lesson 7.1 skills practice - lths...

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Chapter 7 Skills Practice 495

7

Lesson 7.1 Skills Practice

Name Date

Unequal EqualsSolving Polynomial Inequalities

Problem Set

Analyze the graph. Identity the set of x-values to represent when p(x) , 0 and when p(x) . 0.

1.

x

y

21 10 2 3 4

p(x)

222324

1

2

21

22

23

24

3

4

The function p(x) , 0 when {22 , x , 2}.

The function p(x) . 0 when x , 22 x . 2

.

2.

x

y

22 20 4 6 8242628

4

8

24

28

212

216

12

16 p(x)

The function p(x) , 0 when x , 21 3 , x , 5

.

The function p(x) . 0 when 21 , x , 3 x . 5

.

451453_IM3_Skills_CH07_495-528.indd 495 03/12/13 2:45 PM

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496 Chapter 7 Skills Practice

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Lesson 7.1 Skills Practice page 2

3.

x

y

22 20 4 6 8242628

50

100

250

2100

2150

2200

150

200 p(x)

The function p(x) , 0 when 26 , x , 22 1 , x , 5

.

The function p(x) . 0 when x , 26 22 , x , 1

x . 5 .

4.

x

y

22 20 4 6 8242628

20

40

220

240

260

280

60

80

p(x)

The function p(x) , 0 when x , 27 22 , x , 1

6 , x , 9 .

The function p(x) . 0 when 27 , x , 22 1 , x , 6

x . 9 .

5.

x

y

21 10 2 3 4222324

2

4

22

24

26

28

6

8

p(x)

The function p(x) , 0 when x , 0 x . 0

.

The function p(x) is never greater than zero.


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6.

x

y

21 10 2 3 4222324

2

4

22

24

26

28

6

8p(x)

The function p(x) , 0 when {x . 2}.

The function p(x) . 0 when x , 1 __ 2

1 __ 2 , x , 2

.

Use a graphing calculator to solve each inequality. Round decimals to the nearest hundredths.

7. 21 , 3x2 1 1

I graphed y1 5 3 x 2 1 1 and y2 5 21.

Using the intersection function of the calculator, I determined that 21 , 3 x 2 1 1 when

x , 22.58 or x . 2.58.

8. 4x2 2 5 # 9

I graphed y1 5 4 x 2 2 5 and y2 5 9.

Using the intersection function of the calculator, I determined that 4 x 2 2 5 # 9 when

21.87 # x # 1.87.

9. 23 # x3 1 2x 1 6

I graphed y1 5 x 3 1 2x 1 6 and y2 5 23.

Using the intersection function of the calculator, I determined that 23 # x 3 1 2x 1 6 when

x $ 21.76.


Name Date


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10. 210.5 . 21.5x2 2 15.5x

I graphed y1 5 21.5 x 2 2 15.5x and y2 5 210.5.

Using the intersection function of the calculator, I determined that 210.5 . 21.5 x 2 2 15.5x when

x , 210.97 or x . 0.64.

11. 21.2x3 2 4x2 1 15x # 1

I graphed y1 5 21.2 x 3 2 4 x 2 1 15x and y2 5 1.

Using the intersection function of the calculator, I determined that 21.2 x 3 2 4 x 2 1 15x # 1 when

25.59 # x # 0.07 or x $ 2.19.

12. 26.6 , 212.4x2 1 2.2x3 1 0.8x4

I graphed y1 5 212.4 x 2 1 2.2 x 3 1 0.8 x 4 and y2 5 26.6.

Using the intersection function of the calculator, I determined that 26.6 , 212.4 x 2 1 2.2 x 3 1 0.8 x 4 when x , 25.51 or 20.70 , x , 0.81 or x . 2.65.

Solve each inequality by factoring and sketching. Use the coordinate plane to sketch the general graph of the polynomial in order to determine which values satisfy the inequality.

