Answer Key - cloudfront.net

Answer Key - cloudfront.net

  c. Graph the function.   4-1 Graphing Quadratic Functions Complete parts a–c for each quadratic function.   a. Find the y-intercept, the equation ...

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  c. Graph the function.  

4-1 Graphing Quadratic Functions Complete parts a–c for each quadratic function.

  a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.   b. Make a table of values that includes the vertex.   c. Use this information to graph the function.

 

ANSWER:   a. y-int = 0; axis of symmetry: x = 0; x-coordinate = 0   b.

1. 

  SOLUTION:   a. Compare the function  with the  standard form of a quadratic function.

  Here, a = 3, b = 0 and c = 0.

  The y-intercept is 0.

  c.  The equation of the axis of symmetry is

.

  Therefore, x = 0 is the axis of symmetry.

  The x-coordinate of the vertex is

.

    b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

 

  2. 

  SOLUTION:   a. Compare the function with the standard form of a quadratic function.

    c. Graph the function.  

Here, a = –6, b = 0 and c = 0.

  The y-intercept is 0.

  The equation of the axis of symmetry is

.

  Therefore, x = 0 is the axis of symmetry. eSolutions Manual - Powered by Cognero

  The x-coordinate of the vertex is

Page 1

.

The y-intercept is 0.

  4-1 Graphing Quadratic Functions The equation of the axis of symmetry is

.

  c.

  Therefore, x = 0 is the axis of symmetry.

  The x-coordinate of the vertex is

.

  b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

    3. 

  SOLUTION:  

  c. Graph the function .

a. Compare the function  with the  standard form of a quadratic function.

  Here, a = 1, b = –4 and c = 0.

  The y-intercept is 0.

  The equation of the axis of symmetry is .

  Therefore, x = 2 is the axis of symmetry.

  ANSWER:   a. y-int = 0; axis of symmetry: x = 0; x-coordinate = 0   b.

  c.

The x-coordinate of the vertex is

.

  b. Substitute 0, 1, 2, 3 and 4 for x and make the table.

 

  c. Graph the function.  

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Page 2

 

  The equation of the axis of symmetry is

  4-1 Graphing Quadratic Functions c. Graph the function.  

.

  Therefore, x = 1.5 is the axis of symmetry.

  The x-coordinate of the vertex is

.

  b. Substitute 0, –1, –1.5, –2 and –3 for x and make the table.

    ANSWER:   a. y-int = 0; axis of symmetry: x = 2; x-coordinate = 2   b.

  c. Graph the function.  

  c.

  ANSWER:   a. y-int = 4; axis of symmetry: x = –1.5; x-coordinate = –1.5   b.

  4. 

  SOLUTION:   a. Compare the function the standard form of a quadratic function. Here, a = –1, b = –3 and c = 4.

  The y-intercept is 4.

 with 

  c.

  The equation of the axis of symmetry is .

  eSolutions Manual - Powered by Cognero

Page 3

Therefore, x = 1.5 is the axis of symmetry.

   

  c. Graph the function.  

4-1 Graphing Quadratic Functions

  c.

 

 

5. 

ANSWER:   a. y-int = –3; axis of symmetry: x = 0.75; xcoordinate = 0.75

  SOLUTION:   a. Compare the function the standard form of a quadratic function. Here, a = 4, b = –6 and c = –3.

 

 with 

b.

  The y-intercept is –3.

  The equation of the axis of symmetry is .

  c.

  Therefore, x = 0.75.

  The x-coordinate of the vertex is

.

  b. Substitute 0, –1, 0.75, 1.5 and 2.5 for x and make the table.

    6. 

  SOLUTION:  

  c. Graph the function.  

a. Compare the function the standard form of a quadratic function.

 with 

  Here, a = 2, b = –8 and c = 5.

  The y-intercept is 5.

    The equation of the axis of symmetry is .

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Therefore, x = 2 is the axis of symmetry.

 

Page 4

Here, a = 2, b = –8 and c = 5.

  The y-intercept is 5.

  4-1 Graphing Quadratic Functions

  c.

 

The equation of the axis of symmetry is . Therefore, x = 2 is the axis of symmetry.

  The x-coordinate of the vertex is

.

  b. Substitute 0, 1, 2, 3 and 4 for x and make the table.

 

  Determine whether each function has a maximum or minimum value, and find that value. Then state the domain and range of the function.

  7. 

  c. Graph the function.  

  SOLUTION:   Compare the function standard form of a quadratic function.

 with the 

  Here, a = –1, b = 6 and c = –1.

  For this function, a = –1, so the graph opens down and the function has a maximum value.

  The x-coordinate of the vertex is

  ANSWER:   a. y-int = 5; axis of symmetry: x = 2; x-coordinate = 2   b.

  c.

.

  Substitute 3 for x in the function to find the ycoordinate of the vertex.

 

  Therefore, the maximum value of the function is 8.   The domain is all real numbers. D = {all real numbers} . The range is all real numbers less than or equal to the maximum value.  

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Page 5

ANSWER:   max = 8; D = {all real numbers},

ANSWER:   max = 8; D = {all real numbers}, 4-1 Graphing Quadratic Functions

 

  Determine whether each function has a maximum or minimum value, and find that value. Then state the domain and range of the function.

8. 

  SOLUTION:  

 

Compare the function standard form of a quadratic function.

7. 

 

 with the 

  Here, a = 1, b = 3 and c = –12.

SOLUTION:  

 

Compare the function standard form of a quadratic function.

 with the 

For this function, a = 1, so the graph opens up and the function has a minimum value.

  Here, a = –1, b = 6 and c = –1.

   

 

The x-coordinate of the vertex is

For this function, a = –1, so the graph opens down and the function has a maximum value.

.

 

Substitute –1.5 for x in the function to find the ycoordinate of the vertex.

The x-coordinate of the vertex is

 

.

  Substitute 3 for x in the function to find the ycoordinate of the vertex.

  Therefore, the minimum value of the function is – 14.25.   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers greater than or equal to the minimum value.

 

  Therefore, the maximum value of the function is 8.   The domain is all real numbers. D = {all real numbers} . The range is all real numbers less than or equal to the maximum value.

  ANSWER:   min = –14.25; D = {all real numbers},

   

ANSWER:   max = 8; D = {all real numbers},

9. 

 

 

SOLUTION:  

 

Compare the function standard form of a quadratic function.

SOLUTION:  

Here, a = 3, b = 8 and c = 5.

8. 

eSolutions Manualthe - Powered by Cognero Compare function

standard form of a quadratic function.

 

 with the 

   with the 

 

Page 6

or this function, a = 3, so the graph opens up and the function has a minimum value.

 D = {all real numbers}, 

ANSWER:   min = –14.25; D = {all real numbers}, 4-1 Graphing Quadratic Functions

 

 

9. 

10. 

 

 

SOLUTION:  

SOLUTION:  

Compare the function standard form of a quadratic function.

 with the 

Compare the function the standard form of a quadratic function.

 with 

 

 

Here, a = 3, b = 8 and c = 5.

Here, a = –4, b = 10 and c = –6.

 

 

or this function, a = 3, so the graph opens up and the function has a minimum value.

For this function, a = –4, so the graph opens down and the function has a maximum value.

 

 

The x-coordinate of the vertex is

The x-coordinate of the vertex is

.

.

 

  Substitute 1.25 for x in the function to find the ycoordinate of the vertex.

 for x in the function to find the y-

Substitute

 

coordinate of the vertex.

 

  Therefore, the minimum value of the function is   The domain is all real numbers. D = {all real numbers}   The range is all real numbers greater than or equal to the minimum value.

.

  Therefore, the maximum value of the function is 0.25   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers less than or equal to the maximum value.   ANSWER:   max = 0.25; D = {all real numbers},

 

  ANSWER:    D = {all real numbers}, 

11. BUSINESS A store rents 1400 videos per week at $2.25 per video. The owner estimates that they will rent 100 fewer videos for each $0.25 increase in price. What price will maximize the income of the store?

    10. 

