1.4.1MATHPOWERTM 12, WESTERN EDITION

Chapter 1 Transformations1.4

x y

0

1

2

3

f(x) = | x | x y

0

1

2

3

x y

0

1

2

3

f(x) = 2 | x | f(x) = 3 | x |

0

1

2

3

x y

0

1

2

3

x y

0

1

2

3

f (x)

1

2 x

f (x)

1

3 x

1.4.2

Vertical Stretches of Functions

f(x) = | x |

1.4.3

Vertical Stretches of Functions [cont’d]

A stretch can bean

A stretch can be

Given the graph of y = f(x), there is a 1.4.4

Graphing y = af(x)

y = | x |

Given the graph of y = f(x), there is a 1.4.5

Graphing y = af(x)

y = | x |

In general, for any function y = f(x), the graph of a functiony = af(x) is obtained by multiplying the y-value of each point on the graph of y = f(x) by a.

That is, the point (x, y) on the graph of y = f(x) is transformed into the point (x, ay) on the graph of y = af(x).

• If a < 0, the graph is in the x-axis.

1.4.6

Vertical Stretching and Reflecting of y = f(x)

y = f(x)

For y = af(x), there is a vertical stretch.

1.4.7

Vertical Stretching and Reflecting of y = f(x)

x y

0

1

2

3

f(x) = x2

x y

0

0.5

1

1.5

f(x) = (2x)2

0

1

4

9

x y

0

2

4

6

f(x) = (0.5x)2

Each point on the graph of y = (2x)2 is half as far from the y-axis as the related point on the graph of y = x2.

The graph of y = f(2x) is a of the graph of y = f(x) by a factor of

Each point on the graph of y = (0.5x)2 is as the related point on the graph of y = x2.

The graph of y = f(0.5x) is a of the graph of y = f(x) by a factor of 1.4.8

Horizontal Stretching of y = f(x)

y = x2

(-1, 1) (1, 1)

For y = f(kx), there is a

Horizontal Stretching of y = f(kx) when k > 1

1.4.9

(-1, 1) (1, 1)

y = x2

For y = f(kx), there is a

1.4.10

Horizontal Stretching of y = f(kx) when 0 < k < 1

In general, for any function y = f(x), the graph of the functiony = f(kx) is obtained by at each pointon the graph of y = f(x) by k.

That is, the point (x, y) on the graph of the function y = f(x) istransformed into the point on the graph of y = f(kx).

• If k < 0, there is also a in the .

(x

k, y)

Comparing y = f(x) With y = f(kx)

1.4.11

y = f(x)

Graph y = f(2x).

1.4.12

Graphing y = f(kx) and its Reflection

Describe what happens to the graph of a function y = f(x).

a) y = f(3x) b) 3y = f(x)

c) y = f( x) d) -2y = f(x)

e) 2y = f(2x) f) y = f(-3x)

1.4.13

Describing the Horizontal or Vertical Stretch of a Function

1

2

1

4

The graph of the function y = f(x) is transformed as described. Write the new equation in the form y = af(kx).

a) Horizontal stretch factor of one-third, and a vertical stretch factor of two

b) Horizontal stretch factor of two, a vertical stretch by a factor of one-third, and a reflection in the x-axis

c) Horizontal stretch factor of one-fourth, a vertical stretch factor of three, and a reflection in the y-axis

d) Horizontal stretch factor of three, vertical stretch factor of one-half, and a reflection in both axes

1.4.14

Stating the Equation of y = af(kx)

Given the graph ofsketch the graphs with the following transformations. a) Stretch horizontally by a factor of 2.

y 16 x 4 2=

1.4.15

,

State the zeros of this polynomial, and a possible equation of P(x).Graphing a Polynomial and its Transformations

1.4.16

Suggested Questions:Pages 38-401-26, 27-41,45-51

1.4.17