13. x2 2 3x 2 10 , 0

(x 2 5)(x 1 2) 5 0

x 5 5, 22

The boxes represent the x-values where the polynomial is less than zero. The ovals represent the x-values where the polynomial is greater than zero.

The function x2 2 3x 2 10 , 0 when 22 , x , 5.

x

y

22 20 4 6 8242628

4

8

24

28

212

216

12

16


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14. x3 1 3x2 1 x 1 3 $ 0

(x3 1 3x2) 1 (x 1 3) 5 0

x2(x 1 3) 1 1(x 1 3) 5 0

(x2 1 1)(x 1 3) 5 0

x 5 23

The box represents the x-values where the polynomial is less than zero. The oval represents the x-values where the polynomial is greater than zero.

The function x3 1 3x2 1 x 1 3 $ 0 when x $ 23.

x

y

21 10 2 3 4222324

2

4

22

24

26

28

6

8

15. 2x3 1 6x2 2 20x # 0

2x3 1 6x2 2 20x 5 0

2x(x2 1 3x 2 10) 5 0

2x(x 2 2)(x 1 5) 5 0

x 5 0, 2, 25


The function 2x3 1 6x2 2 20x # 0 when x # 25 or 0 # x # 2.

x

y

22 20 4 6 8242628

20

40

220

240

260

280

60

80

16. x3 1 4x2 1 x 2 6 . 0

x3 1 4x2 1 x 2 6 5 0

(x 2 1)(x2 1 5x 1 6) 5 0

(x 2 1)(x 1 2)(x 1 3) 5 0

x 5 1, 22, 23


The function x3 1 4x2 1 x 2 6 . 0 when 23 , x , 22 or x . 1.

x

y

21 10 2 3 4222324

2

4

22

24

26

28

6

8


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17. x4 2 25x2 1 144 $ 0

x4 2 25x2 1 144 5 0

(x2 2 9)(x2 2 16) 5 0

(x 1 3)(x 2 3)(x 1 4)(x 2 4) 5 0

x 5 63, 64


The function x4 2 25x2 1 144 $ 0 when x # 24, 23 # x # 3, or x $ 4.

x

y

21 10 2 3 4222324

40

80

240

280

2120

2160

120

160

18. x4 2 8x3 1 2x2 1 80x 2 75 # 0

x4 2 8x3 1 2x2 1 80x 2 75 5 0

(x2 2 10x 1 25)(x2 1 2x 2 3) 5 0

(x 2 5)2(x 2 1)(x 1 3) 5 0

x 5 5, 5, 1, 23

The box represents the x-values where the polynomial is less than zero. The ovals represent the x-values where the polynomial is greater than zero.

The function x4 2 8x3 1 2x2 1 80x 2 75 # 0 when 23 # x # 1 or x 5 5.

x

y

22 20 4 6 8242628

40

80

240

280

2120

2160

120

160



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Name Date

America’s Next Top Polynomial Model Modeling with Polynomials

Vocabulary

Explain each key term in your own words.

1. regression equation

A regression equation is a function that models the relationship between 2 variables in a scatter plot.

2. coefficient of determination

The coefficient of determination (R2) measures the strength of the relationship between the original data and its regression equation. The value ranges from 0 to 1 with a value of 1 indicating a perfect fit between the curve and the original data.

Problem Set

Create a scatter plot of the data. Predict the type of polynomial that best fits the data. Explain your reasoning.

1. The table of values represents the temperature of 2 liters of water in a teakettle over time as it is set to boil and then cools down.

The data increases, then decreases. So, the data could be represented by a quadratic equation.

Time (minutes) Temperature (°C)

0 15

10 40

15 90

20 100

30 80

45 50

60 25

x

y

100 20 30 40

10

20

30

40

50

Time (minutes)

60 70 80 90

50

Tem

per

atur

e (°

C)

60

70

80

90


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2. The table of values represents the number of work hours for which Jay was hired throughout the year.

The data increases, then decreases. So, the data could be represented by a quadratic equation.