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SOLUTION:  

SOLUTION:   Let x be the number of increase in price and let f (x) be the income. Page 7

 

of symmetry, and the x-coordinate of the vertex.

ANSWER:   max = 0.25; D = {all real numbers},

  b. Make a table of values that includes the vertex.

4-1 Graphing Quadratic Functions

  11. BUSINESS A store rents 1400 videos per week at $2.25 per video. The owner estimates that they will rent 100 fewer videos for each $0.25 increase in price. What price will maximize the income of the store?

  c. Use this information to graph the function.

  12. 

 

 

SOLUTION:  

SOLUTION:   Let x be the number of increase in price and let f (x) be the income.

 

 

Here, a = 4, b = 0 and c = 0.

  The y-intercept is 0.

  The equation of the axis of symmetry is

.

  Therefore, x = 0 is the axis of symmetry.

  Here, a = −25, b = 125, and c = 3150.

 

  The x-coordinate of the vertex is

.

  b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

 

  The function gets maximum value at 2.5. That is, 2.5 number of increase in price will maximize the income. $2.25 + (2.5)(0.25) ≈ $2.88

  So, the price of $2.88 per video will maximize the income of the store.

  c. Graph the function.  

  ANSWER:   $2.88

  Complete parts a–c for each quadratic function.

  a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.

  b. Make a table of values that includes the vertex.

  c. Use this information to graph the function. eSolutions   Manual - Powered by Cognero

12. 

 

  ANSWER:   a. y-int = 0; axis of symmetry: x = 0; x-coordinate = 0   Page 8 b.

 

The x-coordinate of the vertex is

.

 

ANSWER:   a. y-int = 0;Quadratic 4-1 Graphing Functions axis of symmetry: x = 0; x-coordinate = 0   b.

b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

 

  c.

  c. Graph the function.  

 

 

 

ANSWER:   a. y-int = 0; axis of symmetry: x = 0; x-coordinate = 0   b.

13.  SOLUTION:   a. Compare the function  with the  standard form of a quadratic function.

  Here, a = –2, b = 0 and c = 0.

  The y-intercept is 0.

  The equation of the axis of symmetry is

  c.

.

  Therefore, x = 0 is the axis of symmetry.

  The x-coordinate of the vertex is

.

  b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

    eSolutions Manual - Powered by Cognero

Page 9

14. 

 

  4-1 Graphing Quadratic Functions

ANSWER:   a. y-int = –5; axis of symmetry: x = 0; x-coordinate = 0   b.

  14. 

  SOLUTION:   a. Compare the function  with the  standard form of a quadratic function.

  Here, a = 1, b = 0 and c = –5.

  The y-intercept is –5.

  c.

  The equation of the axis of symmetry is

.

  Therefore, x = 0 is the axis of symmetry.

  The x-coordinate of the vertex is

.

  b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

 

  15. 

  SOLUTION:  

  c. Graph the function.  

a. Compare the function  with the  standard form of a quadratic function. Here, a = 4, b = 0 and c = –3.

  The y-intercept is –3.

  The equation of the axis of symmetry is

.

  Therefore, x = 0 is the axis of symmetry. The x-coordinate of the vertex is

.

  b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

  ANSWER:   a. y-int = –5; axis of symmetry: x = 0; x-coordinate = 0 eSolutions   Manual - Powered by Cognero b.

 

Page 10

The x-coordinate of the vertex is

.

  b. Substitute –2, –1, 0,Functions 1 and 2 for x and make the 4-1 Graphing Quadratic table.

 

  16. 

  SOLUTION:  

    c. Graph the function.  

a. Compare the function  with the  standard form of a quadratic function. Here, a = 1, b = 0 and c = 3.

  The y-intercept is 3.

  The equation of the axis of symmetry is

.

  Therefore, x = 0 is the axis of symmetry.

  The x-coordinate of the vertex is

.

   

b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

  ANSWER:   a. y-int = –3; axis of symmetry: x = 0; x-coordinate = 0   b.

  c. Graph the function.     c.

 

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16. 

 

ANSWER:   a. y-int = 3; axis of symmetry: x = 0; x-coordinate = 0   Page 11 b.

The x-coordinate of the vertex is

 

.

  ANSWER:   4-1 Graphing Quadratic Functions a. y-int = 3; axis of symmetry: x = 0; x-coordinate = 0   b.

b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

 

  c. Graph the function .

  c.

 

 

ANSWER:   a. y-int = 5; axis of symmetry: x = 0; x-coordinate = 0   b.

17. 

  SOLUTION:   a. Compare the function standard form of a quadratic function.

 with the 

  Here, a = –3, b = 0 and c = 5.

  The y-intercept is 5. The equation of the axis of symmetry is

.

  c.

  Therefore, x = 0 is the equation of axis of symmetry. The x-coordinate of the vertex is

.

  b. Substitute –2, –1, 0, 1 and 2 for x and make the table.

 

  eSolutions Manual - Powered by Cognero

Page 12

18. 

 

 

ANSWER:   a. y-int = 8; axis of symmetry: x = 3; x-coordinate = 3   b.

4-1 Graphing Quadratic Functions

  18. 

  SOLUTION:   a. Compare the function standard form of a quadratic function.

 with the 

  c.

  Here, a = 1, b = –6 and c = 8. The y-intercept is 8. The equation of the axis of symmetry is . Therefore, x = 3 is the equation of the axis of symmetry. The x-coordinate of the vertex is

.

  b. Substitute 1, 2, 3, 4 and 5 for x and make the table.

  19. 

  SOLUTION:  

  c. Graph the function.  

a. Compare the function standard form of a quadratic function. Here, a = 1, b = –3 and c = –10.

 with the

  The y-intercept is –10.

  The equation of the axis of symmetry is . Therefore, x = 1.5 is the equation of the axis of symmetry. The x-coordinate of the vertex is

.

   

b. Substitute 0, 1, 1.5, 2 and 3 for x and make the table.

ANSWER:   a. y-int = 8; axis of symmetry: x = 3; x-coordinate = 3   b.

 

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Page 13

 

The x-coordinate of the vertex is

.

  b. Substitute 0, 1, 1.5, Functions 2 and 3 for x and make the 4-1 Graphing Quadratic table.

 

  20. 

  SOLUTION:  

  c. Graph the function.  

a. Compare the function the standard form of a quadratic function. Here, a = –1, b = 4 and c = –6. The y-intercept is –6. The equation of the axis of symmetry is

 with 

. Therefore, x = 2 is the equation of the axis of symmetry. The x-coordinate of the vertex is

.

b. Substitute 0, 1, 2, 3 and 4 for x and make the table.

  ANSWER:   a. y-int = –10; axis of symmetry: x = 1.5; xcoordinate =1.5   b.

  c.

c. Graph the function.

  ANSWER:   a. y-int = –6; axis of symmetry: x = 2; x-coordinate = 2   b.

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20. 

Page 14

c.  

ANSWER:   a. y-int = –6; axis of symmetry: x = 2; x-coordinate = 2 4-1 Graphing Quadratic Functions   b.

  c. Graph the function.

c.   ANSWER:   a. y-int = 9; axis of symmetry: x = 0.75; x-coordinate = 0.75   b.

  21. 

 

c.

SOLUTION:   a. Compare the function the standard form of a quadratic function. Here, a = –2, b = 3 and c = 9. The y-intercept is 9. The equation of the axis of symmetry is

 with 

. The equation of the axis of symmetry is x = 0.75.

 

The x-coordinate of the vertex is

Determine whether each function has a maximum or minimum value, and find that value. Then state the domain and range of the function.

.

  b. Substitute –1, 0, 0.75, 1.5 and 2.5 for x and make the table.

  22. 

  SOLUTION:   Compare the function form of a quadratic function.

  c. Graph the function. eSolutions Manual - Powered by Cognero

 with the standard 

  Here, a = 5, b = 0 and c = 0.

  For this function, a = 5, so the graph opens up and Page 15 the function has a minimum value.

