Transformations of Functions and their Graphs

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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

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:

• Graphically
• Numerically
• Symbolically
• Verbally
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.

Translations

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

This does it!

+

=

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 = ???

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.

x

|x|

2

2

1

1

0

0

1

1

2

2

Vertical Translation Example

Graphy = |x|

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

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

y = 3x2

y = 3x2 – 3

y = 3x2 + 2

Example Vertical Translations
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?

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!!!

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!!!!!!

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.

y = 3x2

y = 3(x+2)2

y = 3(x-2)2

Horizontal Translation
Horizontal Shift Animation

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

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).

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

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:

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).

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).

f(x) = x + 1

f(-x) = -x + 1

Example Reflection over y-axis

http://www.mathgv.com/

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)

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

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

What is this?

Base Function

y = |x|

y = ????

y = -2|x -1| + 4

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

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

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
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.

Graphing with More than One Transformation
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) 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.

Horizontal Shear for k a constant

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

Vertical Shear for k a constant

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

Shears