Time Since December (months)

Work Time (hours)

1 40

3 100

5 160

7 140

9 160

11 60

x

y

20 4 6 8

20

40

60

80

10

Time Since December (months)

12 14 16 18

100

Wo

rk T

ime

(ho

urs)

120

140

160

180

3. The table of values represents the download speed in kilobytes per second (kBps) of Sue’s Internet connection throughout the day.

The data increases, then decreases, then increases, then decreases, and finally increases. So, the data could be represented by a quintic equation.

Time Since 7:00 am (hours)

Download Speed (kBps)

1 5775

3 7000

5 4505

7 6855

9 6540

11 5020

13 3780

15 4250 x

y

20 4 6 8

1000

2000

3000

4000

10

Time Since 7:00 AM (hours)

12 14 16 18

5000

Do

wnl

oad

Sp

eed

(kB

ps)

6000

7000

8000

9000


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Name Date

4. The table of values represents the annual attendance in hundred thousands at a theme park.

The data increases, then decreases. So, the data could be represented by a quadratic function.

Time Since 1998 (years)

Attendance (hundred thousands)

0 13.4

1 17.9

2 19.2

3 22.1

4 18.3

5 16.8

6 11.2

x

y

10 2 3 4

3

6

9

12

5


6 7 8 9

15

Att

end

ance

(hun

dre

d t

hous

and

s)

18

21

24

27

5. The table of values represents the natural gas usage in quadrillion BTU in the US over several decades.

The data increases, then decreases, and finally increases. So, the data could be represented by a cubic function.


Gas Usage(quadrillion BTU)

0 12.4

10 21.8

20 20.4

30 19.3

37 22.6

50 24.6

x

y

50 10 15 20

3

6

9

12

25


30 35 40 45

15

Gas

Usa

ge

(qua

dri

llio

n B

TU

)

18

21

24

27


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6. The table of values represents the number of US $20 bills produced each year.

The data decreases, then increases, then decreases, then increases, then decreases, and finally increases. So, the data could be represented by a 6th degree polynomial.


Number of U.S. $20 Bills Produced

(hundred thousands)

0 30.6

1 8.9

2 19.7

3 6.3

4 7.2

5 22.7

6 9.0

7 15.7

x

y

10

2 3 4

212

26

6

5


6 7 8 9

12

Num

ber

of

U.S

. $20

Bill

s P

rod

uced

(hun

dre

d t

hous

and

s)

18

24

30

36

Use a graphing calculator to determine the regression equations for the data from Problems 1 through 6. Round decimals to the nearest thousandth. Sketch each regression equation on the coordinate plane with the corresponding scatter plot. How well does each regression equation model the data? Explain your reasoning.

7. Regression equation for Problem 1: The regression equation is approximately f(x) 5 20.078x2 1 4.632x 1 19.100 with a coefficient of determination of 0.760. The equation is an acceptable fit for the data.

See graph.

8. Regression equation for Problem 2: The regression equation is approximately f(x) 5 24.286x2 1 55.143x 2 16.571 with a coefficient of determination of 0.891. The equation is a pretty good fit for the data.

See graph.

9. Regression equation for Problem 3: The regression equation is approximately f(x) 5 0.675x5 2 25.706x4 1 348.561x3 2 2039.792x2 1 4851.468x 1 2747.452 with a coefficient of determination of 0.733. The equation is an acceptable fit for the data.

See graph.


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Name Date

10. Regression equation for Problem 4: The regression equation is approximately f(x) 5 20.927x2 1 5.218x 1 13.388 with a coefficient of determination of 0.945. The equation is a very good fit for the data.

See graph.

11. Regression equation for Problem 5: The regression equation is approximately f(x) 5 0.0005x3 2 0.044x2 1 1.09x 1 12.969 with a coefficient of determination of 0.873. The equation is a pretty good fit for the data.

See graph.