 

ANSWER:   min = 0; D = {all real numbers}, 4-1 Graphing Quadratic Functions

 

  Determine whether each function has a maximum or minimum value, and find that value. Then state the domain and range of the function.

23. 

  SOLUTION:  

 

Compare the function  with the  standard form of a quadratic function.

22. 

 

  Here, a = –1, b = 0 and c = –12.

SOLUTION:   Compare the function form of a quadratic function.

   with the standard 

For this function, a = –1, so the graph opens down and the function has a maximum value.

 

 

Here, a = 5, b = 0 and c = 0.

The x-coordinate of the vertex is

 

.

For this function, a = 5, so the graph opens up and the function has a minimum value.

 

  The x-coordinate of the vertex is

Substitute 0 for x in the function to find the ycoordinate of the vertex.

.

  Substitute 0 for x in the function to find the ycoordinate of the vertex.

  Therefore, the maximum value of the function is – 12.   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers less than or equal to the maximum value.

  Therefore, the minimum value of the function is 0.   The domain is all real numbers. D = {all real numbers}   The range is all real numbers greater than or equal to the minimum value.

  ANSWER:   max = –12; D = {all real numbers},

  ANSWER:   min = 0; D = {all real numbers},

  24. 

 

  SOLUTION:  

23. 

 

Compare the function standard form of a quadratic function.

SOLUTION:  

 

Compare the function  with the  standard form of a quadratic function.

  eSolutions Manual - Powered Cognero Here, a = –1, b = 0by and c = –12.

  For this function, a = –1, so the graph opens down

 with the 

Here, a = 1, b = –6 and c = 9.

  For this function, a = 1, so the graph opens up and the function has a minimum value.

   

Page 16

ANSWER:   max = –12; D = {all real numbers},

ANSWER:   min = 0; D = {all real numbers},

4-1 Graphing Quadratic Functions

 

 

24. 

25. 

 

 

SOLUTION:  

SOLUTION:  

Compare the function standard form of a quadratic function.

 with the 

Compare the function standard form of a quadratic function.

 with the 

 

 

Here, a = 1, b = –6 and c = 9.

Here, a = –1, b = –7 and c = 1.

 

 

For this function, a = 1, so the graph opens up and the function has a minimum value.

For this function, a = –1, so the graph opens down and the function has a maximum value.

   

  The x-coordinate of the vertex is

The x-coordinate of the vertex is

.

.

 

Substitute 3 for x in the function to find the ycoordinate of the vertex.

Substitute –3.5 for x in the function to find the ycoordinate of the vertex.

 

  Therefore, the minimum value of the function is 0.   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers greater than or equal to the minimum value.

 

 

 

Therefore, the maximum value of the function is 13.25.

  The domain is all real numbers. D = {all real numbers}.

  The range is all real numbers less than or equal to the maximum value.

ANSWER:   min = 0; D = {all real numbers},

ANSWER:   max = 13.25; D = {all real numbers},

 

 

25. 

26. 

 

 

SOLUTION:   Compare the function standard form of a quadratic function.

SOLUTION:    with the 

  Here, a = –1, b = –7 and c = 1.

  For this function, a = –1, so the graph opens down and the function has a maximum value. eSolutions Manual - Powered by Cognero

 

The x-coordinate of the vertex is

  Compare the function standard form of a quadratic function. Here, a = –3, b = 8 and c = 2.

 with the 

  For this function, a = –3, so the graph opens down Page 17 and the function has a maximum value.

  The x-coordinate of the vertex is

max =

ANSWER:   max = 13.25; D = {all real numbers},

 D = {all real numbers}, 

4-1 Graphing Quadratic Functions

 

 

26. 

27. 

 

 

SOLUTION:  

SOLUTION:  

  Compare the function standard form of a quadratic function. Here, a = –3, b = 8 and c = 2.

Compare the function standard form of a quadratic function.

 with the 

 with the 

  Here, a = –2, b = –4 and c = 5.

 

 

For this function, a = –3, so the graph opens down and the function has a maximum value.

For this function, a = –2, so the graph opens down and the function has a maximum value.

 

 

The x-coordinate of the vertex is

The x-coordinate of the vertex is

.

.

 

 

Substitute

Substitute –1 for x in the function to find the ycoordinate of the vertex.

 for x in the function to find the y-

 

coordinate of the vertex.

 

  Therefore, the maximum value of the function is 7.   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers less than or equal to the maximum value.

Therefore, the maximum value of the function is .   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers less than or equal to the maximum value.

  ANSWER:   max = 7; D = {all real numbers},

   

28. 

ANSWER:   max =

 D = {all real numbers}, 

  SOLUTION:   Compare the function  with the  standard form of a quadratic function.

 

  Here, a = –5, b = 0 and c = 15.

 

27. 

eSolutions Manual - Powered by Cognero

 

SOLUTION:  

For this function, a = –5, so the graph opens down Page 18 and the function has a maximum value.

 

ANSWER:   max = 15; D = {all real numbers},

ANSWER:   max = 7; D = {all real numbers}, 4-1 Graphing Quadratic Functions

 

 

28. 

29. 

 

 

SOLUTION:  

SOLUTION:  

Compare the function  with the  standard form of a quadratic function.

Compare the function standard form of a quadratic function.

 

 

Here, a = –5, b = 0 and c = 15.

Here, a = 1, b = 12 and c = 27.

 

 

For this function, a = –5, so the graph opens down and the function has a maximum value.

For this function, a = 1, so the graph opens up and the function has a minimum value.

 

 with the 

The x-coordinate of the vertex is

The x-coordinate of the vertex is

.

 

.

Substitute –6 for x in the function to find the ycoordinate of the vertex.

  Substitute 0 for x in the function to find the ycoordinate of the vertex.

 

  Therefore, the maximum value of the function is 15.   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers less than or equal to the maximum value.

 

 

 

ANSWER:   max = 15; D = {all real numbers},

ANSWER:   min = –9; D = {all real numbers},

 

 

Therefore, the minimum value of the function is –9.

  The domain is all real numbers. D = {all real numbers}.

  The range is all real numbers greater than or equal to the minimum value.

30. 

29. 

 

 

SOLUTION:  

SOLUTION:  

Compare the function standard form of a quadratic function.

 with the 

 

 

For this function, a = 1, so the graph opens up and the function has a minimum value.

 

 with 

Here, a = –1, b = 10 and c = 30.

 

The x-coordinate of the vertex is

Compare the function the standard form of a quadratic function.

 

Here, a = 1, b = 12 and c = 27.

eSolutions Manual - Powered by Cognero

 

For this function, a = –1, so the graph opens down and the function has a maximum value. .

  The x-coordinate of the vertex is

Page 19

ANSWER:   min = –9; D = {all real numbers},

ANSWER:   max = 55; D = {all real numbers}, 

4-1 Graphing Quadratic Functions

 

 

30. 

31. 

 

 

SOLUTION:  

SOLUTION:  

  Compare the function the standard form of a quadratic function.

 with 

Compare the function the standard form of a quadratic function.

 with 

 

 

Here, a = 2, b = –16 and c = –42.

Here, a = –1, b = 10 and c = 30.

 

 

For this function, a = 2, so the graph opens up and the function has a minimum value.

For this function, a = –1, so the graph opens down and the function has a maximum value.

 

  The x-coordinate of the vertex is

The x-coordinate of the vertex is

.

 

.

Substitute 4 for x in the function to find the ycoordinate of the vertex.

  Substitute 5 for x in the function to find the ycoordinate of the vertex.

 

   

Therefore, the minimum value of the function is –74.

Therefore, the maximum value of the function is 55.

 

 

The domain is all real numbers. D = {all real numbers}.

The domain is all real numbers. D = {all real numbers}.

 

  The range is all real numbers less than or equal to the maximum value.

The range is all real numbers greater than or equal to the minimum value. .

   

ANSWER:   min = –74; D = {all real numbers},

ANSWER:   max = 55; D = {all real numbers}, 

    32. CCSS MODELING A financial analyst determined that the cost, in thousands of dollars, of producing

31. 

