Transformations of Functions and their Graphs
Ms. P
Linear Transformations
Translations (shifts) Reflections Dilations (stretches or shrinks)
We examine the mathematics: Graphically Numerically Symbolically Verbally
These are the common linear transformations used in high school algebra courses.
Translations
Translations
How do we get the flag figure in the left graph to move to the position in the right graph?
TranslationsThis picture might help.
Translations
How do we get the flag figure in the left graph to move to the position in the right graph?
Here are the alternate numerical representations of the line graphs above.
1 1
1 2
1 3
2 3
2 2
1 2
4 2
4 3
4 4
5 4
5 3
4 3
Translations
How do we get the flag figure in the left graph to move to the position in the right graph?
This does it!
1 1
1 2
1 3
2 3
2 2
1 2
4 2
4 3
4 4
5 4
5 3
4 3
3 1
3 1
3 1
3 1
3 1
3 1
+
=
Translations
Alternately, we could first add 1 to the y-coordinates and then 3 to the x-coordinates to arrive at the final image.
TranslationsWhat translation could be applied to the left graph to obtain the right graph?
2y x y = ???
Translations
Following the vertex, it appears that the vertex, and hence all the points, have been shifted up 1 unit and right 3 units.
Graphic Representations:
Translations
Numerically, 3 has been added to each x-coordinate and 1 has been added to each y coordinate of the function on the left to produce the function on the right. Thus the graph is shifted up 1 unit and right 3 units.
Numeric Representations:
Translations
To find the symbolic formula for the graph that is seen above on the right, let’s separate our translation into one that shifts the function’s graph up by one unit, and then shift the graph to the right 3 units.
Translations
To find the symbolic formula for the graph that is seen above on the right, let’s separate our translation into one that shifts the function’s graph up by one unit, and then shift the graph to the right 3 units.
The graph on the left above has the equation y = x2.
To translate 1 unit up, we must add 1 to every y-coordinate. We can alternately add 1 to x2 as y and x2 are equal. Thus we have
y = x2 + 1
Translations
We verify our results below:
The above demonstrates a vertical shift up of 1.
y = f(x) + 1 is a shift up of 1 unit that was applied to the graph y = f(x).
How can we shift the graph of y = x2 down 2 units?
Translations
We verify our results below:
The above demonstrates a vertical shift down of 2.
y = f(x) - 2 is a shift down 2 unit to the graph y = f(x)
Vertical Shifts
If k is a real number and y = f(x) is a function, we say that the graph of y = f(x) + k is the graph of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward.
Did you guess to subtract 2 units?
Vertical Translation Example
Graph y = |x|
22
11
00
11
22
|x|x
Aside: y = |x| on the TI83/84
Vertical Translation Example
Graph y = |x| + 2
422
311
200
311
422
|x|+2|x|x
Vertical Translation Example
Graph y = |x| - 1
122
011
-100
011
122
|x| -1|x|x
Example Vertical Translations
y = 3x2
Example Vertical Translations
y = 3x2
y = 3x2 – 3
y = 3x2 + 2
Example Vertical Translations
y = x3
Example Vertical Translations
y = x3
y = x3 – 3
y = x3 + 2
Translations
Vertical Shift Animation:http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/verticalshift.html
Translations
The vertex has been shifted up 1 unit and right 3 units.
Starting with y = x2 we know that adding 1 to x2, that is y = x2 +1 shifts the graph up 1 unit. Now, how to we also shift the graph 3 units to the right, that is a horizontal shift of 3 units?
Getting back to our unfinished task:
Translations
Starting with y = x2 we know that adding 1 to x2, that is y = x2 +1 shifts the graph up 1 unit. Now, how to we also shift the graph 3 units to the right, that is a horizontal shift of 3 units?
We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?
Translations
We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?
x y=x2
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
x y=x2+1
-3 10
-2 5
-1 2
0 1
1 2
2 5
3 10
x+3 y=x2+1
0 10
1 5
2 2
3 1
4 2
5 5
6 10
TranslationsWe need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?
x+3 y=x2+1
0 10
1 5
2 2
3 1
4 2
5 5
6 10
So, let’s try y = (x + 3)2 + 1 ???
Oops!!!
TranslationsWe need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?
x+3 y=x2+1
0 10
1 5
2 2
3 1
4 2
5 5
6 10
So, let’s try y = (x - 3)2 + 1 ???
Hurray!!!!!!
Translations
Vertical Shifts
If k is a real number and y = f(x) is a function, we say that the graph of y = f(x) + k is the graph of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward.
Horizontal Shifts
If h is a real number and y = f(x) is a function, we say that the graph of y = f(x - h) is the graph of f(x) shifted horizontally by h units. If h follows a minus sign, then the shift is right and if h follows a + sign, then the shift is left.
Example Horizontal Translation
Graph g(x) = |x|
22
11
00
11
22
|x|x
Example Horizontal Translation
Graph g(x) = |x + 1|
322
211
100
011
122
|x + 1||x|x
Example Horizontal Translation
Graph g(x) = |x - 2|
022
111
200
311
422
|x - 2||x|x
Horizontal Translation
y = 3x2
Horizontal Translation
y = 3x2
y = 3(x+2)2
y = 3(x-2)2
Horizontal Shift Animation
http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/horizontalshift.html
Summary of Shift Transformations
To Graph: Shift the Graph ofy = f(x) by c units
y = f(x) + c UP
y = f(x) - c DOWN
y = f(x + c) LEFT
y = f(x - c) RIGHT
Translations – Combining ShiftsInvestigate Vertex form of a Quadratic Function: y = x2 + bx + c
y = x2
vertex: (0, 0)
y = (x – 3)2 + 1
vertex: (3, 1)
Vertex Form of a Quadratic Function (when a = 1):
The quadratic function: y = (x – h)2 + k
has vertex (h, k).