12. Regression equation for Problem 6: The regression equation is approximately f(x) 5 0.188x6 2 3.964x5 1 31.443x4 2 116.031x3 1 197.954x2 2 131.327x 1 30.605 with a coefficient of determination of 0.999. The equation is nearly a perfect fit for the data.

See graph.

Use the data and regression equations from Problems 1 through 12 to make predictions for each problem situation. Explain your reasoning.

13. Charlotte wants to make sure the hot chocolate is not too hot for her daughter. She wants to pour the water at about 60°C. Use the regression equation for Problem 1 to predict after how many minutes she should pour the water from the kettle.

Using the regression equation, I solved f(x) 5 60 to predict when the water is about 608C.

Charlotte should pour the water after approximately 11 minutes or 48 minutes.

14. Use the regression equation for Problem 2 to predict how many hours of work Jay will be hired for in October.

Using the regression equation, I calculated f(10) to predict how many hours of work Jay will be hired for in October.

Jay will be hired for approximately 106 hours in October.


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15. Sue gets off work at 7:00 pm and wants to download some music. Use the graph from Problem 3 to predict the download speed she should expect at that time.

Using the graph, I determined f(12) to predict the download speed at 7:00 pm.

The download speed will be approximately 4500 kBps at 7:00 pm.

16. If the theme park in Problem 4 opened in 1995, explain why the regression equation would not give an accurate prediction of attendance that year.

The x-value for 1995 is x 5 23. Using the regression equation, I calculated f(23) 5 210.6 (hundred thousands). It is not possible to have a negative number of people in attendance.

17. Use the graph from Problem 1 to predict the temperature of the water be after 64 minutes. Is this likely? Explain your reasoning.

Using the graph, I determined f(64) to be a negative value. Therefore, after 64 minutes, the water will be below 0°C. It is possible, but not likely that the water will be frozen after an hour. It is unlikely that water cooling in a kettle will reach the freezing point, unless the ambient temperature is at or below 0ºC.

18. Use the regression equation from Problem 5 to predict the amount of natural gas the US used in 2000.

Using the regression equation, I calculated f(40) to predict the amount of natural gas the US used in 2000.

The US used approximately 18.2 quadrillion BTU of natural gas in 2000.

19. Use the regression equation from Problem 5 to predict the amount of natural gas the US will use in 2020.

Using the regression equation, I calculated f(60) to predict the amount of natural gas the US will use in 2020.

The US will use approximately 28 quadrillion BTU of natural gas in 2020.

20. Use the regression equation from Problem 6 to predict the number of $20 bills made in 2004. Is this likely? Do you think the regression equation is a good match for the data? Explain your reasoning.

Using the regression equation, I calculated f(21) to predict the number of $20 bills made in 2004. In 2004, the US made 51,151,200 bills. No, this is not likely because it is much more than the average amount of bills made each year.

The equation fits the data points very well. The regression equation may not be the best equation to use as an equation with a larger degree might have a coefficient of determination closer to 1.



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Connecting PiecesPiecewise Functions

Vocabulary

Write a definition for the term in your own words.

1. piecewise function

A piecewise function includes different functions that represent different parts of the domain.

Problem Set

Sketch each piecewise function on the coordinate plane.

1. p(x) 5 (x 1 3)2, x , 0

(x 2 3)2, x $ 0

x

y

p(x)

22 20 4 6 8242628

2

4

22

24

26

28

6

8

2. b(x) 5 1 __ 4

x2, x # 2

2 1 __ 2 (x 2 2)2, x . 2

x

y

21 10 2 3 4222324

1

2

21

22

23

24

3

4b(x)


Name Date


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3. f(x) 5

2x 1 1, x , 0

(x 2 2)2 2 3, 0 # x # 2

23, x . 2

x

y

21 10 2 3 4222324

1

2

21

22

23

24

3

4f(x)

4. g(x) 5 2x 1 12, x , 23

2x4 1 9x2, 23 # x # 3

7x 2 42, x . 3

x

y

22 20 4 6 8242628

6

12

26

212

218

224

18

24

g(x)

5. t(x) 5 2 1 __ 4 (x 2 2)2 1 3, x # 2

22x 1 4, x . 2

x

y

21 10 2 3 4222324

1

2

21

22

23

24

3

4t(x)


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Name Date

6. m(x) 5 x2, x # 21

2x3 1 3, 21 , x # 1

2x2 1 4, x . 1

x

y

21 10 2 3 4222324

1

2

21

22

23

24

3

4

m(x)

Write the equation of each piecewise function given its graph.