2

 

bicycle frames is C = 0.000025f – 0.04f + 40, where f is the number of frames produced.

SOLUTION:   Compare the function the standard form of a quadratic function.

   with 

a. Find the number of frames that minimizes cost.

 

 

b. What is the total cost for that number of frames?

Here, a = 2, b = –16 and c = –42.

 

  For this function, a = 2, so the graph opens up and the function has a minimum value.

eSolutions Manual - Powered by Cognero

 

SOLUTION:   a. The x-coordinate of the vertex is:

Page 20

 

ANSWER:   min = –74; D = {all real numbers},

Here, a = –3, b = –9 and c = 2.

  The y-intercept is 2.

4-1 Graphing Quadratic Functions

 

 

The equation of the axis of symmetry is  32. CCSS MODELING A financial analyst determined that the cost, in thousands of dollars, of producing 2

.

bicycle frames is C = 0.000025f – 0.04f + 40, where f is the number of frames produced.

 

 

 

a. Find the number of frames that minimizes cost.

Equation of the axis of symmetry is x = –1.5.

 

The x-coordinate of the vertex is

b. What is the total cost for that number of frames?

 

 

b. Substitute –3, –2, –1.5, –1 and 0 for x and make the table.

SOLUTION:   a. The x-coordinate of the vertex is:

.

The number of frames that minimize the cost is 800. b. Substitute 800 for f in the function and simplify.

  c. Graph the function.  

Therefore, the total cost is $24, 000. ANSWER:   a. 800 b. $24,000 Complete parts a–c for each quadratic function.

  a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.

  b. Make a table of values that includes the vertex.

  c. Use this information to graph the function.

ANSWER:   a. y-int = 2; axis of symmetry: x = –1.5; x-coordinate of vertex = –1.5   b.

  33.  SOLUTION:   a. Compare the function the standard form of a quadratic function.

  Here, a = –3, b = –9 and c = 2.

 

 with 

  c.

The y-intercept is 2.

  The equation of the axis of symmetry is  . eSolutions Manual - Powered by Cognero

  Equation of the axis of symmetry is x = –1.5.

Page 21

  c.

  c. Graph the function.  

 

 

SOLUTION:  

ANSWER:   a. y-int = –9; axis of symmetry: x = 1.5; x-coordinate of vertex = 1.5   b.

4-1 Graphing Quadratic Functions

34. 

a. Compare the function the standard form of a quadratic function.

 with 

  Here, a = 2, b = –6 and c = –9.

  The y-intercept is –9. The equation of the axis of symmetry is .

  c.

  Therefore, x = 1.5 is the axis of symmetry. The x-coordinate of the vertex is

.

  b. Substitute 0, 1, 1.5, 2 and 3 for x and make the table.

    35.  SOLUTION:  

  c. Graph the function.  

a. Compare the function standard form of a quadratic function.   Here, a = –4, b = 5 and c = 0.   The y-intercept is 0.   The equation of the axis of symmetry is

 with the 

. eSolutions Manual - Powered by Cognero

 

  Equation of the axis of symmetry is x = .  

Page 22

The y-intercept is 0.   The equation of the axis of symmetry is 4-1 Graphing Quadratic Functions .

c.

  Equation of the axis of symmetry is x =

.

  The x-coordinate of the vertex is

.

  b. Substitute

 for x and make the

table.

 

  36. 

SOLUTION:   a. Compare the function standard form of a quadratic function.   Here, a = 2, b = 11 and c = 0.   The y-intercept is 0.   The equation of the axis of symmetry is

  c. Graph the function.  

 with the 

.   Equation of the axis of symmetry is x = –2.75.   The x-coordinate of the vertex is

.

b. Substitute –4, –3, –2.75, –2.5 and –1.5 for x and make the table.

 

  ANSWER:   a. y -int = 0; axis of symmetry: x =

; x-coordinate

of vertex =   b.

  c. Graph the function.  

c. eSolutions Manual - Powered by Cognero

Page 23

 

The y-intercept is 4. Equation of the axis of symmetry is

  4-1 Graphing Quadratic Functions

.

c. Graph the function.  

  Therefore, x = –6 is the axis of symmetry.   The x-coordinate of the vertex is

.

  b. Substitute –10, –8, –6, –4 and –2 for x and make the table.

  ANSWER:   a. y-int = 0; axis of symmetry: x = –2.75; xcoordinate of vertex = –2.75   b.

  c. Graph the function.  

  c.   ANSWER:   a. y-int = 4; axis of symmetry: x = –6; x-coordinate of  vertex = –6   b.

  37.  SOLUTION:     a. Compare the function the standard form of a quadratic function. Here, a = 0.25, b = 3 and c = 4.   The y-intercept is 4. Equation of the axis of symmetry is

 with

  c.

.   Therefore, x = –6 is the axis of symmetry. eSolutions   Manual - Powered by Cognero The x-coordinate of the vertex is

Page 24

.

 

  c.

  c. Graph the function .

 

 

4-1 Graphing Quadratic Functions

ANSWER:  

38. 

a. y -int = 6; axis of symmetry: x =

SOLUTION:   a. Compare the function

; x-coordinate

of vertex =

 with the standard form of a  quadratic function.   Here, a = –0.75, b = 4 and c = 6.   The y-intercept is 6.   The equation of the axis of symmetry is

  b.

.   Equation of the axis of symmetry is x =

  c.

.

  The x-coordinate of the vertex is

.

  b. Substitute

 for x and make the

table.

    39.  SOLUTION:   a. Compare the function

  c. Graph the function .

 with 

the standard form of a quadratic function.   Here, a =

, b = 4 and c =

.

  eSolutions Manual - Powered by Cognero

The y-intercept is

or –2.5.

  The equation of the axis of symmetry is

Page 25

a. y -int = –2.5; axis of symmetry: x =

the standard form of a quadratic function.  

; x-

coordinate of vertex =

4-1 Graphing Functions Here, a = Quadratic , b = 4 and c= .

 

 

b. or –2.5.

The y-intercept is

  The equation of the axis of symmetry is .

  c.

  Therefore, x =

 is the axis of symmetry.

  The x-coordinate of the vertex is b. Substitute

.  for x and make

the table.

      40.  SOLUTION:  

  c. Graph the function.  

a. Compare the function

 with 

the standard form of a quadratic function.   Here, a =

,b =

 and c = 9.

  The y-intercept is 9.   The equation of the axis of symmetry is .

 

  Therefore, x = 1.75 is the axis of symmetry.

ANSWER:   a. y -int = –2.5; axis of symmetry: x = coordinate of vertex =

; x-

The x-coordinate of the vertex is

.

  b. Substitute 0.5, 1.5, 1.75, 2 and 3 for x and make the table.

  b.

eSolutions Manual - Powered by Cognero

Page 26

  c. Graph the function.

Therefore, x = 1.75 is the axis of symmetry. The x-coordinate of the vertex is

.

  4-1 Graphing Quadratic Functions b. Substitute 0.5, 1.5, 1.75, 2 and 3 for x and make the table.

41. FINANCIAL LITERACY A babysitting club sits for 50 different families. They would like to increase their current rate of $9.50 per hour. After surveying the families, the club finds that the number of families will decrease by about 2 for each $0.50 increase in the hourly rate.

  a. Write a quadratic equation that models this situation.

    c. Graph the function.  

b. State the domain and range of this function as it applies to the situation.

  c. What hourly rate will maximize the club’s income? Is this reasonable?

  d. What is the maximum income the club can expect to make?

  SOLUTION:   a. Let x be the number of increase.

    ANSWER:   a. y-int = 9; axis of symmetry: x = 1.75; x-coordinate of vertex = 1.75   b.

  b. The function is defined in the interval [0, 25]. Therefore, .

  The maximum value of the function is 484. Therefore, .

 

  c.

c. $11; Because the function has a maximum at x = 3, it is in the domain. Therefore, three $0.50 increases is reasonable.

  d. The value of the function at x = 3 is 484. Therefore, the maximum income the club can expect to make is $484.

  ANSWER:   a. I(x) = –x2 + 6x + 475 b.

