1.4 Shifting, Reflecting, and Sketching Graphs• Students will recognize graphs of common functions such

as:

• Students will use vertical and horizontal shifts and reflections to graph functions.

• Students will use nonrigid transformations to graph functions.

f x x( )

f x x( ) f x x( ) 2 f x x( ) 3

f x c( ) f x x( )

Vertical and Horizontal Shifts

Experiment with the following functions to determine how minor changes in the function alter the graphs:

y x1 y x2 2 y x3 2 y x1 y x2 2 y x3 2

Student ExampleIf , make a guess and check with the calculator.

Give the function that would move f(x):

a) down 4 units

b) left 3 units

c) right 2 units and up 5 units

f x x( ) 2

Vertical and Horizontal Shifts

Let c be a positive real number. Vertical and horizontal shifts in the graph of y=f(x) are represented as follows:

1. Vertical shift c units upwards: h(x)=f(x)+c

Ex. Moves up 2 units from

2. Vertical shift c units downward: h(x)=f(x)-cEx. Moves down 2 units from

3. Horizontal shift c units to the right: h(x)=f(x-c)

Ex. Moves right 2 units from

4. Horizontal shift c units to the left: h(x)=f(x+c)

Ex. Moves left 2 units from

f x x( ) 2

f x x( ) 2 2

f x x( ) 2 2

f x x( ) 2

f x x( ) ( ) 2 2 f x x( ) 2

f x x( ) 2f x x( ) ( ) 2 2

Example 1: Compare the graphs of each function with the graph of f x x( ) 3

a g x x. ) ( ) 3 1 b h x x. ) ( ) ( ) 1 3 c k x x. ) ( ) ( ) 2 13

Example 2

The graph of is shown in Figure 1.44. Each of the graphs in Figure 1.45 is a transformation of the graph of f. Find an equation for each function.

y=g(x) y=h(x)

f x x( ) 2

f x x( ) 2

Student Example:What must be done to the point (x,y) to reflect over the x-axis and the y-axis.

y

» (x,y).

» x

Reflections in the Coordinate AxesReflections in the coordinate axes of the graph of y = f(x) are

represented as follows.

1. Reflection in the x – axis: h(x) = -f(x)

2. Reflection in the y – axis: h(x) = f(-x)

Student Example

f x x x x( ) 3 24 1

Find an equation that will:

a) reflect f(x) over the x-axis.

b) Reflect f(x) over the y-axis.

Example 3

The graph of is shown. Each graph shown is a transformation of the graph of f. Find an equation for each function.

f(x) y=g(x) y=h(x)

f x x( ) 4

Example 4

Compare the graph of each function with the graph of

a. b. c.

f x x( )

g x x( ) h x x( ) k x x( ) 2

Example 5: Nonrigid Transformations

Compare the graph of each function with the graph of

a. b.

f x x( )

h x x( ) 3 g x x( ) 1

3

Example 6

Compare the graph of with the graph of h x f x( ) ( )1

2f x x( ) 2 3

Tuition has risen at private colleges. The table lists the average tuition for selected years. Use a non-rigid transformation of a linear function to best fit the data:

Use the function to predict the cost of tuition during your freshmen year of college. Does it seem accurate?

Year 1980 1985 1990 1995

Tuition $3,617 $6,121 $9,340 $12,432

Year 1982 1984 1986 1988 1990 1992 1994

Fatalities

from AIDS

619 5605 24,593 61,911 120,811 196,283 270,533

Use a non-rigid transformation to adjust a quadratic to best fit the data:

Year 1982 1984 1986 1988 1990 1992 1994

Fatalities

from AIDS

619 5605 24,593 61,911 120,811 196,283 270,533

Use the function to predict the number of AIDS fatalities in 2010.

Year 1982 1984 1986 1988 1990 1992 1994

Fatalities

from AIDS

619 5605 24,593 61,911 120,811 196,283 270,533

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