7.

x

y

22 20 4 6 8242628

2

4

22

24

26

28

6

8 b(x)

b(x) 5 9, x , 23

x2, 23 # x # 3

9, x . 3


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8.

x

y

21 10 2 3 4222324

1

2

21

22

23

24

3

4

c(x)

c(x) 5 (x 1 2)2 2 2, x , 22

x, 22 # x # 2

2(x 2 2)2 1 2, x . 2

9.

x

y

21 10 2 3 4222324

1

2

21

22

23

24

3

4

d(x)

d(x) 5 x3, x # 1

2x 1 1, x . 1

10.

x

y

22 20 4 6 8242628

2

4

22

24

26

28

6

8

f(x)

f(x) 5 23, x # 24

(x 1 4)(x 1 1)(x 2 2), 24 , x , 2

3, x $ 2


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Name Date

11.

x

y

21 10 2 3 4222324

2

4

22

24

26

28

6

8

g(x)

g(x) 5 2(x 1 3)2, x , 22

2x3, 22 # x , 2

(x 2 3)2, x $ 2

12.

x

y

22 20 4 6 8242628

2

4

22

24

26

28

6

8 h(x)

h(x) 5 (x 1 3)2 2 4, x , 21

(x 1 1)2, 21 # x # 1

(x 2 1)2 1 4, x . 1


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Analyze the scatter plot. Determine a regression equation over each interval to write a piecewise function that models the data. Round decimals to the nearest thousandth. Then, graph the piecewise function on the scatter plot.

13.

x

y

1

3

0 2 3 4

4

5

6

7

5 6 7 8 9 10 11

8

9

10

11

12

Answers will vary.

f(x) 5

20.286x2 2 1.4x 1 9.7 , 0.5 # x , 3.5

0.438x 1 1.574 , 3.5 # x , 9

0.826x2 2 15.245x 1 77.265 , 9 # x # 12

14.

x

y

20 4 6 8

5

10

15

20

10 12 14 16 18

25

30

35

40

45

Answers will vary.

f(x) 5

4.167x2 2 5.883x 1 25 , 0 # x # 3

23.214x 1 55.357 , 3 , x # 8

0.024x4 2 1.371x3 1 29.496x2 2 279.42x 1 993.553 , 8 , x # 20


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Name Date

15.

x

y

0.10 0.2 0.3 0.4

2

4

6

8

0.5 0.6 0.7 0.8 0.9

10

12

14

16

18

Answers will vary.

f(x) 5

21777.278x3 1 1712.187x2 2 495.635x 1 51.743 , 0 , x # 0.5

146.667x2 2 170x 1 53.133 , 0.5 , x # 0.8

14 , 0.8 , x # 1.0

16.

x

y

20

8

0 30 40 50

12

16

20

24

10 60 70 80 90 100

28

32

36

40

44

Answers will vary.

f(x) 5

0.343x 1 8.286 , 10 # x # 35

20.047x2 1 5.014x 2 88.405 , 35 , x # 75

0.0189x2 2 4.120x 1 235.06 , 75 , x # 110


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17.

x

y

20 4 6 8

2

4

6

8

10 12 14 16 18

10

12

14

16

18

Answers will vary.