;

  41. FINANCIAL LITERACY A babysitting club sits for 50 different families. They would like to increase their current rate of $9.50 per hour. After surveying the families, the club finds that the number of families will decrease by about 2 for each $0.50 increase in the hourly rate.

eSolutions Manual - Powered by Cognero

 

a. Write a quadratic equation that models this situation.

c. $11; Because the function has a maximum at x = 3, it is in the domain. Therefore, three $0.50 increases is reasonable. d. $484

  Page 27

42. ACTIVITIES Last year, 300 people attended the Franklin High School Drama Club’s winter play. The

ANSWER:   a. $11.50   b. $2645

c. $11; Because the function has a maximum at x = 3, it is in the domain. Therefore, three $0.50 increases is reasonable. 4-1 Graphing Quadratic Functions d. $484

 

  42. ACTIVITIES Last year, 300 people attended the Franklin High School Drama Club’s winter play. The ticket price was $8. The advisor estimates that 20 fewer people would attend for each $1 increase in ticket price.

CCSS TOOLS  Use a calculator to find the  maximum or minimum of each function. Round to the nearest hundredth if necessary.   43. 

 

SOLUTION:  

a. What ticket price would give the greatest income for the Drama Club?

Enter as Y1. KEYSTROKES: Y=   1  2  

 

2

 X    –   2   

1     +   8 Fix the left and right bounds. KEYSTROKES:   2nd    [CALC]    3    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER  

b. If the Drama Club raised its tickets to this price, how much income should it expect to bring in?

  SOLUTION:   a. Let x be the number of increase.

 

  So, the maximum value of the function is –1.19.

The function gets maximum value at 3.5.

 

ANSWER:   min = –1.19 

Therefore, $11.50 will give the greatest income for the Drama Club.

  b. Substitute 3.5 for x in the function and simplify.

 

44.  SOLUTION:   Enter

as Y1.

  KEYSTROKES: Y=     (–)  

  The Drama Club will get $2645.

2   

 

 

ANSWER:   a. $11.50   b. $2645

Fix the left and right bounds.

 

 

2

   X   – 1   

   +   1   9  

  KEYSTROKES:   2nd    [CALC]    4    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER 

CCSS TOOLS  Use a calculator to find the  maximum or minimum of each function. Round to the nearest hundredth if necessary.   43.  SOLUTION:   Enter as Y1. eSolutions Manual - Powered by Cognero KEYSTROKES: Y=   1  2   1  

  +   8

  2

 X    –   2   

So, the maximum value of the function is 23.

  ANSWER:  

Page 28

So, the maximum value of the function is 23.

So, the maximum value of the function is –1.19.

 

ANSWER:   4-1 Graphing Quadratic Functions min = –1.19 

ANSWER:   max = 23  45. 

44. 

SOLUTION:  

SOLUTION:   Enter

Enter

as Y1.

as Y1

 

  2

KEYSTROKES: Y=     (–)   2   

KEYSTROKES: Y=     (–)   8    .   3 

   X   – 1   

2

   X   + 1   4   

   +   1   9  

   –   6  

  Fix the left and right bounds.   KEYSTROKES:   2nd    [CALC]    4    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER  

  Fix the left and right bounds.

  KEYSTROKES:   2nd    [CALC]    4    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER 

 

 

 

 

So, the maximum value of the function is –0.01 .

So, the maximum value of the function is 23.

 

ANSWER:   max = –0.01

ANSWER:   max = 23  46.  45. 

SOLUTION:   SOLUTION:   Enter

Enter as Y1

  KEYSTROKES: Y=     (–)   8    .   3  2

   X   + 1   4   

as Y1.

 

   –   6  

  Fix the left and right bounds.   KEYSTROKES:   2nd    [CALC]    4    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER  

 

eSolutions   Manual - Powered by Cognero

So, the maximum value of the function is –0.01 .

KEYSTROKES: Y=   9  .   7   3  

2

 X    –   1

  –   9

  Fix the left and right bounds.

  KEYSTROKES:   2nd    [CALC]    3    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER 

 

 

Page 29

So, the maximum value of the function is –13.36.

 

So, the maximum value of the function is –13.36.

So, the maximum value of the function is –0.01 .

  ANSWER:   min = –13.36 

ANSWER:   4-1 Graphing Quadratic Functions max = –0.01 47. 

46. 

SOLUTION:  

SOLUTION:   Enter

Enter

as Y1.

  KEYSTROKES: Y=   9  .   7   3  

as Y1.

  2

KEYSTROKES: Y=    2  8  

 X    –   1

8    

  –   9

   X

  –  1  5 –   1  

2

 

 

Fix the left and right bounds.

Fix the left and right bounds.

 

 

KEYSTROKES:   2nd    [CALC]    3    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER 

KEYSTROKES:   2nd    [CALC]    4    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER  

 

 

 

 

So, the maximum value of the function is –13.36.

So, the maximum value of the function is –4.11.

 

 

ANSWER:   min = –13.36 

ANSWER:   max = –4.11  48. 

47. 

SOLUTION:  

SOLUTION:   Enter

Enter

as Y1.

  KEYSTROKES: Y=    2  8   8    

   X

  –  1  5 –   1  

2

  Fix the left and right bounds.

  KEYSTROKES:   2nd    [CALC]    4    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER  

 

eSolutions Manual - Powered by Cognero

 

So, the maximum value of the function is –4.11.

 

as Y1.

  KEYSTROKES: Y=    (–)  1   6    –  1    4  

  –  1  2 

   X

2

  Fix the left and right bounds.

  KEYSTROKES:   2nd    [CALC]    4    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER  

 

 

Page 30

So, the maximum value of the function is –11.92.

 

So, the maximum value of the function is –4.11.

So, the maximum value of the function is –11.92.

 

  ANSWER:   max = –11.92 

ANSWER:   4-1 Graphing Quadratic Functions max = –4.11 

Determine whether each function has a maximum or minimum value, and find that value. Then state the domain and range of the function.

48.  SOLUTION:   Enter

as Y1.

 

  KEYSTROKES: Y=    (–)  1   6    –  1    4  

  –  1  2 

   X

49. 

2

SOLUTION:  

 

Compare the function  with the  standard form of a quadratic function.   Here, a = –5, b = 4 and c = –8.   For this function, a = –5, so the graph opens down and the function has a maximum value.   The x-coordinate of the vertex is

Fix the left and right bounds.

  KEYSTROKES:   2nd    [CALC]    4    ◄  ◄    ◄   ENTER   ►  ►  ►  ►   ►   ►  ►      ENTER   ENTER  

 

.   Substitute 0.4 for x in the function to find the ycoordinate of the vertex.

 

 

So, the maximum value of the function is –11.92.

  ANSWER:   max = –11.92 

  Therefore, the maximum value of the function is –7.2.

Determine whether each function has a maximum or minimum value, and find that value. Then state the domain and range of the function.

 

 

The domain is all real numbers. D = {all real numbers}.

SOLUTION:  

The range is all real numbers less than or equal to the maximum value.

 

49. 

Compare the function  with the  standard form of a quadratic function.   Here, a = –5, b = 4 and c = –8.   For this function, a = –5, so the graph opens down and the function has a maximum value.   The x-coordinate of the vertex is .   Substitute 0.4 for x in the function to find the ycoordinate of the vertex.

eSolutions Manual - Powered by Cognero

 

  ANSWER:   max = –7.2; D = {all real numbers},

50.  SOLUTION:     Compare the function  with the  standard form of a quadratic function. Page 31 Here, a = –4, b = –3 and c = 2.   For this function, a = –4, so the graph opens down

  ANSWER:   max = –7.2; D = {all real numbers}, 4-1 Graphing Quadratic Functions

50. 

ANSWER:   max = 2.5625; D = {all real numbers},

51.  SOLUTION:     Compare the function  with the  standard form of a quadratic function. Here, a = –4, b = –3 and c = 2.   For this function, a = –4, so the graph opens down and the function has a maximum value. The x-coordinate of the vertex is  

SOLUTION:     Compare the function  with the  standard form of a quadratic function.   Here, a = 6, b = 3 and c = –9.   For this function, a = 6, so the graph opens up and the function has a minimum value.

