Math 30-1 1

1.2A Stretches1. The graph of y + 3 = f(x) is the graph of f(x) translated…

up 3 units left 3 units down 3 units right 3 units

2. The graph of f(x) + 4 is the graph of f(x) translated…

4 units up 4 units left 4 units down 4 units right

3. The graph of f(x – 7) + 6 is the graph of f(x) that has been translated..

7 units left, 6 units up 7 units left, 6 units down

7 units right, 6 units down 7 units right, 6 units up

4. In general, the graph of f(x – h) + k, where h and k are positive, as compared to the parent function graph f(x), is translated

h units left and k units up h units right and k units up h units left and k units down h units right and k units down

x

x

x

x

Math 30-1 2

5. When the output of a function y = f(x) is multiplied by -1, the result, y = -f(x), is a reflection of the graph in the a. x – axisb. y – axisc. line y = x.

6. When y = f(x) is transformed to y = f(-x), then (x, y) is transformed to d. (-x, -y)e. (x , -y)f. (-x, y)

Math 30-1 3

Function Transformations: Vertical Stretch

1.2A Vertical Stretches

Vertical Stretches: The effect of parameter a. ( )y af xA stretch changes the shape of the graph. (Translations changed the position of a graph.)

( )y af x|a| describes a vertical stretch about the x-axis.

An Invariant point is a point on a graph that remains unchanged after a transformation.

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In general, for any function y = f(x), the graph of the function y =a f(x) has been vertically stretched about the x-axis by a factor of |a| .

The point (x, y) → (x, ay). Only the y coordinates are affected.

Vertical Stretches

Invariant points are on the line of stretch, the x-axis. are the x-intercepts.

When |a| > 1, the points on the graph move farther away from the x-axis.

When |a|< 1, the points on the graph move closer to the x-axis.

( )y k fa x h

3 ( )y f x Vertical stretch by a factor of 3

1( )

3y f x Vertical stretch by a factor of ⅓

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Vertical Stretching y = af(x), |a| > 1y = f(x)A vertical stretch

about the x-axis by a factor of 2.

y = 2f(x)

Key Points

(x, y) → (x, 2y)

(-2, 0) → (-2, 0)

(-1, -7) → (-1, -14)

(1/2, 1) → (1/2, 2)

Invariant Points

(-2, 0)

(0, 0)

(1, 0)

(2, 0)

Domain and Range

Math 30-1 6

Consider

Write the equation of the function after a vertical stretch about the x-axis by a factor of ½.

Write the coordinates of the image of the point (-7, 7)

Write the transformation in function notation.

The point (x, y) maps to

y 1

2f (x)

(-7, 3.5)

List any invariant points.

How are the domain and range affected?

Vertical Stretches about the x-axis y = af(x), |a| < 1y x

1

2y x

1,2

x y

0,0

2y x

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The graph of g(x) is a transformation of f(x).

Is the transformation a translation?

Is the transformation a vertical stretch?

Invariant Point

(2, 0)→ (1, 0)

On the y-axis

(x, y)→(½x, y)

Horizontal stretchBy a factor of ½

(2 )y f x

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In general, for any function y = f(x), the graph of the function

y = f(bx) has been horizontally stretched by a factor of .

The point (x, y) →

Horizontal Stretches

1

b

x

b, y

Invariant points are on the line of stretch, the y-axis. are the y-intercepts.

When |b| > 1, the points on the graph move closer to the y-axis.

When |b|< 1, the points on the graph move farther away from the x-axis.

( )y bk af x h

(3 )y f x Horizontal stretch by a factor of ⅓

1

4y f x

Horizontal stretch by a factor of 4

Only the x coordinates are affected.

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Consider y xWrite the equation of the function after a horizontal stretch about the y-axis by a factor of 3.

y 1

3x

Write the coordinates of the image of the point (-3, 3 )

Write the transformation in function notation. y f1

3x

Write the coordinates of the image of the point (x, y) 3x, y

List any invariant points.

How are the domain and range affected?

Characteristics of Horizontal Stretches about the y-axis

3y x

Can it be written in any other way?

→ (-9, 3)

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Is This a Horizontal or Vertical Stretch of y = f(x)?

y = f(x)

The graph y = f(x) is stretched vertically about the x-axis bya factor of .

1

2

2 ( )y f x

1( )

2y f x

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Has the graph been stretched Horizontally or Vertically?

(1, 1)

2(2 )y x

24y x

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Zeros of a Function

1. What are the zeros of the function?

2. Use transformations to determine the zeros of the following functions.

b) 2 ( )y P x

1c)

2y P x

a) ( 1)y P x x = -1, 1, 4

x = -2, 0, 3

x = -2, 0, 3

x = -4, 0, 6

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Describe the transformation in words compared to the graph of a function y = f(x).

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

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

Stretched horizontally by a factor of 2 about the y-axis.

Describing the Horizontal or Vertical Stretch of a Function

Stretched horizontally by a factor of about the y-axis.

1

3

Stretch vertically by a factor

of about the x-axis.1

3

1

2Stretched vertically by a factor

of about the x-axis.1

2

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Consider the graph of a function y = f(x)

The transformation described by y = f(2x+4) is horizontal stretch about the y-axis by a factor of ½.

The translation described by y = f(2x + 4) is horizontal shift of 4 units to the left.

Translations must be factored ( ( ))y k af b x h

y = f(2(x + 2))

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

a) Stretched horizontal by a factor of one-third about the y-axis, and stretched vertically about the x-axis by a factor of two.

y = 2f(3x)

b) Stretched horizontally by a factor of two about the y-axis and translated four units to the left.

Stating the Equation of y = af(kx)

1

2y f x

14

2y f x

12

2y f x

Math 30-1 16

AssignmentPage 282, 5a,b, 6, 7a,c, 8, 13, 14c, d