TranslationsCompare the following 2 graphs by explaining what to do to the graph of the first function to obtain the graph of the second function.
f(x) = x4
g(x) = (x – 3)4 - 2
Warm-up
If 0 < x < 1, rank the following in order from smallest to largest:
Warm-up
Reflections
Reflections
How do we get the flag figure in the left graph to move to the position in the right graph?
Reflections
How do we get the flag figure in the left graph to move to the position in the right graph? The numeric representations of the line graphs are:
1 1
1 2
1 3
2 3
2 2
1 2
1 -1
1 -2
1 -3
2 -3
2 -2
1 -2
ReflectionsSo how should we change the equation of the function, y = x2 so that the result will be its reflection (across the x-axis)?
Try y = - (x2) or simply y = - x2 (Note: - 22 = - 4 while (-2)2 = 4)
Reflection:Reflection: (across the x-axis)
The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x).
Example Reflection over x-axis
f(x) = x2
Example Reflection over x-axis
f(x) = x2
f(x) = -x2
Example Reflection over x-axis
f(x) = x3
Example Reflection over x-axis
f(x) = x3
f(x) = -x3
Example Reflection over x-axis
f(x) = x + 1
Example Reflection over x-axis
f(x) = x + 1
f(x) = -(x + 1) = -x - 1
More Reflections
Reflection in x-axis: 2nd coordinate is negated
Reflection in y-axis: 1st coordinate is negated
Reflection:Reflection: (across the x-axis)
The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x).
Reflection: (across the y-axis)
The graph of the function, y = f(-x) is the reflection of the graph of the function y = f(x).
Example Reflection over y-axis
f(x) = x2
Example Reflection over y-axis
f(x) = x2
f(-x) = (-x)2 = x2
Example Reflection over y-axis
f(x) = x3
Example Reflection over y-axis
f(x) = x3
f(-x) = (-x)3 = -x3
Example Reflection over y-axis
f(x) = x + 1
Example Reflection over y-axis
f(x) = x + 1
f(-x) = -x + 1
http://www.mathgv.com/
Dilations
How do we get the flag figure in the left graph to move to the position in the right graph?
Dilations (Vertical Stretches and Shrink)
1 1
1 2
1 3
2 3
2 2
1 2
1 2
1 4
1 6
2 6
2 4
1 4
Dilations (Stretches and Shrinks)Definitions: Vertical Stretching and Shrinking
The graph of y = af(x) is obtained from the graph of y = f(x) by
a). shrinking the graph of y = f ( x) by a when a > 1, or
b). stretching the graph of y = f ( x) by a when 0 < a < 1.
Vertical Stretch Vertical Shrink
Example Vertical Stretching/Shrinking
y = |x|
Example: Vertical Stretching/Shrinking
y = |x|
y = 0.5|x|
y = 3|x|
Vertical Stretching / Shrinking Animation
http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/verticalstretch.html
What is this?
Base Function
y = |x|
y = ????
y = -2|x -1| + 4
Warm-up
Explain how the graph of
can be obtained from the graph of .
How do we get the flag figure in the left graph to move to the position in the right graph?
Dilations (Horizontal Stretches and Shrink)
1 1
1 2
1 3
2 3
2 2
1 2
2 1
2 2
2 3
4 3
4 2
2 2
Horizontal Stretching / Shrinking Animation
http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/horizontalstretch.html
Procedure: Multiple TransformationsGraph a function involving more than one transformation in
the following order:1. Horizontal translation2. Stretching or shrinking3. Reflecting4. Vertical translation
Multiple Transformations
Graphing with More than One Transformation
Graph -|x – 2| + 1First graph f(x) = |x|
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1First graph f(x) = |x|
1. Perform horizontal translation: f(x) = |x-2|The graph shifts 2 to the right.
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1First graph f(x) = |x|
1. Perform horizontal translation: f(x) = |x-2|The graph shifts 2 to the right.
2. There is no stretch3. Reflect in x-axis:
f(x) = -|x-2|
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1First graph f(x) = |x|
1. Perform horizontal translation: f(x) = |x-2|The graph shifts 2 to the right.
2. There is no stretch3. Reflect in x-axis:
f(x) = -|x-2|4. Perform vertical translation:
f(x) = -|x-2| + 1The graph shifts up 1 unit.
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1First graph f(x) = |x|
1. Perform horizontal translation: f(x) = |x-2|The graph shifts 2 to the right.
2. There is no stretch3. Reflect in x-axis:
f(x) = -|x-2|4. Perform vertical translation:
f(x) = -|x-2| + 1The graph shifts up 1 unit.
Questions?
Time for worksheet
Other Transformation: Shears
(x, y) (x+y, y)
Can we Apply this Shear to y = x2?
Look at a line graph first!
Apply the shear:
(x, y) (x+y, y)
Can we Apply this Shear to y = x2?
Apply the shear:
(x, y) (x+y, y)
Can we Apply this Shear to y = x2?Apply the shear:
(x, y) (x+y, y)
Yes we CAN Apply this Shear to y = x2.Apply the shear:
(x, y) (x+y, y)
BUT…Can we write the symbolic equation in terms of x and y?
Shear Example
2y t t y
Apply the shear:
(x, y) (x+y, y) to y = x2
Parametrically we have:
x = t + t2 Our job is to eliminate t.
y = t2 We will use the substitution method.
Now substitute t back into the x equation and we have.
2
2
2
2
2
2
22
2 0
x y y
x y y
x y y
x y y
x xy y y
x xy y y
Shears
Horizontal Shear for k a constant
(x, y ) (x+ky, y)
Vertical Shear for k a constant
(x, y ) (x, kx+y)
Other Linear Transformations?
Rotations