 

.

The x-coordinate of the vertex is

  Substitute

.  for x in the function to find the y-

  Substitute –0.25 for x in the function to find the ycoordinate of the vertex.

coordinate of the vertex .

 

  Therefore, the minimum value of the function is – 9.375.

  Therefore, the maximum value of the function is 2.5625.   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers less than or equal to the maximum value.

  The domain is all real numbers. D = {all real numbers}.

  The range is all real numbers greater than or equal to the minimum value.

    ANSWER:   min = –9.375; D = {all real numbers},

ANSWER:   max = 2.5625; D = {all real numbers}, 52.  51. 

SOLUTION:  

SOLUTION:     Compare the function  with the  standard form of a quadratic function.   Here, a = 6, b = 3 and c = –9.   For this function, a = 6, so the graph opens up and eSolutions Manual - Powered by Cognero the function has a minimum value.

  The x-coordinate of the vertex is

Compare the function standard form of a quadratic function.

 with the 

  Here, a = –4, b = 2 and c = –5.

  For this function, a = –4, so the graph opens down and the function has a maximum value.

 

Page 32

The x-coordinate of the vertex is .

 

 

ANSWER:   min = –9.375; D = {all real numbers}, 4-1 Graphing Quadratic Functions

ANSWER:   max = –4.75; D = {all real numbers},

52. 

53.  SOLUTION:   Compare the function standard form of a quadratic function.

SOLUTION:  

 with the 

 with the

Compare the function

 

standard form of a quadratic function.

Here, a = –4, b = 2 and c = –5.

 

 

For this function, a = –4, so the graph opens down and the function has a maximum value.

Here, a =

 

 

The x-coordinate of the vertex is

, b = 6 and c = –10.

For this function, a =

.

, so the graph opens up and

the function has a minimum value.

 

 

The x-coordinate of the vertex is

Substitute 0.25 for x in the function to find the ycoordinate of the vertex.

.

   

Substitute –4.5 for x in the function to find the ycoordinate of the vertex.

 

  Therefore, the maximum value of the function is – 4.75.   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers less than or equal to the maximum value.

  Therefore, the minimum value of the function is – 23.5.   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers greater than or equal to the minimum value.

  ANSWER:   max = –4.75; D = {all real numbers},

  ANSWER:   min = –23.5; D = {all real numbers},

53.  SOLUTION:    with the

Compare the function standard form of a quadratic function.

  Here, a =

54.  SOLUTION:  

, b = 6 and c = –10.

  Manual - Powered by Cognero eSolutions For this function, a =

, so the graph opens up and

Compare the function standard form of a quadratic function.

 

 with the Page 33

  max =

ANSWER:   min = –23.5; D = {all real numbers}, 4-1 Graphing Quadratic Functions

; D = {all real numbers},

Determine the function represented by each graph.

54. 

 

SOLUTION:    with the

Compare the function standard form of a quadratic function.

  , b = 4 and c = –8.

Here, a =

55. 

  For this function, a =

, so the graph opens down

SOLUTION:   Given graph is a parabola. Therefore, the function 2

and the function has a maximum value.

must be in the form of f (x) = ax + bx + c.   Substitute the points (0, –5) and (2, –9) in the function.

  The x-coordinate of the vertex is .

 

   for x in the function to find the y-

Substitute

 

coordinate of the vertex.

  The vertex of the graph is (2,–9).

  Therefore, the maximum value of the function is

Therefore, the x-coordinate of the vertex is

.

.   The domain is all real numbers. D = {all real numbers}.   The range is all real numbers less than or equal to the maximum value.

  Substitute –4a for b in the first equation and solve for a.

 

   

Substitute 1 for a in the second equation and solve for b.

ANSWER:  

 

max =

; D = {all real numbers},

  Therefore, the required function is 2 f(x) = x – 4x – 5. eSolutions Manual - Powered by Cognero Determine the function represented

by each

Page 34

graph.

ANSWER:  

 

f(x) = x – 4x – 5

2

for b.

   

4-1 Graphing Quadratic Functions Therefore, the required function is 2 f(x) = x – 4x – 5.

  Therefore, the required function is

ANSWER:  

2

f(x) = x + 2x – 6.

2

f(x) = x – 4x – 5 ANSWER:   2

f(x) = x + 2x – 6

56.  SOLUTION:   Given graph is a parabola. Therefore, the function 2 must be in the form of f (x) = ax + bx + c.

 

57.  SOLUTION:   Given graph is a parabola. Therefore, the function 2

Substitute the points (–1, –7) and (0, –6) in the function.

must be in the form of f (x) = ax + bx + c.

 

Substitute the points (0, 8) and (3, –1) in the function.

   

   

  The vertex of the graph is (–1,–7).

 

 

Therefore, the x-coordinate of the vertex is

The vertex of the graph is (3,–1).

 

.

Therefore, the x-coordinate of the vertex is

 

.

  Substitute 2a for b in the first equation and solve for a.

 

 

Substitute –6a for b in the first equation and solve for a.

 

 

Substitute 1 for a in the second equation and solve for b.

Substitute 1 for a in the second equation and solve for b.

 

 

 

 

eSolutions Manual -the Powered by Cognero Therefore, required function

2

f(x) = x + 2x – 6.

is

Therefore, the required function is 2 f(x) = x – 6x + 8.

Page 35

Therefore, the required function is 2

f(x) = x + 2x – 6. ANSWER:   4-1 Graphing Quadratic Functions 2 f(x) = x + 2x – 6

Therefore, the required function is 2 f(x) = x – 6x + 8. ANSWER:   2

f(x) = x – 6x + 8 58. MULTIPLE REPRESENTATIONS Consider f 2 2 (x) = x – 4x + 8 and g(x) = 4x – 4x + 8.

  a. TABULAR Make a table of values for f (x) and g (x) if

  b. GRAPHICAL Graph f (x) and g(x).

57.  SOLUTION:   Given graph is a parabola. Therefore, the function

 

must be in the form of f (x) = ax + bx + c.

c. VERBAL Explain the difference in the shapes of the graphs of f (x) and g(x). What value was changed to cause this difference?

 

 

Substitute the points (0, 8) and (3, –1) in the function.

d. ANALYTICAL Predict the appearance of the

 

graph of h(x) = 0.25x – 4x + 8. Confirm your prediction by graphing all three functions if

2

2

 

 

SOLUTION:   a.

  The vertex of the graph is (3,–1).

  Therefore, the x-coordinate of the vertex is .

  Substitute –6a for b in the first equation and solve for a.

b.

  Substitute 1 for a in the second equation and solve for b.

 

  Therefore, the required function is 2 f(x) = x – 6x + 8. ANSWER:  

c. Sample answer: g(x) is much narrower than f (x). The value of a changed from 1 to 4. d. Sample answer: The graph of h(x) will be wider than f (x).

2

f(x) = x – 6x + 8 eSolutions Manual - Powered by Cognero REPRESENTATIONS 58. MULTIPLE

Consider f 2 2 (x) = x – 4x + 8 and g(x) = 4x – 4x + 8.

 

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c. Sample answer: g(x) is much narrower than f (x). The value Quadratic of a changed from 1 to 4. 4-1 Graphing Functions d. Sample answer: The graph of h(x) will be wider than f (x). 59. VENDING MACHINES Omar owns a vending machine in a bowling alley. He currently sells 600 cans of soda per week at $0.65 per can. He estimates that he will lose 100 customers for every $0.05 increase in price and gain 100 customers for every $0.05 decrease in price. (Hint: The charge must be a multiple of 5.)

  a. Write and graph the related quadratic equation for a price increase. ANSWER:   a.

  b. If Omar lowers the price, what price should he charge in order to maximize his income?

  c. What will be his income per week from the vending machine?

  SOLUTION:   a. Let x be the number of increase.