f(x) 5

0.361x3 2 2.69x2 1 6.829x 2 0.119 , 0 # x # 5

2x3 1 21.5x2 2 153.5x 1 370 , 5 , x # 9

0.333x3 2 11.5x2 1 132.17x 2 500 , 9 , x # 14

20.5x3 1 23.5x2 2 368x 1 1926 , 14 , x # 18

18.

x

y

840

12 16 20

1

2

3

4

24 28 32 36 40

5

6

7

8

9

Answers will vary.

f(x) 5

0.643x 2 2.643 , 4 # x , 8

20.00966x4 1 0.541x3 2 11.149x2 1 100.5x 2 323.61, 8 # x , 20

4.5 , 20 # x , 32

0.1014x2 2 7.412x 1 135.837 , 32 # x # 44


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Name Date

Modeling GigModeling Polynomial Data

Problem Set

Use Data Sets A through F to solve the following problems.

A: The table shows the average share price of WXY company stock since 1994.


WXY Share Price (dollars)

1 5

3 10

5 25

7 15

9 15

11 40

13 100

15 150

17 300

19 500

B: The table shows the number of less than 100- mile trips in the US over the Thanksgiving holiday.

Time Since Monday before Thanksgiving

(days)

Number of Less Than 100-Mile Trips

(millions)

1 12

2 19

3 27

4 23

5 24

6 18

C: The table shows the relationship between J. Company’s advertising spending and their profit.

Advertising Spending (hundred dollars)

Profit (ten thousand dollars)

0 2

2 6

6 14

10 18

12 20

14 16

16 12

18 8

20 4

D: The table shows the number of tons of apples harvested per acre since 1990.


Tons of Apples (thousands)

1 4.9

3 5.4

5 5.2

7 5.4

9 5.9

11 6.3

13 7.1

15 9.7


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E: The table shows the average home mortgage interest rate since 1999.


Interest Rate (%)

0 6.5

1 8.5

2 7.0

3 6.5

4 6.0

5 5.5

6 6.0

7 7.0

8 5.0

9 4.5

F: The table shows the relationship between shell length of a turtle and number of eggs laid per clutch.

Shell Length (millimeters)

Number of Eggs Laid per Clutch

285 3

290 7

300 9

305 10

310 10

315 9

320 7

330 5

335 2

Create a scatter plot for the data.

1. Data Set A

x

y

310

5 7 9

50

100

150

200

11 13 15 17 19

250

300

350

400

450

Time Since 1994 (Years)

WX

Y S

hare

Pri

ce (d

olla

rs)


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Name Date

2. Data Set B

x

y

10 2 3 4

3

6

9

12

5 6 7 8 9

15

18

21

24

27

Time Since Monday before Thanksgiving (days)

Num

ber

of

Less

Tha

n10

0-M

ile T

rip

s (m

illio

ns)

3. Data Set C

x

y

20 4 6 8

2

4

6

8

10 12 14 16 18

10

12

14

16

18

Advertising Spending (hundred dollars)


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4. Data Set D

x

y

20 4 6 8

1

2

3

4

10 12 14 16 18

5

6

7

8

9


To

ns o

f A

pp

les

(tho

usan

ds)

5. Data Set E

x

y

10 2 3 4

1

2

3

4

5 6 7 8 9

5

6

7

8

9


Inte

rest

Rat

e (%

)


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Name Date

6. Data Set F

x

y

2850

295 305

1

2

3

4

315 325

5

6

7

8

9

Shell Length (millimeters)

Num

ber

of

Eg

gs

Laid

Per

Clu

tch

Analyze each data set and its scatter plot and describe the polynomial function that best models the data. Explain your reasoning.

7. Data Set A: The data increases, decreases, then increases again. A cubic function models the data.

8. Data Set B: The data increases, reaches a maximum, then decreases. A quadratic function models the data.


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9. Data Set C: The data increases, reaches a maximum, then decreases. A quadratic function models the data.

10. Data Set D: The data increases, then decreases, then increases again. A cubic function models the data.

11. Data Set E: The data increases, then decreases, then increases, and finally decreases. A quartic function models the data.

