Transformations of
Functions and their Graphs
Ms. P 
Linear Transformations
These are the common linear transformations used in
high school algebra courses.
 Translations (shifts)
 Reflections
 Dilations (stretches or shrinks)
We examine the mathematics:
 Graphically
 Numerically
 Symbolically
 Verbally
Translations
Translations
How do we get the flag figure in the left graph to move to the position in the
right graph?
Translations
This 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
4
2
1
2
4
3
1
3
4
4
2
3
5
4
2
2
5
3
1
2
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
3
1
4
2
1
2
3
1
4
3
1
3
3
1
4
4
2
3
3
1
5
4
2
2
3
1
5
3
1
2
3
1
4
3
+
=
Translations
Alternately, we could first add 1 to the y-coordinates and then 3 to
the x-coordinates to arrive at the final image.
Translations
What translation could be applied to the left graph to obtain the right graph?
y  x2
y = ???
Translations
Graphic Representations:
Following the vertex, it appears that the vertex, and hence all the points,
have been shifted up 1 unit and right 3 units.
Translations
Numeric Representations:
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.
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
Did you guess to subtract 2 units?
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.
Vertical Translation Example
Graph y = |x|
x
|x|
2
2
1
1
0
0
1
1
2
2
Aside: y = |x| on the TI83/84
Vertical Translation Example
Graph y = |x| + 2
x
|x|
|x|+2
2
2
4
1
1
3
0
0
2
1
1
3
2
2
4
Vertical Translation Example
Graph y = |x| - 1
x
|x|
|x| -1
2
2
1
1
1
0
0
0
-1
1
1
0
2
2
1
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
Getting back to our unfinished task:
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?
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
-3
y=x2
9
x
-3
y=x2+1
10
x+3 y=x2+1
0
10
-2
-1
0
4
1
0
-2
-1
0
5
2
1
1
2
3
5
2
1
1
2
3
1
4
9
1
2
3
2
5
10
4
5
6
2
5
10
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+3 y=x2+1
0
10
1
2
3
4
5
2
1
2
5
6
5
10
So, let’s try
y = (x + 3)2 + 1
Oops!!!
???
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+3 y=x2+1
0
10
1
2
3
4
5
2
1
2
5
6
5
10
So, let’s try
y = (x - 3)2 + 1
Hurray!!!!!!
???
Translations
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.
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.
Example Horizontal Translation
Graph g(x) = |x|
x
|x|
2
2
1
1
0
0
1
1
2
2
Example Horizontal Translation
Graph g(x) = |x + 1|
x
|x|
|x + 1|
2
2
1
1
1
0
0
0
1
1
1
2
2
2
3
Example Horizontal Translation
Graph g(x) = |x - 2|
x
|x|
|x - 2|
2
2
4
1
1
3
0
0
2
1
1
1
2
2
0
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:
y = f(x) + c
Shift the Graph of
y = f(x) by c units
UP
y = f(x) - c
DOWN
y = f(x + c)
LEFT
y = f(x - c)
RIGHT
Translations – Combining Shifts
Investigate Vertex form of a Quadratic Function: y = x2 + bx + c
y = x2
y = (x – 3)2 + 1
vertex: (0, 0)
vertex: (3, 1)
Vertex Form of a Quadratic Function (when a = 1):
The quadratic function:
has vertex (h, k).
y = (x – h)2 + k
Translations
Compare 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
-1
1
2
1
-2
1
3
1
-3
2
3
2
-3
2
2
2
-2
1
2
1
-2
Reflections
So 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
Dilations (Vertical Stretches and Shrink)
How do we get the flag figure in the left graph to move to the position in the
right graph?
1
1
1
2
1
2
1
4
1
3
1
6
2
3
2
6
2
2
2
4
1
2
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 .
Dilations (Horizontal Stretches and Shrink)
How do we get the flag figure in the left graph to move to the position in the
right graph?
1
1
2
1
1
2
2
2
1
3
2
3
2
3
4
3
2
2
4
2
1
2
2
2
Horizontal Stretching / Shrinking Animation
http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/horizontalstretch.html
Multiple Transformations
Procedure: Multiple Transformations
Graph a function involving more than one transformation in
the following order:
1. Horizontal translation
2. Stretching or shrinking
3. Reflecting
4. Vertical translation
Graphing with More than One Transformation
Graph -|x – 2| + 1
First graph f(x) = |x|
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1
First 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| + 1
First graph f(x) = |x|
1. Perform horizontal
translation: f(x) = |x-2|
The graph shifts 2 to the
right.
2. There is no stretch
3. Reflect in x-axis:
f(x) = -|x-2|
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1
First graph f(x) = |x|
1. Perform horizontal
translation: f(x) = |x-2|
The graph shifts 2 to the right.
2. There is no stretch
3. Reflect in x-axis:
f(x) = -|x-2|
4. Perform vertical
translation:
f(x) = -|x-2| + 1
The graph shifts up 1 unit.
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1
First graph f(x) = |x|
1. Perform horizontal
translation: f(x) = |x-2|
The graph shifts 2 to the right.
2. There is no stretch
3. Reflect in x-axis:
f(x) = -|x-2|
4. Perform vertical
translation:
f(x) = -|x-2| + 1
The 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
Apply the shear:
(x, y)
(x+y, y)
Parametrically we have:
x = t + t2
Our job is to eliminate t.
y = t2
We will use the substitution method.
y  t2  t   y
Now substitute t back into the x equation and we have.

x y  y

2
x yy
x y  y
 x  y
2

  y

2
x 2  2 xy  y 2  y
x 2  2 xy  y  y 2  0
to y = x2
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