13 5 12 1

x x

x x x x

A. 1

9

B. 3 1

2 3

x

x

C. 3 42 3 x x

D.

3 2 2 7

5 4 2 7

x x

x x

E.

2 5 1 2

3 1 4 1 2 5

x x

x x x

F.

2

3

4

x

x

G. 2

3

4

x

x

H.

23 7 5

2 2 1

x x

x x x

I.

2

6

x

J.

26 5 12

2 2 3

x x

x x

K.

24

3 2 1

x

x x

L.

3 22 2 12 9

3 2 2 3 3

x x x

x x x

M. 3 2

3 1

x

x

N.

23 20 7

2 3 4

x x

x x

O. 4 1

8 1

x

x

G

M

C

I

A

D

K

O

F

B

J

N

H

L

E

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Instructor’s Key:

Modeling with a Radical Function

Distance to the Horizon

Using your favorite internet browser, go to http://illuminations.nctm.org/Activity.aspx?id=4128 . Alternatively,

perform a search for: illuminations distance horizon . It should be the first link.

The site illustrates a natural phenomenon. As you increase your elevation, the distance that you can see to thehorizon also increases. In this activity, you will

use the website to collect realistic data points create a scatter plot of the data points use the data points to determine an algebraic function that models the data Use the algebraic function to answer some questions

1. Use the website to collect realistic data points.

Move the slider to various elevation levels (Height above sea level). Record the elevation level and thecorresponding Distance to Horizon in the table of values. Collect at least 10 distinct data points, keeping theelevation levels in numerical order and under 20000 ft. One data point has been pre recorded for you.

Height above sea level (ft.) Distance to Horizon (miles)

0 0

2150 56.8

4032 77.8

5914 94.2

8065 110.0

9947 122.2

12098 134.8

13980 144.9

15862 154.3

18012 164.5

19894 172.9

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2. Create a scatter plot of the data points

Using the axes provided, plot the data points from the table. Consider the Height above sea level as theinput (x variable) and the Distance to Horizon as the output (y variable). Be sure to indicate the scale thatyou use on each axis.

x

y

3. Use the data points to determine an algebraic function that models the data

A. Considering the table of values that you have recorded, could a linear function reasonably model thedata? Why or why not?

No.

B. Considering the scatter plot you have drawn, could a linear function reasonably model the data? Why orwhy not?

Yes. If we disregard the first point, the plotted points look linear.

It is known that this phenomenon can be modeled by a function of the form ( ) f x a x , where x represents the Height above sea level and y represents the Distance to Horizon. The a is a parameter that

can be determined from the data set.

C. Is ( ) f x a x a linear function? Why or why not? (What does a linear function look like,

symbolically? )

No.

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Now we will determine the value of the parameter a . We do this by substituting a data point into the model

( ( ) f x a x ) and solving for a . For example, if one of the data points was (500,10), we could write that

10 500a . Solving for a , we find that10

.447500

a

D. Determine the value of a for each of the data points that you collected [exclude the point (0,0)]. Roundeach a value to three decimal places. Write these a values to the right of the data points in the table of

values. If you do this correctly, you should find that the a values will all be close to a certain number.

What is the a value? 1.225a

E. Now that we have the a value, we can revise our model. Replace the a in the model ( ) f x a x withthe specific value of a that you determined in D). Write the revised model!

( ) 1.225 f x x

4.

Use the algebraic function to answer some questions. Use the model you wrote in part 3E to answer thesequestions. Show your work!

A. Assuming perfect weather conditions, how far on the horizon could you see from the top of Mt. McKinley(North America’s tallest peak), which is 20320 feet tall?

(20320) 1.225 20320 174.62 . f mi

B. The distance from Orlando, Florida to Lakeland, Florida is 54 miles. Assuming perfect weatherconditions, how high above Orlando would you need to be to see Lakeland on the horizon?

54 1.225

1943.19 .

x

x ft

C. Use an internet search to determine whether or not any buildings in Orlando are tall enough to allow youto see Lakeland on the horizon.

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Instructor’s Key:

Imaginary numbers

The Cycle of i

9 This problem asks the following question: What number if it was multiplied by itselfwould equal 9. The answer will be 3 because if 3 were multiplied by itself you would get 9.

Some examples of square roots are: 25 5 ; 81 9; 49 7

Previously we also learned that multiplying 2 numbers with the same sign would always bepositive.

1 This problem presents a mathematical riddle. 1 1 This seems easy enough but it isnot possible to find a number to multiply by itself and come up with −1. The mathematical worldhas chosen to use the word “ imaginary” when confronted with this riddle. We will use “i” toreference this imaginary answer.

1i

1i

2

2

1 1i 4 1i 2 1i

3 2 1

3 1 1 1 1i i i

3i i

4 2 2

4 1 1 1 1 1 1i

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Simplify: Use information from the previous page:

1. 45 1i i ii i 5.

49 4 1 1i i i ii i

2. 46 2 1 1 1i ii 6.

410 2 2( ) 1i i i

3. 47 3 1i i i ii 7.

411 2 3( )i i i i

4.

4 48

1 1 1i ii 8. 312 4

( ) 1ii

Mathematics is full of PATTERNS. Looking at your work on the 8 questions above and the notesfrom the previous page what pattern(s) do you and your group find?

Simplify:

9.

730 4 2 7

1 1 1i i i

10. 3

413i ii i

11.

10 242 4 1ii i

12.

103

41 4 24 1i i i

13. 2

112 8

4 1ii

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Simplify: Put your answer into standard form: a+bi .Examples:

2 3 7 5i i

2 7 3 5i i 9 2i

2

2 3 7 5

14 10 21 15

14 11 15 1

14 11 15

29 11

i i

i i i

i

i

i

2

2

2 3 2 3 7 5

7 5 7 5 7 5

2 3 7 5

7 5 7 5

14 10 21 1549 35 35 25

1 31

74

1 31

74 74

i i i

i i i

i i

i i

i i i

i i i

i

i

14. 5 2 7 33 17i ii

15. 6 4 7 2 13 2i i i

16. 6 2 6 2 40i i

17. 85 3 x i x ii

Documents

MAT1033C - 2014-2015 - Instructor