12. Data Set F: The data increases, reaches a maximum, then decreases. A quadratic function models the data.

Use a graphing calculator to determine the regression equation that best models the data. Round decimals to the nearest thousandth.

13. Data Set A: The function y 5 0.229x3 2 4.106x2 1 22.496x 2 18.003 models the data.

14. Data Set B: The function y 5 21.661x2 1 12.796x 1 0.9 models the data.


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15. Data Set C: The function y 5 20.159x2 1 3.29x 1 1.018 models the data.

16. Data Set D: The function y 5 0.005x3 2 0.085x2 1 0.487x 1 4.484 models the data.

17. Data Set E: The function y 5 20.013x4 1 0.24x3 2 1.343x2 1 2.189x 1 6.743 models the data.

18. Data Set F: The function y 5 20.01093x2 1 6.737x 2 1028.5 models the data.

Use the regression equations from Problems 13 through 18 to answer each question.

19. Susan bought 25 shares of WXY stock in 2006. How much money did she pay for her shares?

In the year 2006, 25 shares cost $56.40. I used the regression equation to determine the output value for the input x 5 12.

20. Approximately how many people travel less than 100 miles on the Monday after Thanksgiving?

Approximately 9.1 million people travel less than 100 miles on the Monday after Thanksgiving. I used the regression equation to determine the output value for the input x 5 7.


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21. What is the optimal amount of money the J. Company should spend on advertising to maximize profit?

The J. Company should spend approximately $1040 on advertising to maximize their profit at $180,366. I determined the vertex of the regression equation at (10.346, 18.037).

22. How many tons of apples were harvested in 2007?

Approximately 12,800 tons of apples were harvested in 2007. I used the regression equation to determine the output value for the input x 5 17.

23. Predict the home mortgage interest rate in 2015. Is this likely? Explain your reasoning.

The interest rate in 2015 would be about 2171%. I used the regression equation to determine the output value for the input x 5 16. This figure is not realistic because it is a negative number. The regression equation does not make an accurate prediction for future interest rates.

24. What shell size is best for laying the largest clutch of eggs? Why might larger size shells be associated with smaller clutches of eggs?

The shell size that would produce the largest clutch size is 308 mm. I determined the vertex of the regression equation at (308.188, 9.632). Larger shells might belong to older turtles and older turtles may not be able to produce as many eggs.


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The Choice Is Yours Comparing Polynomials in Different Representations

Problem Set

Analyze each pair of representations. Then, answer each question and justify your reasoning.

1. Which polynomial function has a greater degree?

A polynomial function b(x) with 2 absolute minimums and 1 relative maximum.

c(x) 5 22(3 2 x2)(x 2 4) 1 9

The function b(x) has a greater degree.

A function with 2 absolute minimums and 1 relative maximum must have a degree greater than 3. The first function is at least a quartic function. The second function is a cubic function.

2. Which polynomial function has a greater number of real zeros?

d(x) 5 x2 2 x – 6 x f(x)

25 28

24 21

23 0

22 1

21 8

0 27

1 64

The function f(x) has a greater number of real zeros.

The Fundamental Theorem of Algebra states that the number of zeros must be equal to the degree of the function. Therefore, d(x) has 2 zeros. If I factor d(x) to (x 2 3)(x 1 2), I can determine that d(x) has 2 real zeros at 3 and 22. I know that the function f(x) is a cubic function because it approaches negative infinity as x approaches negative infinity and it approaches positive infinity as x approaches positive infinity but it does not have a constant rate of change. There is a real zero at 23, and because the function is cubic, it is a triple root.


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3. Which function has an odd degree?

x

y

22 20 4 6 8242628

2

4

22

24

26

28

6

8g(x)

A polynomial function h(x) with 2 real zeros and an imaginary zero.

The function g(x) has an odd degree.

