5- 8 Curve Fitting with Quadratic Models 377

GUIDED PRACTICE 1. Vocabulary How does a quadratic model differ from a linear model?

SEE EXAMPLE 1 p. 374

Determine whether each data set could represent a quadratic function. Explain.

2. x -2 -1 0 1 2

y 16 8 0 -8 -16

3. x 1 2 3 4 5

y 1 3 9 27 81

4. x 2 4 6 8 10

y 4 -5 -8 -5 4

SEE EXAMPLE 2 p. 375

Write a quadratic function that fits each set of points.

5. (-2, 5) , (0, -3) , and (3, 0) 6. (0, 1) , (2, -1) , and (3, -8)

7. (-1, 8) , (0, 4) , and (2, 2) 8. (-4, 9) , (0, -7) , and (1, -1)

9. (2, 3) , (6, 3) , and (8, -3) 10. (-1, -12) , (1, 0) , and (2, 9)

SEE EXAMPLE 3 p. 376

11. Hobbies The cost of mounting different-sized photos is shown in the table. Find a quadratic model for the cost given the average side length. (For an 8 in. ! 10 in. photo, the average side length is 8 + 10 _____ 2 = 9 in.) Estimate the cost of mounting a 24 in. ! 36 in. photo.

PRACTICE AND PROBLEM SOLVING Determine whether each data set could represent a quadratic function. Explain.

12. x 0 2 4 6 8

f (x) -1 2 11 26 47

13. x 0 1 2 3 4

f (x) 10 9 6 1 -6

14. x 1 2 3 4 5

f (x) -3 0 3 6 9

KEYWORD: MB7 Parent

KEYWORD: MB7 5-8

5-8

THINK AND DISCUSS 1. Describe how to determine if a data set is quadratic.

2. Explain whether a quadratic function is a good model for the path of an airplane that ascends, descends, and rises again out of view.

3. GET ORGANIZED Copy and complete the graphic organizer. Compare the different quadratic models presented in the lesson.

ExercisesExercises

Costs of Mounting Photos

Size (in.) Cost ($)

8 ! 10 10

14 ! 18 16

16 ! 20 19

24 ! 30 27

32 ! 40 39

a207se_c05l08_0374_0381.indd 377a207se_c05l08_0374_0381.indd 377 9/29/05 1:46:59 PM9/29/05 1:46:59 PM

           

     

     

378 Chapter 5 Quadratic Functions

Write a quadratic function that fits each set of points.

15. (-2, 5) , (-1, 0) , and (1, -2) 16. (1, 2) , (2, -1) , and (5, 2) 17. (-4, 12) , (-2, 0) , and (2, -12) 18. (-1, 2.6) , (1, 4.2) , and (2, 14)

19. Gardening The table shows the amount spent on water gardening in the United States between 1999 and 2003. Find a quadratic model for the annual amount in millions of dollars spent on water gardening based on number of years since 1999. Estimate the amount that people in the United States will spend on water gardening in 2015.

Write a function rule for each situation, and identify each relationship as linear, quadratic, or neither.

20. the circumference C of a bicycle wheel, given its radius r

21. the area of a triangle A with a constant height, given its base length b

22. the population of bacteria P in a petri dish doubling every hour t

23. the area of carpet A needed for square rooms of length s

24. Physics In the past, different mathematical descriptions of falling objects were proposed.a. Which rule shows the

greatest increase in the distance fallen per second and thus the greatest rate of increase in speed?

b. Identify each rule as linear, quadratic, or neither.

c. Describe the differences in da Vinci’s rule, and compare it with the differences in Galileo’s.

d. The most accurate rule is sometimes described as the odd-number law. Which rule shows an odd-number pattern of first differences and correctly describes the distance for falling objects?

Find the missing value for each quadratic function.

25. x -1 0 1 2 3

f (x) 0 1 0 -8

26. x -3 -2 -1 0 1

f (x) 12 2 0 8

27. x -2 0 2 4 6

f (x) -2 2 7 14

Relative Distance Fallen (units)

Time Interval (s)

Aristotle’s Rule

da Vinci’s Rule

Galileo’s Rule

0 0 0 0

1 1 1 1

2 2 3 4

3 3 6 9

4 4 10 16

Water Gardening

YearAmount Spent

(million $)

1999 806

2000 943

2001 1205

2002 1441

2003 1565

Length (in.) Area ( in 2 )

4 28

6 54

8 88

10 130

For See Exercises Example

12–14 1 15–18 2 19 3

Independent Practice

Skills Practice p. S13Application Practice p. S36

Extra Practice

28. This problem will prepare you for the Multi-Step Test Prep on page 390. A home-improvement store sells several sizes of rectangular tiles, as shown in the table.a. Find a quadratic model for the area of a tile based

on its length.b. The store begins selling a new size of tile with a

length of 9 in. Based on your model, estimate the area of a tile of this size.

a207se_c05l08_0374_0381.indd 378a207se_c05l08_0374_0381.indd 378 9/28/05 6:23:23 PM9/28/05 6:23:23 PM

               

378 Chapter 5 Quadratic Functions

Write a quadratic function that fits each set of points.

