Transformations Transformations Transformations 24 Transformations of Functions and Graphs We will be looking at simple functions and seeing how various modifications to the functions transform them ID: 265168
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Slide1
Transformations
Transformations
Transformations
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.Slide2
VERTICAL TRANSLATIONS
Above is the graph of
What would
f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them).
What would
f
(
x
) - 3 look like? (This would mean taking all the function values and subtracting 3 from them).
As you can see, a number added or subtracted from a function will cause a vertical shift or
translation
in the function.Slide3
VERTICAL TRANSLATIONS
Above is the graph of
What would f(
x) + 2 look like?
So 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
What would
f
(
x
) - 4 look like? Slide4
Above is the graph of
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
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).Slide5
HORIZONTAL TRANSLATIONS
Above is the graph of
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).
What would
f
(
x
-3) 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).
So shift along the
x
-axis by 3
shift right 3Slide6
We could have a function that is transformed or translated both vertically AND horizontally.
Above is the graph of
What would the graph of look like?
up 3
left 2Slide7
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:Slide8
Above is the graph of
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
.
What would
2
f
(
x
) look like?
What would
4
f
(
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.Slide9
Above is the graph of
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
.
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.
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.
What if the value of
a
was positive but less than 1?
What would
1/4
f
(
x
) look like?
What would
1/2
f
(
x
) look like? Slide10
Above is the graph of
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)
What if the value of
a
was negative?
What would
-
f
(
x
) look like?
Notice any
x
on the new (red) graph has a
y
value that is the negative of the original (blue) graph's
y
value.Slide11
Above is the graph of
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.
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)Slide12
Summary of Transformations So Far
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
)Slide13
Graph using transformations
We know what the graph would look like if it was
from our library of functions.
moves up 1
moves right 2
reflects about the
x
-axisSlide14
There is one more Transformation we need to know.
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
bSlide15
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.