Transcript 1.6 Transformation of Functions

Transformation of Functions
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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
f ( x)  x
2
The square root function
f ( x)  x
The absolute value function
f ( x)  x
The cubing function
f ( x)  x
3
The cube root function
f ( x)  x
3
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
f ( x)  x
Vertical Translation
For b > 0,
the graph of y = f(x) + b
is the graph of y = f(x)
shifted up b units;
2
f ( x)  x2  3
the graph of y = f(x)  b
is the graph of y = f(x)
shifted down b units.
f ( x)  x2  2
Horizontal Translation
f ( x)  x2
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.
y   x  2
2
y   x  2
2
• 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.
y  f (x  d )  b
Numbers added or subtracted
inside translate left or right, while
numbers added or subtracted
outside translate up or down.
Recognizing the shift from the
equation, examples of shifting the
2
function f(x) =x
• Vertical shift of 3 units up
f ( x)  x , h( x)  x  3
2
2
• Horizontal shift of 3 units left (HINT: x’s go the
opposite direction that you might believe.)
f ( x)  x , g ( x)  ( x  3)
2
2
Points represented by (x , y) on the
graph of f(x) become
x
d , y  b for the function 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
(3,1)
Use the basic graph to sketch the
following:
f ( x)  x  3
f ( x)  x  3
f ( x)  ( x  2)
3
f ( x)  x  5
2
Combining a vertical & horizontal
shift
• Example of function
that is shifted down 4
units and right 6 units
from the original
function.
f ( x)  x ,
g ( x)  x  6  4
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
f ( x)   x
2
f ( x)  x
Vertical Stretching and Shrinking
The graph of af(x) can be obtained from the
graph of f(x) by
stretching vertically for |a| > 1, or
shrinking 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!)
f ( x)  x  4
2
f ( x)  3 x  4
2
1 2
f ( x)  x  4
2
Horizontal Stretching or Shrinking
The graph of y = f(cx) can be obtained from the
graph of y = f(x) by
shrinking horizontally for |c| > 1, or
stretching 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.)
g ( x)  (3 x)  4
2
f ( x)  x  4
2
1 2
f ( x)  ( x)  4
3
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)  x
3
3 f ( x  2)  1  3( x  2)3  1
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)
Graph of Example
f ( x)  x3
(-1,-1), (0,0), (1,1)
g ( x)  3 f ( x  2)  1  3( x  2)3  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)
• (-10, 4)
• g(x)= 4f(x)
• (-12, 16)
• g(x) = f(½x)
• (-24, 4)
• g(x) = -f(x)
• (-12, -4)