  Convert the price into cents.

b.  

c. Sample answer: g(x) is much narrower than f (x). The value of a changed from 1 to 4. d. Sample answer: The graph of h(x) will be wider thanf (x).

  b. Let x be the number of decrease. Convert the price into cents.

 

  The function is maximum at 3.5.

 

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59. VENDING MACHINES Omar owns a vending machine in a bowling alley. He currently sells 600 cans of soda per week at $0.65 per can. He

Therefore, Omar should charge (65 – 5(3.5) = 475) 45 cents or 50 cents.   Page 37 c. Suppose the number of decrease is 3. Then his income is:

  The function is maximum at 3.5.

 

4-1 Graphing Quadratic Functions Therefore, Omar should charge (65 – 5(3.5) = 475) 45 cents or 50 cents.   c. Suppose the number of decrease is 3. Then his income is:

  f(x) = (65 – 5(3))(600 + 100(3)) = 50 × 900 = 45000 cents or $450

  Suppose the number of decrease is 4. Then his income is:

 

b. Omar can charge at 45 cents or 50 cents. c. $450 per week 60. BASEBALL Lolita throws a baseball into the air and the height h of the ball in feet at a given time t in seconds after she releases the ball is given by the function

  2

h(t) = –16t + 30t + 5.

  a. State the domain and range for this situation.   b. Find the maximum height the ball will reach.

f(x) = (65 – 5(4))(600 + 100(4)) = 45 × 1000 = 45000 cents or $450 Omar’s income per week from the vending machine is $450.

 

ANSWER:  

 

a. f (x) = 39,000 – 3500x – 500x

2

SOLUTION:   a. Time t is always positive. So, t is greater than or equal to zero. The t-intercept of the function is 2.09. Therefore,

.

  The x-coordinate of the vertex is

  The maximum of the function.

 

b. Omar can charge at 45 cents or 50 cents. c. $450 per week 60. BASEBALL Lolita throws a baseball into the air and the height h of the ball in feet at a given time t in seconds after she releases the ball is given by the function

  2

h(t) = –16t + 30t + 5.

  a. State the domain and range for this situation.   b. Find the maximum height the ball will reach.

  SOLUTION:   a. Time t is always positive. So, t is greater than or equal to zero. The t-intercept of the function is 2.09.

  Therefore,

.

  b. The maximum height the ball will reach is 19.0625 ft.

  ANSWER:   a. b. 19.0625 ft 61. CCSS CRITIQUE  Trent thinks that the function f (x) graphed below, and the function g(x) described

next to it have the same maximum. Madison thinks that g(x) has a greater maximum. Is either of them correct? Explain your reasoning.

  Therefore,

.

 

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The x-coordinate of the vertex is

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  ANSWER:   a. 4-1 Graphing Quadratic Functions b. 19.0625 ft 61. CCSS CRITIQUE  Trent thinks that the function f (x) graphed below, and the function g(x) described

next to it have the same maximum. Madison thinks that g(x) has a greater maximum. Is either of them correct? Explain your reasoning.

ANSWER:   Sample answer: Always; the coordinates of a quadratic function are symmetrical, so x-coordinates equidistant from the vertex will have the same ycoordinate. 63. CHALLENGE The table at the right represents some points on the graph of a quadratic function.

  a. Find the values of a, b, c, and d.

  b. What is the x-coordinate of the vertex?

  c. Does the function have a maximum or a minimum?

 

 

      SOLUTION:   Sample answer: Madison. Sample answer: f (x) has a maximum of –2. g(x) has a maximum of 1. When Trent found the x-coordinate of the vertex, he multiplied two negatives and mistakenly kept a negative. ANSWER:   Sample answer: Madison. When Trent found the xcoordinate of the vertex, he multiplied two negatives and mistakenly kept a negative. 62. REASONING Determine whether the following is sometimes, always, or never true. Explain your reasoning. In a quadratic function, if two x-coordinates are equidistant from the axis of symmetry, then they will have the same y-coordinate. SOLUTION:   Sample answer: Always; the coordinates of a quadratic function are symmetrical, so x-coordinates equidistant from the vertex will have the same ycoordinate. ANSWER:   Sample answer: Always; the coordinates of a quadratic function are symmetrical, so x-coordinates equidistant from the vertex will have the same ycoordinate.

  SOLUTION:   a. Substitute the points (–20, –377), (–5, –2), and (–1, 22) from the table in the general quadratic function f 2 (x) = ax + bx + c to get a system of three equations in three variables.

  400a – 20b + c = –377 25a – 5b + c = –2

  a – b + c = 22

  The solution of the system is a = –1, b = 0, and c = 2

23. So, the quadratic function is f (x) = –x + 23. Substitute (5, a – 24) into f (x) to find a.

 

  Substitute (7, –b) into f (x) to find b.

 

63. CHALLENGE The table at the right represents quadratic function.

eSolutions - Powered by Cognero someManual points on the graph of a

Page 39

 

 

a. Find the values of a, b, c, and d.

Substitute (c, –13) into f (x) to find c.

  4-1 Graphing Quadratic Functions Substitute (7, –b) into f (x) to find b.  

ANSWER:   a. a = 22; b = 26; c = –6; d = 2 b. 0 c. maximum 64. OPEN ENDED Give an example of a quadratic function with a

  a. maximum of 8.

 

 

Substitute (c, –13) into f (x) to find c.

 

b. minimum of –4.

  c. vertex of (–2, 6).

  SOLUTION:   a. Sample answer: f (x) = –x2 + 8

 

 

Substitute (d – 1, a) or (d – 1, 22) into f (x) to find d.

b. Sample answer: f (x) = x – 4

 

2

  c. Sample answer: f (x) = x2 + 4x + 10

  ANSWER:   2

a. Sample answer: f (x) = –x + 8 b. Sample answer: f (x) = x2 – 4 2

c. Sample answer: f (x) = x + 4x + 10

  So, a = 22, b = 26, c = –6, and d = 2.

  b. Because b = 0, the x-coordinate of the vertex is 0.

  c. For this function, a = –1, so the graph opens down and the function has a maximum value.

  ANSWER:   a. a = 22; b = 26; c = –6; d = 2 b. 0 c. maximum 64. OPEN ENDED Give an example of a quadratic function with a

  a. maximum of 8.

 

65. WRITING IN MATH Why can the discriminant be used to confirm the number and the type of solutions

of a quadratic equation? SOLUTION:  

Sample answer: If the discriminant is positive, the Quadratic Formula will result in two real solutions because you are adding and subtracting the square root of a positive number in the numerator of the expression. If the discriminant is zero, there will be one real solution because you are adding and subtracting the square root of zero. If the discriminant is negative, there will be two complex solutions because you are adding and subtracting the square root of a negative number in the numerator of the expression.  

b. minimum of –4.

ANSWER:  

 

Sample answer: If the discriminant is positive, the Quadratic Formula will result in two real solutions because you are adding and subtracting the square root of a positive number in the numerator of the expression. If the discriminant is zero, there will be one real solution because you are adding and Page 40 subtracting the square root of zero. If the discriminant is negative, there will be two complex solutions because you are adding and subtracting

c. vertex of (–2, 6).

  SOLUTION:   a. Sample answer: f (x) = –x2 + 8

  Manual - Powered by Cognero eSolutions 2

b. Sample answer: f (x) = x – 4

 

ANSWER:   2

a. Sample answer: f (x) = –x + 8 2 b. Sample Quadratic answer: f (x)Functions =x –4 4-1 Graphing 2 c. Sample answer: f (x) = x + 4x + 10 65. WRITING IN MATH Why can the discriminant be used to confirm the number and the type of solutions

subtracting the square root of zero. If the discriminant is negative, there will be two complex solutions because you are adding and subtracting the square root of a negative number in the numerator of the expression. 66. Which expression is equivalent to

of a quadratic equation?

 

SOLUTION:  

A

Sample answer: If the discriminant is positive, the Quadratic Formula will result in two real solutions because you are adding and subtracting the square root of a positive number in the numerator of the expression. If the discriminant is zero, there will be one real solution because you are adding and subtracting the square root of zero. If the discriminant is negative, there will be two complex solutions because you are adding and subtracting the square root of a negative number in the numerator of the expression.  

