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

2)( xxf

xxf )(

xxf )(

3)( xxf

3( )f x x

Vertical TranslationFor 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 TranslationFor 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.

2( )f x x

22y x 2

2y 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

( )y f x d b

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

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

)1,3(

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

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

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

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).

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

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

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

The 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.)

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

The 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.)

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

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

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)

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)