Section 1.3

Shifting, Reflecting

, and Stretching Graphs

In this section, you will learn how to:

• Shift graphs (left, right, up, down)• Reflect graphs (across the x or y

axis)• Stretch graphs (or shrink)

You may also look at a graph and be asked to explain what happened to the “original” graph.

You must first recognize the graphs of common functions.

p. 100 Common Functions

cxf =)( xxf =)( xxf =)(

xxf =)( 2)( xxf = 3)( xxf =

Vertical and Horizontal ShiftsLet 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 upward: h(x)= f(x) + c• 2. Vertical shift c units down: h(x)= f(x) - c• 3. Horizontal shift c units right: h(x)= f(x - c)• 4. Horizontal shift c units left: h(x)= f(x + c)***Notice on the horizontal shift, you move

OPPOSITE of what you think you would.

If the shift happens outside the function, it is vertical. If the shift happens inside the function, it is horizontal.

Example 1 Compare the graph of each function with the graph of

.)( 3xxf =

1)( ) 3 −=xxga

3)1()( ) −= xxgb

1)2()( ) 3 ++= xxgcShifts down 1.

Shifts to the right 1.

Shifts to the left 2 and up 1.

The graphs shown are shifts of the graph of Find equations for g and h.

4)( 2 +=xxg

.)( 2xxf

y=g(x)

y=h(x)

1)2()( 2 −+= xxh

y=f(x)

Reflections

Reflections•h(x)= - f(x) reflects across the x axis.

•h(x)= f(-x) reflects across the y axis.

If it happens outside the function, reflects across the x axis.

If it happens inside the function, reflects across the y axis.

function.each ofequation an Find

.xf(x) ofgraph theofation transforma isshown graphs theofEach 4=

2)( 4 +−= xxg

y=g(x)

y=h(x)

4)3()( −−= xxh

Example 4 p. 104

.)( xxf =Compare the graph of each function with the graph of

2)()

)()

)()

+−=

−=

−=

xxkc

xxhb

xxga Reflects across the x axis.

Reflects across the y axis.

Reflects across the x axis.

Shifts to the left 2 units.

Transformations

• Rigid transformations don’t change the basic shape of the graph.

• Nonrigid transformations do change or distort the basic shape of the graph.

• The graph of y = f (x) is represented by y = cf (x). “c” represents a constant #.Vertical Stretch if the function is

multiplied by a number >1.Vertical Shrink if the function is multiplied

by a fraction between 0 and 1.

Example.xf(x) ofgraph ith thefunction weach ofgraph theCompare =

h(x) = 3|x| Vertical stretch

Example.xf(x) ofgraph ith thefunction weach ofgraph theCompare =

€

g(x) =1

3x Vertical shrink

Specify the sequence of transformations that will yield the graph of the given function from the graph of the function

This graph is going down 10 units, followed by a vertical stretch.

25)( )2(5)( 22 −=−= xxhxxg

This graph is going down 2 units followed by a vertical stretch.

.)( 2xxf =

“Nobody ever uses this!!!”

•Tilt-Shift Photography

•Tilt Shift Video•The mathematics…