transformations 2.4: transformations of functions and graphs we will be looking at simple functions...
TRANSCRIPT
TransformationsTransformationsTransformations
Transformations
2.4: Transformations of Functions and Graphs
We will be looking at simple functions and seeing how various modifications to the functions transform them.
VERTIC
AL
TR
AN
SLA
TIO
NS
Above is the graph of 2xxf
x
y
What would f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them).
x
y
11 2 xxf
What would f(x) - 3 look like? (This would mean taking all the function values and subtracting 3 from them).
x
y
33 2 xxf
2xxf As you can see, a number added or
subtracted from a function will
cause a vertical shift or
translation in the function.
VERTIC
AL
TR
AN
SLA
TIO
NS
Above is the graph of xxf What would f(x) + 2 look like?
22 xxfSo the graph
f(x) + k, where k is any real
number is the graph of f(x) but vertically
shifted by k. If k is positive it will shift up. If k is negative it will shift down
x
y
x
y
x
y
44 xxf
xxf
What would f(x) - 4 look like?
Above is the graph of 2xxf
x
y
What would f(x+2) look like? (This would mean taking all the x values and adding 2 to them before putting them in the function).
As you can see, a number added or
subtracted from the x will cause
a horizontal shift or
translation in the function but opposite way of the sign of the
number.
HORIZONTAL TRANSLATIONS
x
y
x
y 2xxf
211 xxf
222 xxf
What would f(x-1) look like? (This would mean taking all the x values and subtracting 1 from them before putting them in the function).
HORIZONTAL TRANSLATIONS
Above is the graph of 3xxf What would f(x+1) look like?
So the graph f(x-h), where h is
any real number is the graph of f(x) but horizontally
shifted by h. Notice the negative.
(If you set the stuff in parenthesis = 0 & solve it will tell you how to shift
along x axis).
x
y
x
y
x
y
311 xxf 3xxf
What would f(x-3) look like?
333 xxf
03 x
So the graph f(x-h), where h is
any real number is the graph of f(x) but horizontally
shifted by h. Notice the negative.
(If you set the stuff in parenthesis = 0 & solve it will tell you how to shift
along x axis).
3xSo shift along the x-axis by 3
shift right 3
x
y
x
y
x
y
We could have a function that is transformed or translated both vertically AND horizontally.
Above is the graph of xxf What would the graph of look like? 3)2( xxf
up
3
left 2
and
If we multiply a function by a non-zero real number it has the effect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number.
Let's try some functions multiplied by non-zero real numbers to see this.
DILATION:
Above is the graph of xxf
So the graph a f(x), where a
is any real number
GREATER THAN 1, is the graph of f(x) but vertically stretched or dilated by a factor of a.
x
y
x
y
x
y
xxf
xxf 22 xxf 44
What would 2f(x) look like?
What would 4f(x) look like?
Notice for any x on the graph, the new (red) graph has a y value that is 2 times as much as the original (blue) graph's y value.
Notice for any x on the graph, the new (green) graph has a y value that is 4 times as much as the original (blue) graph's y value.
Above is the graph of xxf
So the graph a f(x), where a
is 0 < a < 1, is the graph of
f(x) but vertically
compressed or dilated by a factor of a.
x
y
x
y
Notice for any x on the graph, the new (red) graph has a y value that is 1/2 as much as the original (blue) graph's y value.
x
y
Notice for any x on the graph, the new (green) graph has a y value that is 1/4 as much as the original (blue) graph's y value.
xxf4
1
4
1
What if the value of a was positive but less than 1?
xxf
xxf2
1
2
1
What would 1/4 f(x) look like?
What would 1/2 f(x) look like?
Above is the graph of xxf
So the graph - f(x) is a reflection about the
x-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the
function over the x-axis)
x
y
What if the value of a was negative?
What would - f(x) look like?
x
y
xxf
xxf
Notice any x on the new (red) graph has a y value that is the negative of the original (blue) graph's y value.
x
y
Above is the graph of 3xxf
There is one last transformation we want to look at.
Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value.
x
y
3xxf 3xxf
What would f(-x) look like? (This means we are going to take the negative of x before putting in the function)
So the graph f(-x) is a
reflection about the
y-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the
function over the y-axis)
Summary of Transformations So Far
khxfa
horizontal translation of h (opposite sign of number with the x)
If a > 1, then vertical dilation or stretch by a factor of a
vertical translation of k
If 0 < a < 1, then vertical dilation or compression by a factor of a
f(-x) reflection about y-axis
**Do reflections and dilations BEFORE vertical and horizontal translations**
If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a)
Graph using transformations 12
1
xxf
We know what the graph would look like if it wasfrom our library of functions.
x
xf1
x
y
moves up 1
moves right 2
reflects about the
x -axis
x
y
x
y
x
y
x
y
There is one more Transformation we need to know.
kb
hxfa
)(
horizontal translation of h (opposite sign of number with the x)
If a > 1, then vertical dilation or stretch by a factor of a
vertical translation of k
If 0 < a < 1, then vertical dilation or compression by a factor of a
f(-x) reflection about y-axis
Do reflections and dilations BEFORE vertical and horizontal translations
If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a)
horizontal dilation by a factor of b
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified.