Lesson 7.1 Skills Practice - Davis School District / … · water at about 60°C. Use the regression equation for Problem 1 to predict after how many minutes Use the regression equation

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Chapter 7 Skills Practice 495

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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)

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)

4.

x

y

22 20 4 6 8242628

20

40

220

240

260

280

60

80

p(x)

5.

x

y

21 10 2 3 4222324

2

4

22

24

26

28

6

8

p(x)


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6.

x

y

21 10 2 3 4222324

2

4

22

24

26

28

6

8p(x)

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

9. 23 # x3 1 2x 1 6


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10. 210.5 . 21.5x2 2 15.5x

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

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

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

x

y

21 10 2 3 4222324

2

4

22

24

26

28

6

8

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

x

y

22 20 4 6 8242628

20

40

220

240

260

280

60

80

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

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

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

x

y

22 20 4 6 8242628

40

80

240

280

2120

2160

120

160



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America’s Next Top Polynomial Model Modeling with Polynomials

Vocabulary

Explain each key term in your own words.

1. regression equation

2. coefficient of determination

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.

Time Since December (months)

Work Time (hours)

1 40

3 100

5 160

7 140

9 160

11 60

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

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


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

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

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


Gas Usage(quadrillion BTU)

0 12.4

10 21.8

20 20.4

30 19.3

37 22.6

50 24.6


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


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

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:



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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.


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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.

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.

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

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

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



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Connecting PiecesPiecewise Functions

Vocabulary

Write a definition for the term in your own words.

1. piecewise function

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

4


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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

4

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

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

4


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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

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

9.

x

y

21 10 2 3 4222324

1

2

21

22

23

24

3

4

d(x)

d(x) 5

10.

x

y

22 20 4 6 8242628

2

4

22

24

26

28

6

8

f(x)

f(x) 5


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11.

x

y

21 10 2 3 4222324

2

4

22

24

26

28

6

8

g(x)

g(x) 5

12.

x

y

22 20 4 6 8242628

2

4

22

24

26

28

6

8 h(x)

h(x) 5


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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

f(x) 5

, 0 # x # 3

, 3 , x # 8

, 8 , x # 20


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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

f(x) 5

, 0 , x # 0.5

, 0.5 , x # 0.8

, 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

f(x) 5

, 10 # x # 35

, 35 , x # 75

, 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

f(x) 5

, 0 # x # 5

, 5 , x # 9

, 9 , x # 14

, 14 , x # 18

18.

x

y

840

12 16 20

1

2

3

4

24 28 32 36 40

5

6

7

8

9

f(x) 5

, 4 # x , 8

, 8 # x , 20

, 20 # x , 32

, 32 # x # 44


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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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2. Data Set B

3. Data Set C


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4. Data Set D

5. Data Set E


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6. Data Set F

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:


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9. Data Set C:

10. Data Set D:

11. Data Set E:

12. Data Set F:

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:


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15. Data Set C:

16. Data Set D:

17. Data Set E:

18. Data Set F:

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?


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

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

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

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?


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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


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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.

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.


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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

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


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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).

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.


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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)

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


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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

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


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Lesson 7.1 Skills Practice - Davis School District / … · water at about 60°C. Use the regression equation for Problem 1 to predict after how many minutes Use the regression equation