Math 1014: Precalculus with Transcendentals Ch. 2: Functions and Graphs Sec. 2.5 Transformations of Functions I. Transformations of Functions

-1 1 2 3 4 5 6 7 8 9x

1

2

3y

fHxL= x

standard function

A. Vertical Shift

Let f be a function and c a positive real number. • The graph of y = f (x)+ c is the graph of y = f (x) shifted c units vertically upward. • The graph of y = f (x)− c is the graph of y = f (x) shifted c units vertically

downward.

-1 1 2 3 4 5 6 7 8 9x

123456y

fHxL= x +3

-1 1 2 3 4 5 6 7 8 9x

-3

-2

-1

yfHxL= x -3

vertical shift up 3 units vertical shift down 3 units B. Horizontal Shift

Let f be a function and c a positive real number. • The graph of y = f (x + c) is the graph of y = f (x) shifted to the left c units. • The graph of y = f (x − c) is the graph of y = f (x) shifted to the right c units.

-3 -2 -1 1 2 3 4 5 6 7 8 9x

123

yfHxL= x + 3

1 2 3 4 5 6 7 8 9x

1

2

yfHxL= x - 3

horizontal shift left 3 units horizontal shift right 3 units

C. Reflection about x-axis

The graph of y = − f (x) is the graph of y = f (x) reflected about the x-axis.

-2 -1 1 2 3 4 5 6 7 8 9x

-3

-2

-1

yfHxL=- x

D. Reflection about y-axis

The graph of y = f (−x) is the graph of y = f (x) reflected about the y-axis.

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2x

1

2

3y

fHxL= -x

E. Vertical Stretching and Shrinking

Let f be a function and c a positive real number. • If c >1 , the graph of y = c f (x) is the graph of y = f (x) vertically stretched by

multiplying each of its y-coordinates by c.

• If 0 < c <1 , the graph of y = c f (x) is the graph of y = f (x) vertically shrunk by multiplying each of its y-coordinates by c.

-1 1 2 3 4 5 6 7 8 9x

123456789y

fHxL=3 x

-1 1 2 3 4 5 6 7 8 9x

1y

fHxL=13

x

vertical stretching vertical shrinking

F. Horizontal Stretching and Shrinking Let f be a function and c a positive real number. • If c >1 , the graph of y = f (cx) is the graph of y = f (x) horizontally shrunk by

dividing each of its x-coordinates by c.

• If 0 < c <1 , the graph of y = f (cx) is the graph of y = f (x) horizontally stretched by dividing each of its x-coordinates by c.

-1 1 2 3 4 5 6 7 8 9x

1

2

3

4

5

yfHxL= 3 x

-1 1 2 3 4 5 6 7 8 9x

1

y

fHxL= 13x

horizontal shrinking horizontal stretching II. Graphing Using Sequence of Transformations

A. Use the graph of f (x) = x2 to graph g(x) = − 12 (x −1)

2 + 3 .

-3-2-1 1 2 3x

123456789y

fHxL=x2

-2-1 1 2 3 4x

123456789yy=Hx-1L2

-2 -1 1 2 3 4x

-8-7-6-5-4-3-2-1

yy=-Hx-1L2

standard equation horizontal shift reflection about x-axis

-2 -1 1 2 3 4x

-4

-3

-2

-1

y

y=-12Hx-1L2

-2 -1 1 2 3 4

x

-1

1

2

3y

gHxL=-12Hx-1L2+3

vertical shrinkage vertical shift

B. Example 1. Given the graph of y = g(t) below. Graph the following:

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6t

-5

-4

-3

-2

-1

1y=gHtL

a. y = g(−t) b. y = −g(t)

c. g(t)+ 3 d. y = 1− g(t)

c. g(t − 2) d. y = −g(t − 4)

2. Use the graph of f (x) = x3 to graph g(x) = 2 x + 33 − 4 .

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9x

-3

-2

-1

1

2

3y

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9x

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3y

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9x

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3y

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9x

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3y

III. Summary of Transformations (p. 279)