1.6 Transformation of Functions• Recognize graphs of common functions

• Use shifts to graph functions

• Use reflections to graph functions

• Use stretching & shrinking to graph functions

• Graph functions w/ sequence of transformations

The following basic graphs will be used extensively in this section. It is important to be able to sketch

these from memory.

The identity function f(x) = x

The squaring function

2)( xxf

xxf )(

The square root function

xxf )(The absolute value function

3)( xxf

The cubing function

The cube root function

3( )f x x

We will now see how certain transformations (operations)

of a function change its graph. This will give us a better idea of how to quickly sketch the

graph of certain functions. The transformations are (1)

translations, (2) reflections, and (3) stretching.

Vertical Translation

Vertical Translation

For b > 0,

the graph of y = f(x) + b is the graph of y = f(x) shifted up b units;

the graph of y = f(x) b is the graph of y = f(x) shifted down b units.

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

2( ) 2f x x

Horizontal Translation

Horizontal Translation

For d > 0,

the graph of y = f(x d) is the graph of y = f(x) shifted right d units;

the graph of y = f(x + d) is the graph of y = f(x) shifted left d units.

22y x 2

2y x

2( )f x x

• Vertical shifts– Moves the graph up or

down– Impacts only the “y”

values of the function– No changes are made

to the “x” values

• Horizontal shifts– Moves the graph left

or right– Impacts only the “x”

values of the function– No changes are made

to the “y” values

The values that translate the graph of a function will occur as a number added or subtracted either inside or

outside a function.

Numbers added or subtracted inside translate left or right, while

numbers added or subtracted outside translate up or down.

( )y f x d b

Recognizing the shift from the equation, examples of shifting the

function f(x) =

• Vertical shift of 3 units up

• Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

3)(,)( 22 xxhxxf

22 )3()(,)( xxgxxf

2x

Points represented by (x , y) on the graph of f(x) become

, for the function ( )x d y b f x d b

If the point (6, -3) is on the graph of f(x),find the corresponding point on the graph of f(x+3) + 2

)1,3(

Use the basic graph to sketch the following:

( ) 3f x x 2( ) 5f x x 3( ) ( 2)f x x ( ) 3f x x

Combining a vertical & horizontal shift

• Example of function that is shifted down 4 units and right 6 units from the original function.

( ) 6

)

4

( ,

g x x

f x x

Reflections• The graph of f(x) is the reflection of the

graph of f(x) across the x-axis.

• The graph of f(x) is the reflection of the graph of f(x) across the y-axis.

• If a point (x, y) is on the graph of f(x), then

(x, y) is on the graph of f(x), and

• (x, y) is on the graph of f(x).

Reflecting

• Across x-axis (y becomes negative, -f(x))

• Across y-axis (x becomes negative, f(-x))

Use the basic graph to sketch the following:

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

Vertical Stretching and ShrinkingThe graph of af(x) can be obtained from the graph of f(x) by

stretching vertically for |a| > 1, orshrinking vertically for 0 < |a| < 1.

For a < 0, the graph is also reflected across the x-axis.

(The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)

VERTICAL STRETCH (SHRINK)

• y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

2( ) 3 4f x x

2( ) 4f x x

21( ) 4

2f x x

Horizontal Stretching or ShrinkingThe graph of y = f(cx) can be obtained from the graph of y = f(x) by

shrinking horizontally for |c| > 1, orstretching horizontally for 0 < |c| < 1.

For c < 0, the graph is also reflected across the y-axis.

(The x-coordinates of the graph of y = f(cx) can be obtained by dividing the x-coordinates of the graph of y = f(x) by c.)

Horizontal stretch & shrink• We’re MULTIPLYING

by an integer (not 1 or 0).

• x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)

2( ) (3 ) 4g x x

2( ) 4f x x

21( ) ( ) 4

3f x x

Sequence of transformations• Follow order of operations.• Select two points (or more) from the original function and

move that point one step at a time.

f(x) contains (-1,-1), (0,0), (1,1)1st transformation would be (x+2), which moves the function left

2 units (subtract 2 from each x), pts. are now (-3,-1), (-2,0), (-1,1)2nd transformation would be 3 times all the y’s, pts. are now (-3,-3), (-2,0), (-1,3)3rd transformation would be subtract 1 from all y’s, pts. are now

(-3,-4), (-2,-1), (-1,2)

1)2(31)2(3

)(3

3

xxf

xxf

Graph of Example

3

3

( )

( ) 3 ( 2) 1 3( 2) 1

f x x

g x f x x

(-1,-1), (0,0), (1,1)

(-3,-4), (-2,-1), (-1,2)

The point (-12, 4) is on the graph of y = f(x). Find a point on the graph

of y = g(x).

• g(x) = f(x-2)

• g(x)= 4f(x)

• g(x) = f(½x)

• g(x) = -f(x)

• (-10, 4)

• (-12, 16)

• (-24, 4)

• (-12, -4)