15. (-2, 5) , (-1, 0) , and (1, -2) 16. (1, 2) , (2, -1) , and (5, 2) 17. (-4, 12) , (-2, 0) , and (2, -12) 18. (-1, 2.6) , (1, 4.2) , and (2, 14)

19. Gardening The table shows the amount spent on water gardening in the United States between 1999 and 2003. Find a quadratic model for the annual amount in millions of dollars spent on water gardening based on number of years since 1999. Estimate the amount that people in the United States will spend on water gardening in 2015.

Write a function rule for each situation, and identify each relationship as linear, quadratic, or neither.

20. the circumference C of a bicycle wheel, given its radius r

21. the area of a triangle A with a constant height, given its base length b

22. the population of bacteria P in a petri dish doubling every hour t

23. the area of carpet A needed for square rooms of length s

24. Physics In the past, different mathematical descriptions of falling objects were proposed.a. Which rule shows the

greatest increase in the distance fallen per second and thus the greatest rate of increase in speed?

b. Identify each rule as linear, quadratic, or neither.

c. Describe the differences in da Vinci’s rule, and compare it with the differences in Galileo’s.

d. The most accurate rule is sometimes described as the odd-number law. Which rule shows an odd-number pattern of first differences and correctly describes the distance for falling objects?

Find the missing value for each quadratic function.

25. x -1 0 1 2 3

f (x) 0 1 0 -8

26. x -3 -2 -1 0 1

f (x) 12 2 0 8

27. x -2 0 2 4 6

f (x) -2 2 7 14

Relative Distance Fallen (units)

Time Interval (s)

Aristotle’s Rule

da Vinci’s Rule

Galileo’s Rule

0 0 0 0

1 1 1 1

2 2 3 4

3 3 6 9

4 4 10 16

Water Gardening

YearAmount Spent

(million $)

1999 806

2000 943

2001 1205

2002 1441

2003 1565

Length (in.) Area ( in 2 )

4 28

6 54

8 88

10 130

For See Exercises Example

12–14 1 15–18 2 19 3

Independent Practice

Skills Practice p. S13Application Practice p. S36

Extra Practice

28. This problem will prepare you for the Multi-Step Test Prep on page 390. A home-improvement store sells several sizes of rectangular tiles, as shown in the table.a. Find a quadratic model for the area of a tile based

on its length.b. The store begins selling a new size of tile with a

length of 9 in. Based on your model, estimate the area of a tile of this size.

a207se_c05l08_0374_0381.indd 378a207se_c05l08_0374_0381.indd 378 9/28/05 6:23:23 PM9/28/05 6:23:23 PM

                 

378 Chapter 5 Quadratic Functions

Write a quadratic function that fits each set of points.

15. (-2, 5) , (-1, 0) , and (1, -2) 16. (1, 2) , (2, -1) , and (5, 2) 17. (-4, 12) , (-2, 0) , and (2, -12) 18. (-1, 2.6) , (1, 4.2) , and (2, 14)

19. Gardening The table shows the amount spent on water gardening in the United States between 1999 and 2003. Find a quadratic model for the annual amount in millions of dollars spent on water gardening based on number of years since 1999. Estimate the amount that people in the United States will spend on water gardening in 2015.

Write a function rule for each situation, and identify each relationship as linear, quadratic, or neither.

20. the circumference C of a bicycle wheel, given its radius r

21. the area of a triangle A with a constant height, given its base length b

22. the population of bacteria P in a petri dish doubling every hour t

23. the area of carpet A needed for square rooms of length s

24. Physics In the past, different mathematical descriptions of falling objects were proposed.a. Which rule shows the

greatest increase in the distance fallen per second and thus the greatest rate of increase in speed?

b. Identify each rule as linear, quadratic, or neither.

c. Describe the differences in da Vinci’s rule, and compare it with the differences in Galileo’s.

d. The most accurate rule is sometimes described as the odd-number law. Which rule shows an odd-number pattern of first differences and correctly describes the distance for falling objects?

Find the missing value for each quadratic function.

