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Transformations of Functions and their Graphs. Mary Dwyer Wolfe, Ph.D. July 2009. 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:

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transformations of functions and their graphs

Transformations of Functions and their Graphs

Mary Dwyer Wolfe, Ph.D.

July 2009

linear transformations
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
translations4
Translations

How do we get the flag figure in the left graph to move to the position in the right graph?

translations5
Translations

This picture might help.

translations6
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.

translations7
Translations

How do we get the flag figure in the left graph to move to the position in the right graph?

This does it!

+

=

translations8
Translations

Alternately, we could first add 1 to the y-coordinates and then 3 to the x-coordinates to arrive at the final image.

translations9
Translations

What translation could be applied to the left graph to obtain the right graph?

y = ???

translations10
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.

translations11
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.

translations12
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.

translations13
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

translations14
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?

translations15
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

x

|x|

2

2

1

1

0

0

1

1

2

2

Vertical Translation Example

Graphy = |x|

vertical translation example18

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| + 2

vertical translation example19

x

|x|

|x| -1

2

2

1

1

1

0

0

0

-1

1

1

0

2

2

1

Vertical Translation Example

Graph y = |x| - 1

example vertical translations21
y = 3x2

y = 3x2 – 3

y = 3x2 + 2

Example Vertical Translations
translations24
Translations

Vertical Shift Animation:

http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/verticalshift.html

translations25
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?

translations26
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?

translations27
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?

translations28
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?

So, let’s try y = (x + 3)2 + 1 ???

Oops!!!

translations29
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?

So, let’s try y = (x - 3)2 + 1 ???

Hurray!!!!!!

translations30
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.

horizontal translation35
y = 3x2

y = 3(x+2)2

y = 3(x-2)2

Horizontal Translation
horizontal shift animation
Horizontal Shift Animation

http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/horizontalshift.html

translations combining shifts
Translations – Combining Shifts

Investigate 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).

translations39
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

reflections41
Reflections

How do we get the flag figure in the left graph to move to the position in the right graph?

reflections42
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:

reflections43
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:

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).

more reflections
More Reflections

Reflection in x-axis: 2nd coordinate is negated

Reflection in y-axis: 1st coordinate is negated

reflection52
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 axis58
f(x) = x + 1

f(-x) = -x + 1

Example Reflection over y-axis

http://www.mathgv.com/

dilations vertical stretches and shrink
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?

dilations stretches and shrinks
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). stretching the graph of y = f ( x) by a when a > 1, or

b). shrinking the graph of y = f ( x) by a when 0 < a < 1.

Vertical Stretch

Vertical Shrink

vertical stretching shrinking animation
Vertical Stretching / Shrinking Animation

http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/verticalstretch.html

what is this
What is this?

Base Function

y = |x|

y = ????

y = -2|x -1| + 4

dilations horizontal stretches and shrink
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?

horizontal stretching shrinking animation
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:

Horizontal translation

Stretching or shrinking

Reflecting

Vertical translation

Multiple Transformations
graphing with more than one transformation70
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
graphing with more than one transformation71
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
graphing with more than one transformation72
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
graphing with more than one transformation73
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
can we apply this shear to y x 2
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 x 276
Can we Apply this Shear to y = x2?

Apply the shear:

(x, y) (x+y, y)

can we apply this shear to y x 277
Can we Apply this Shear to y = x2?

Apply the shear:

(x, y) (x+y, y)

yes we can apply this shear to y x 2
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
Shear Example

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.

shears

Horizontal Shear for k a constant

(x, y ) (x+ky, y)

Vertical Shear for k a constant

(x, y ) (x, kx+y)

Shears
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