Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various...

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Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility to create a scatter plot of the data. b) Use the regression feature of a graphing utility to find a quadratic model that best fits the data. c) Use the model to predict the speed that gives the greatest mileage. Speed, x Mileage, y 10, 21.3 45, 31.9 15, 23.7 50, 29.5 20, 25.9 55, 27.6 25, 27.6 60, 25.3 30, 29.4 65, 23.0 35, 31.0 70, 20.0 40, 31.7

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Transcript of Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various...

Page 1: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

Example 2The following table gives the mileage y, in miles per gallon, of a certain

car at various speeds x (in miles per hour).a) Use a graphing utility to create a scatter plot of the data.b) Use the regression feature of a graphing utility to find a quadratic

model that best fits the data.c) Use the model to predict the speed that gives the greatest mileage.Speed, x Mileage, y10, 21.3 45, 31.915, 23.7 50, 29.520, 25.9 55, 27.625, 27.6 60, 25.330, 29.4 65, 23.035, 31.0 70, 20.040, 31.7

Page 2: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

Example 2The following table gives the mileage y, in miles per gallon, of a certain

car at various speeds x (in miles per hour).a) Use a graphing utility to create a scatter plot of the data.b) Use the regression feature of a graphing utility to find a quadratic

model that best fits the data.c) Use the model to predict the speed that gives the greatest mileage.Speed, x Mileage, y10, 21.3 45, 31.915, 23.7 50, 29.520, 25.9 55, 27.625, 27.6 60, 25.330, 29.4 65, 23.035, 31.0 70, 20.040, 31.7

Page 3: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

Example 3

For the data points below, determine whether a linear model or a quadratic model best fits the data.(1, 5)(2, 6)(3, 8)(4, 9)(5, 11)(6, 10)(7, 11)(8, 12)(9, 14)(10, 16)

Page 4: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

Example 3

For the data points below, determine whether a linear model or a quadratic model best fits the data.(1, 5)(2, 6)(3, 8)(4, 9)(5, 11)(6, 10)(7, 11)(8, 12)(9, 14)(10, 16)

Page 5: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#1

Sketch the graph of

by hand and identify the vertex and the intercepts.

f x x x( ) 2 6 5

Page 6: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x( ) 2 6 5

Page 7: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#2

Find the number of units that produce a

minimum cost C if C x x x( ) . , . 01 90 15 0002

Page 8: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

C x x x( ) . , . 01 90 15 0002

Page 9: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#3

Find the quadratic function that has a maximum at (1, 7) and passes through the point (2, 5).

Page 10: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

maximum at (1, 7) and passes through the point (2, 5).

Page 11: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#4

Find two quadratic functions that have

x-intercepts (2,0) and 4

30,F

HGIKJ

Page 12: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

x-intercepts (2,0) and 4

30,F

HGIKJ

Page 13: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#5

Use the leading Coefficient Test to determine the right-hand and left-hand behavior of the graph of the polynomial functionf x x x( ) 3 2 175 3

Page 14: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x( ) 3 2 175 3

Page 15: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#6

Find all the real zeros of Verify your answer with a graphing utility.

f x x x x( ) 5 35 4

Page 16: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x x( ) 5 35 4

Page 17: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#7

Find a polynomial function with 0, 3,

and -2 as zeros.

Page 18: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

0, 3, and -2 as zeros.

Page 19: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#8

Sketch by hand.f x x x( ) 3 12

Page 20: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x( ) 3 12

Page 21: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#9

Divide by x - 3

using long division.

3 7 2 104 2x x x

Page 22: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

3 7 2 104 2x x x x - 3

Page 23: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#10

Divide byx3 11 x x2 2 1 .

Page 24: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

x3 11x x2 2 1 .

Page 25: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#11

Use synthetic division to divide

by x + 5.3 13 12 15 4x x x

Page 26: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

3 13 12 15 4x x x x + 5

Page 27: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#12

Use synthetic division to find

f(-6) when f x x x x( ) 7 40 12 153 2

Page 28: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x x( ) 7 40 12 153 2f(-6)

Page 29: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#13

Find the real zeros of f x x x( ) 3 19 30

Page 30: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x( ) 3 19 30

Page 31: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#14

Find the real zeros of f x x x x x( ) . 4 3 28 9 9

Page 32: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x x x( ) . 4 3 28 9 9

Page 33: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#15

List all possible rational zeros of the functionf x x x x( ) . 6 5 4 153 2

Page 34: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x x( ) . 6 5 4 153 2

Page 35: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#16

Write

as a product of linear factors.

f x x x x x( ) 4 3 23 5 10

Page 36: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x x x( ) 4 3 23 5 10

Page 37: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#17

Write in standard form.2

1 i

Page 38: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

2

1 i

Page 39: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#18

Write in standard form.3

2

1

4

i i

Page 40: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

3

2

1

4

i i

Page 41: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#19

Find a polynomial with real coefficients

that has 2, 3 + i, and 3 - 2i as zeros.

Page 42: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

2, 3 + i, and 3 - 2i as zeros

Page 43: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#20

Use synthetic division to show that 3i is

a zero of f x x x x( ) 3 24 9 36

Page 44: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f x x x x( ) 3 24 9 363i

Page 45: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#21

Sketch the graph of and label all

intercepts and asymptotes.

f xx

x( )

1

2

Page 46: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f xx

x( )

1

2

Page 47: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#22

Sketch the graph of and label all

intercepts and asymptotes.

f xx

x( )

3 42

Page 48: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f xx

x( )

3 42

Page 49: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#23

Find all the asymptotes of f xx

x( )

8 9

1

2

2

Page 50: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f xx

x( )

8 9

1

2

2

Page 51: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#24

Find all the asymptotes of f xx x

x( ) .

4 2 7

1

2

Page 52: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f xx x

x( ) .

4 2 7

1

2

Page 53: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

#25

Sketch the graph of f xx

x x( )

5

202

Page 54: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

f xx

x x( )

5

202

Page 55: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.
Page 56: Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

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