  B

  C

  D

  SOLUTION:  

ANSWER:  

Therefore, option B is the correct answer.

Sample answer: If the discriminant is positive, the Quadratic Formula will result in two real solutions because you are adding and subtracting the square root of a positive number in the numerator of the expression. If the discriminant is zero, there will be one real solution because you are adding and subtracting the square root of zero. If the discriminant is negative, there will be two complex solutions because you are adding and subtracting the square root of a negative number in the numerator of the expression.

ANSWER:   B 67. SAT/ACT The price of coffee beans is d dollars for 6 ounces, and each ounce makes c cups of coffee. In terms of c and d, what is the cost of the coffee beans required to make 1 cup of coffee?

  F

  G

66. Which expression is equivalent to

 

 

H  

 

A

J 6cd

 

 

B



  C

 

 

SOLUTION:  

D

 

The cost of the 1 ounce coffee beans is

.

SOLUTION:   ounce of coffee beans is need to make 1cup of coffee. Therefore, the cost of the coffee bean required to Therefore, the correct eSolutions Manual -option PoweredBbyisCognero ANSWER:   B

answer.

make 1 cup of coffee is

.

Page 41

make 1 cup of coffee is

Therefore, option B is the correct answer. ANSWER:   4-1 Graphing Quadratic Functions B

.

ANSWER:   K

67. SAT/ACT The price of coffee beans is d dollars for 6 ounces, and each ounce makes c cups of coffee. In terms of c and d, what is the cost of the coffee beans required to make 1 cup of coffee?

 

68. SHORT RESPONSE Each side of the square base of a pyramid is 20 feet, and the pyramid’s height is 90 feet. What is the volume of the pyramid? SOLUTION:   Volume of a right regular pyramid is

F

.

 

 

G

Base area = 20 × 20 = 400 ft

 

 

2

H  

  J 6cd

 

 

3

Therefore, the volume of the pyramid is 12000 ft .



 

 

ANSWER:  

SOLUTION:  

12,000 ft

The cost of the 1 ounce coffee beans is

69. Which ordered pair is the solution of the following system of equations?

.

ounce of coffee beans is need to make 1cup of coffee. Therefore, the cost of the coffee bean required to make 1 cup of coffee is

3

  3x – 5y = 11 3x – 8y = 5

  A  (2, 1)

.

  B   (7, –2)

ANSWER:   K

  C  (7, 2)

68. SHORT RESPONSE Each side of the square base of a pyramid is 20 feet, and the pyramid’s height is 90 feet. What is the volume of the pyramid?

  D

 

SOLUTION:   Volume of a right regular pyramid is

SOLUTION:   Subtract the second equation from the first equation.

.

  Base area = 20 × 20 = 400 ft

2

 

Substitute 2 for y in the first equation and solve for x.

  3

Therefore, the volume of the pyramid is 12000 ft .

 

eSolutions Manual - Powered by Cognero

ANSWER:   3

The solution is (7, 2). Therefore, option C is the correct answer. ANSWER:  

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Therefore, the volume of the pyramid is 12000 ft .

The solution is (7, 2). Therefore, option C is the correct answer.

  ANSWER:   4-1 Graphing Quadratic Functions 3 12,000 ft

ANSWER:   C

69. Which ordered pair is the solution of the following system of equations?

Find the inverse of each matrix, if it exists.

 

  70. 

3x – 5y = 11 3x – 8y = 5

 

SOLUTION:  

A  (2, 1)

 

Let

B   (7, –2)

.

 

 

det (A) = 5

C  (7, 2)

 

  D

  SOLUTION:   Subtract the second equation from the first equation.

  ANSWER:   Substitute 2 for y in the first equation and solve for x.

The solution is (7, 2). Therefore, option C is the correct answer.

71. 

ANSWER:   C

SOLUTION:   Let

Find the inverse of each matrix, if it exists.

 

.

det (A) = –24

70.  SOLUTION:   Let

.

  det (A) = 5

ANSWER:  

 

eSolutions Manual - Powered by Cognero

 

72. 

Page 43

SOLUTION:  

  ANSWER:   45

4-1 Graphing Quadratic Functions

74. 

72. 

SOLUTION:  

SOLUTION:   Let

.

  det (A) = 14

 

  ANSWER:   22 75. 

 

SOLUTION:  

ANSWER:  

  Evaluate each determinant.

  73.  SOLUTION:  

ANSWER:   0 76. MANUFACTURING The Community Service Committee is making canvas tote bags and leather tote bags for a fundraiser. They will line both types of bags with canvas and use leather handles on both. For the canvas bags, they need 4 yards of canvas and 1 yard of leather. For the leather bags, they need 3 yards of leather and 2 yards of canvas. The committee leader purchased 56 yards of leather and 104 yards of canvas.

 

 

ANSWER:   45

a. Let c represent the number of canvas bags, and let  represent the number of leather bags. Write a  system of inequalities for the number of bags that can be made.

  74. 

b. Draw the graph showing the feasible region.

  SOLUTION:  

c. List the coordinates of the vertices of the feasible region.

  d. If the club plans to sell the canvas bags at a profit of $20 each and the leather bags at a profit of $35 each, write a function for the total profit on the bags. eSolutions   Manual - Powered by Cognero

ANSWER:  

 

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e . How can the club make the maximum profit?

 

  c. List the coordinates of the vertices of the feasible region.

  4-1 Graphing Quadratic Functions d. If the club plans to sell the canvas bags at a profit of $20 each and the leather bags at a profit of $35 each, write a function for the total profit on the bags.   e . How can the club make the maximum profit?

  f. What is the maximum profit?

  The club makes the maximum profit if they produce 20 canvas tote bags and 12 leather tote bags.

  f. The maximum profit is $820.

  ANSWER:   a. b.

  SOLUTION:   a.

  b.

c. (0, 0), (26, 0), (20, 12), d. e. Make 20 canvas tote bags and 12 leather tote bags. f. $820

  c. The vertices of the solution region is (0, 0), (26, 0), (20, 12) and

.

State whether each function is a linear function. Write yes or no. Explain.

  2

 

77. y = 4x – 3x

d. The optimal function is

.

  e . Substitute the points (0, 0), (26, 0), (20, 12) and in the function.

 

SOLUTION:   No. It cannot be written as y = mx + b.

  ANSWER:   No; it cannot be written as y = mx + b. 78. y = –2x – 4 SOLUTION:   Yes. It is written in y = mx + b form.

 

  The club makes the maximum profit if they produce 20 canvas tote bags and 12 leather tote bags.

  f. The maximum profit is $820.

  Manual - Powered by Cognero eSolutions

ANSWER:   Yes; it is written in y = mx + b form. 79. y = 4 SOLUTION:   Yes. It is written in y = mx + b form, m = 0.  

 

ANSWER:   ANSWER:  

Page 45

Yes. It is written in y = mx + b form.

 

  ANSWER:   4-1 Graphing Quadratic Functions Yes; it is written in y = mx + b form.

ANSWER:   –13 2

79. y = 4

82. f (x) = 6x + 18, x = –5

SOLUTION:   Yes. It is written in y = mx + b form, m = 0.  

 

 

 

SOLUTION:   Substitute –5 for x in the function and evaluate.

ANSWER:   Yes; it is written in y = mx + b form, m = 0. Evaluate each function for the given value.

  2

80. f (x) = 3x – 4x + 6, x = –2 SOLUTION:   Substitute –2 for x in the function and evaluate.

  ANSWER:   168

 

  ANSWER:   26 2

81. f (x) = –2x + 6x – 5, x = 4 SOLUTION:   Substitute 4 for x in the function and evaluate.

 

  ANSWER:   –13 2

82. f (x) = 6x + 18, x = –5

  SOLUTION:   Substitute –5 for x in the function and evaluate.

 

  eSolutions Manual - Powered by Cognero ANSWER:  

168

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