Download - Transformation of Functions
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)