25. x -1 0 1 2 3

f (x) 0 1 0 -8

26. x -3 -2 -1 0 1

f (x) 12 2 0 8

27. x -2 0 2 4 6

f (x) -2 2 7 14

Relative Distance Fallen (units)

Time Interval (s)

Aristotle’s Rule

da Vinci’s Rule

Galileo’s Rule

0 0 0 0

1 1 1 1

2 2 3 4

3 3 6 9

4 4 10 16

Water Gardening

YearAmount Spent

(million $)

1999 806

2000 943

2001 1205

2002 1441

2003 1565

Length (in.) Area ( in 2 )

4 28

6 54

8 88

10 130

For See Exercises Example

12–14 1 15–18 2 19 3

Independent Practice

Skills Practice p. S13Application Practice p. S36

Extra Practice

28. This problem will prepare you for the Multi-Step Test Prep on page 390. A home-improvement store sells several sizes of rectangular tiles, as shown in the table.a. Find a quadratic model for the area of a tile based

on its length.b. The store begins selling a new size of tile with a

length of 9 in. Based on your model, estimate the area of a tile of this size.

a207se_c05l08_0374_0381.indd 378a207se_c05l08_0374_0381.indd 378 9/28/05 6:23:23 PM9/28/05 6:23:23 PM

         

378 Chapter 5 Quadratic Functions

Write a quadratic function that fits each set of points.

15. (-2, 5) , (-1, 0) , and (1, -2) 16. (1, 2) , (2, -1) , and (5, 2) 17. (-4, 12) , (-2, 0) , and (2, -12) 18. (-1, 2.6) , (1, 4.2) , and (2, 14)

19. Gardening The table shows the amount spent on water gardening in the United States between 1999 and 2003. Find a quadratic model for the annual amount in millions of dollars spent on water gardening based on number of years since 1999. Estimate the amount that people in the United States will spend on water gardening in 2015.

Write a function rule for each situation, and identify each relationship as linear, quadratic, or neither.

20. the circumference C of a bicycle wheel, given its radius r

21. the area of a triangle A with a constant height, given its base length b

22. the population of bacteria P in a petri dish doubling every hour t

23. the area of carpet A needed for square rooms of length s

24. Physics In the past, different mathematical descriptions of falling objects were proposed.a. Which rule shows the

greatest increase in the distance fallen per second and thus the greatest rate of increase in speed?

b. Identify each rule as linear, quadratic, or neither.

c. Describe the differences in da Vinci’s rule, and compare it with the differences in Galileo’s.

d. The most accurate rule is sometimes described as the odd-number law. Which rule shows an odd-number pattern of first differences and correctly describes the distance for falling objects?

Find the missing value for each quadratic function.

25. x -1 0 1 2 3

f (x) 0 1 0 -8

26. x -3 -2 -1 0 1

f (x) 12 2 0 8

27. x -2 0 2 4 6

f (x) -2 2 7 14

Relative Distance Fallen (units)

Time Interval (s)

Aristotle’s Rule

da Vinci’s Rule

Galileo’s Rule

0 0 0 0

1 1 1 1

2 2 3 4

3 3 6 9

4 4 10 16

Water Gardening

YearAmount Spent

(million $)

1999 806

2000 943

2001 1205

2002 1441

2003 1565

Length (in.) Area ( in 2 )

4 28

6 54

8 88

10 130

For See Exercises Example

12–14 1 15–18 2 19 3

Independent Practice

Skills Practice p. S13Application Practice p. S36

Extra Practice

28. This problem will prepare you for the Multi-Step Test Prep on page 390. A home-improvement store sells several sizes of rectangular tiles, as shown in the table.a. Find a quadratic model for the area of a tile based

on its length.b. The store begins selling a new size of tile with a

length of 9 in. Based on your model, estimate the area of a tile of this size.

a207se_c05l08_0374_0381.indd 378a207se_c05l08_0374_0381.indd 378 9/28/05 6:23:23 PM9/28/05 6:23:23 PM

NAME:  __________________________________  HR:  _____  

ALG  2B  –  LESSON  5:8    #  12-­‐19,28-­‐31  

 SCORE:      _______    /12  

Explanation:  

YES  or  NO  

Explanation:  

YES  or  NO  

Explanation:  

YES  or  NO  

 

378 Chapter 5 Quadratic Functions

Write a quadratic function that fits each set of points.

15. (-2, 5) , (-1, 0) , and (1, -2) 16. (1, 2) , (2, -1) , and (5, 2) 17. (-4, 12) , (-2, 0) , and (2, -12) 18. (-1, 2.6) , (1, 4.2) , and (2, 14)

19. Gardening The table shows the amount spent on water gardening in the United States between 1999 and 2003. Find a quadratic model for the annual amount in millions of dollars spent on water gardening based on number of years since 1999. Estimate the amount that people in the United States will spend on water gardening in 2015.

Write a function rule for each situation, and identify each relationship as linear, quadratic, or neither.