I know that function g(x) is odd because the graph approaches positive infinity as x approaches negative infinity and the graph approaches negative infinity as x approaches positive infinity. The graph of g(x) shows 3 real zeros, one zero at 21 with a multiplicity of 2 and another zero at 8 with a multiplicity of 1. The function h(x) is even because imaginary zeros are always in pairs, so the function has 4 zeros.

4. Which function has the greater output as x approaches infinity?

j(x) 5 2x4 1 3x2 1 120 A quintic function k(x) with a . 0.

The function k(x) has the greatest output as x approaches infinity.

The function j(x) is an even function with a , 0. Therefore, as x approaches infinity, j(x) approaches negative infinity. The function k(x) is an odd function with a . 0. Therefore, as x approaches infinity, k(x) approaches infinity.


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5. Which function has the smaller output as x approaches negative infinity?

A quadratic equation m(x) with y-intercept of (0, 212) and imaginary roots.

n(x) 5 22(x 1 3)5 2 25

Function m(x) has the smaller output as x approaches negative infinity.

The function m(x) has imaginary roots so it does not cross the x-axis. Also, because it has a negative y-intercept, a , 0 so m(x) approaches negative infinity as x approaches negative infinity. The function n(x) is an odd function with a , 0, so n(x) approaches positive infinity as x approaches negative infinity.

6. Which function has a greater y-intercept?

x p(x)

26 16

24 0

22 28

0 28

2 0

4 16

q(x) 5 (x 1 2)3 2 9

Function q(x) has a greater y-intercept.

The y-intercept of p(x) is (0, 28). I can calculate the y-intercept of function q(x) by substituting 0 into the function, producing (0 1 2 ) 3 2 9 5 21.


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7. Which function has a greater average rate of change over the interval (22, 2)?

x

y

21 10 2 3 4222324

1

2

21

22

23

24

3

4

r(x)

A quadratic equation s(x) with a vertex of (22, 24) and a y-intercept of (0, 0).

Function s(x) has the greater average rate of change over the interval (22, 2).

The function r(x) has an average rate of change of 0 between 22 and 2. The function s(x) has a positive rate of change because the vertex is negative and the y-intercept is 0.

8. Which function has a greater relative maximum?

A quartic function t(x) with a . 0 and 4 distinct real roots.

A cubic function u(x) with y-intercept (0, 212) and 1 real root at 23 and 2 imaginary roots.

Function t(x) has a greater relative maximum.

Function t(x) has a . 0 and 4 distinct real roots, so it decreases below the x-axis, increases to a relative maximum above the x-axis, decreases below the x-axis again before finally increasing across the x-axis. The function u(x) has a relative maximum below the x-axis, in order to intersect at (23, 0) and (0, 212) and have two imaginary roots.


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9. Which function’s axis of symmetry has a greater x-value?

A quadratic function z(x) with zeros at 24 and 4.

x

y

22 20 4 6 8242628

2

4

22

24

26

28

6

8a(x)

The function a(x) has a greater x-value for its axis of symmetry.

The axis of symmetry of function a(x) is x 5 4. The function z(x) has an axis of symmetry at x 5 0 because the zeros are equidistant from x 5 0.

10. Which function has a greater output for a given input?

The basic cubic function f(x) 5 x3. d(x) 5 f(x 2 1) 2 5

The function f(x) has a greater output for a given input.

The transformation shifts the cubic function vertically down 5, so the output values decrease.


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11. Which function has a lower minimum?

x g(x)

22 4

21 1

0 0

1 1

2 4

h(x) 5 4g(x 2 3) 2 8

The function h(x) has a lower minimum.

The function g(x) is the basic quadratic function with a minimum at 0. The transformation in h(x) shifts the minimum output down 8 units.

12. Which function has a greater input for a given output?

x

y

22 20 4 6 8242628

2

4

22

24

26

28

6

8 m(x)

n(x) 5 m(x 1 4) 1 1

The function m(x) has a greater input for a given output.

For a given y-value, the function n(x) is shifted to the left 4 units, so the x-value is less.


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