20. the circumference C of a bicycle wheel, given its radius r

21. the area of a triangle A with a constant height, given its base length b

22. the population of bacteria P in a petri dish doubling every hour t

23. the area of carpet A needed for square rooms of length s

24. Physics In the past, different mathematical descriptions of falling objects were proposed.a. Which rule shows the

greatest increase in the distance fallen per second and thus the greatest rate of increase in speed?

b. Identify each rule as linear, quadratic, or neither.

c. Describe the differences in da Vinci’s rule, and compare it with the differences in Galileo’s.

d. The most accurate rule is sometimes described as the odd-number law. Which rule shows an odd-number pattern of first differences and correctly describes the distance for falling objects?

Find the missing value for each quadratic function.

25. x -1 0 1 2 3

f (x) 0 1 0 -8

26. x -3 -2 -1 0 1

f (x) 12 2 0 8

27. x -2 0 2 4 6

f (x) -2 2 7 14

Relative Distance Fallen (units)

Time Interval (s)

Aristotle’s Rule

da Vinci’s Rule

Galileo’s Rule

0 0 0 0

1 1 1 1

2 2 3 4

3 3 6 9

4 4 10 16

Water Gardening

YearAmount Spent

(million $)

1999 806

2000 943

2001 1205

2002 1441

2003 1565

Length (in.) Area ( in 2 )

4 28

6 54

8 88

10 130

For See Exercises Example

12–14 1 15–18 2 19 3

Independent Practice

Skills Practice p. S13Application Practice p. S36

Extra Practice

28. This problem will prepare you for the Multi-Step Test Prep on page 390. A home-improvement store sells several sizes of rectangular tiles, as shown in the table.a. Find a quadratic model for the area of a tile based

on its length.b. The store begins selling a new size of tile with a

length of 9 in. Based on your model, estimate the area of a tile of this size.

a207se_c05l08_0374_0381.indd 378a207se_c05l08_0374_0381.indd 378 9/28/05 6:23:23 PM9/28/05 6:23:23 PM

378 Chapter 5 Quadratic Functions

Write a quadratic function that fits each set of points.

15. (-2, 5) , (-1, 0) , and (1, -2) 16. (1, 2) , (2, -1) , and (5, 2) 17. (-4, 12) , (-2, 0) , and (2, -12) 18. (-1, 2.6) , (1, 4.2) , and (2, 14)

19. Gardening The table shows the amount spent on water gardening in the United States between 1999 and 2003. Find a quadratic model for the annual amount in millions of dollars spent on water gardening based on number of years since 1999. Estimate the amount that people in the United States will spend on water gardening in 2015.

Write a function rule for each situation, and identify each relationship as linear, quadratic, or neither.

20. the circumference C of a bicycle wheel, given its radius r

21. the area of a triangle A with a constant height, given its base length b

22. the population of bacteria P in a petri dish doubling every hour t

23. the area of carpet A needed for square rooms of length s

24. Physics In the past, different mathematical descriptions of falling objects were proposed.a. Which rule shows the

greatest increase in the distance fallen per second and thus the greatest rate of increase in speed?

b. Identify each rule as linear, quadratic, or neither.

c. Describe the differences in da Vinci’s rule, and compare it with the differences in Galileo’s.

d. The most accurate rule is sometimes described as the odd-number law. Which rule shows an odd-number pattern of first differences and correctly describes the distance for falling objects?

Find the missing value for each quadratic function.

25. x -1 0 1 2 3

f (x) 0 1 0 -8

26. x -3 -2 -1 0 1

f (x) 12 2 0 8

27. x -2 0 2 4 6

f (x) -2 2 7 14

Relative Distance Fallen (units)

Time Interval (s)

Aristotle’s Rule

da Vinci’s Rule

Galileo’s Rule

0 0 0 0

1 1 1 1

2 2 3 4

3 3 6 9

4 4 10 16

Water Gardening

YearAmount Spent

(million $)

1999 806

2000 943

2001 1205

2002 1441

2003 1565

Length (in.) Area ( in 2 )

4 28

6 54

8 88

10 130

For See Exercises Example

12–14 1 15–18 2 19 3

Independent Practice

Skills Practice p. S13Application Practice p. S36

Extra Practice

28. This problem will prepare you for the Multi-Step Test Prep on page 390. A home-improvement store sells several sizes of rectangular tiles, as shown in the table.a. Find a quadratic model for the area of a tile based

on its length.b. The store begins selling a new size of tile with a

length of 9 in. Based on your model, estimate the area of a tile of this size.

a207se_c05l08_0374_0381.indd 378a207se_c05l08_0374_0381.indd 378 9/28/05 6:23:23 PM9/28/05 6:23:23 PM

                   

           

 

         

(If  yes)  Equation:  

YES  or  NO  

(If  yes)  Equation:  

YES  or  NO