Transformations of Functions

Transformations of Functions

Distortions and Reflections

Distortion of parent functions

• A graph can be stretched which makes the new graph narrower than the parent function

• A graph can be compressed which makes the new graph wider than the parent function.

• A parent function has a scale factor of 1.

Stretch or Compression is given by the "a" value.

• The "a" value on a parent function is 1.

•  $f\left(x\right)\ =\ x^2$   OR   $f\left(x\right)\ =\ \left|x\right|$  OR  $f\left(x\right)\ =\ x$  

• The "a" value is the coefficient.

Stretch (vertically)

• If a > 1, then the graph will be stretched.

• The higher the "a" value becomes the more narrow the graph becomes.

• The higher the "a" value becomes the closer the graph gets to the y-axis.

Compress/Shrink (vertically)

• If 0 < a < 1, then the graph will be compressed.

• The closer to zero the "a" value becomes the wider the graph becomes.

• The closer to zero the "a" value becomes the closer the graph gets to the x-axis.

Multiple Choice

 $y=3\left|x\right|$  

Which graph represents this transformed function

Multiple Choice

The original function has equation f(x). State the transformations given the new function f(x) = ⅔(x - 7)²

• Vertical Shrink of ⅔

  Right 7

• Horizontal Shrink of ⅔

  Left 7

• Left ⅔

  Vertical stretch of 7

• Right ⅔

  Horizontal stretch of 7

Multiple Choice

• g(x)=2x²
• g(x)=0.5x²
• g(x)=x²+2
• g(x)=(x+2)²

Multiple Choice

Identify the transformations.

• horizontal shift to the left 2 and vertical stretch by a factor of 3

• horizontal shift to the right 2 and vertical stretch by a factor of 3

• horizontal shift to the right 2 and vertical shrink by a factor of 3

• horizontal shift to the left 2 and vertical shrink by a factor of 3

Multiple Choice

 $y=-2\left|x\right|$  

Which graph represents this transformed function

Reflections

• the graph of  $f\left(x\right)$   across the          x-axis is given by

   $-f\left(x\right).$  

•  $f\left(x\right)\ =\ x^2\ \ \ \Longrightarrow\ \ \ f\left(x\right)\ =\ -x^2$  

•  $f\left(x\right)\ =\sqrt{x\ }\ \ \Longrightarrow\ \ \ f\left(x\right)\ =\ -\sqrt{x}$  

Reflections

• the graph of  $f\left(x\right)$   across the y-axis is given by

   $f\left(-x\right).$ 

•  $f\left(x\right)\ =\sqrt{x\ }\ \ \Longrightarrow\ \ \ f\left(x\right)\ =\ \sqrt{-x}$  

Multiple Choice

• $y=\ \left|x\right|\ -\ 1$

• $y=\ -\ \left|x\right|\$

• $y=\ \left|x\right|\$

• $y=\ \left|x-1\right|$

Multiple Choice

g(x)=-f(x)

Describe the transformation to f(x) that results in g(x).

• f(x) has been reflected over the x-axis.

• f(x) has been reflected over the y-axis.

• f(x) has been reflected over y -x

• f(x) has been rotated

Multiple Choice

 $y=-2\left|x\right|$  

Which graph represents this transformed function

Multiple Choice

If the blue function is f(x)=√x, the red function must be

• g(x)=√(-x)

• g(x)=√(x-1)

• g(x)=√x-1

• g(x)=-√x

Multiple Choice

If the blue graph is f(x) (the original function) and the red graph is the transformation, what is the equation of the red graph?

• -f(x)

• f(-x)

• f^-1(x)

• f(2x)

3 main letters representing transformations

• "h" - represents horizontal shift - think OPPOSITE

• "k" - represents vertical shifts - think same

• "a" - has two jobs - reflections AND distortions

Quadratic Function

 $f\left(x\right)\ =\ a\left(x-h\right)^2\ +\ k$  

Absolute Value Function

 $f\left(x\right)\ =\ a\left|x-h\right|\ +\ k$  

Cubic Function

 $f\left(x\right)\ =\ a\left(x-h\right)^3\ +\ k$  

Square Root Function

 $f\left(x\right)\ =\ a\sqrt{x\ -\ h}+\ k$  

Linear Function

•  $f\left(x\right)\ =\ ax\ +\ k$  

•  $f\left(x\right)\ =\ mx\ +\ b$    (slope-intercept)
  

Multiple Choice

Match the equation with the graph:

• y = (x - 2)² + 4

• y = -(x - 4)² + 2

• y = -(x - 2)² + 4

• y = -(x + 2)² + 4

Multiple Choice

Describe the transformations that maps

y = g(x) to y = - g(x + 6) - 10

• Reflect over y-axis

  Shifted down 10 units

  Shifted left 6 units

• Reflect over x-axis

  Shifted up 10 units

  Shifted left 6 units

• Reflect over y-axis

  Shifted up 10 units

  Shifted right 6 units

• Reflect over x-axis

  Shifted down 10 units

  Shifted left 6 units

Multiple Choice

Identify the transformation from the graph of f(x)=x² to the graph g(x)= -(x+5)²

• shifted up five units

  reflection in x-axis

• shifted down five units

  reflection in x-axis

• shifted left 5 units

  reflection x-axis

• shifted right 5 units

  reflection x-axis

Multiple Choice

Given f(x) = (x-3)² + 5. What transformations took place from the original function f(x)?

• Left 3 and up 5

• Right 3 and down 5

• Left 3 and down 5

• Right 3 and up 5

Multiple Choice

• y = Ix + 2I - 1
• y = -Ix + 2I - 1
• y = -Ix - 2I - 1
• y = -Ix + 1I - 2

Multiple Choice

Describe the parent graph transformation

• stretch by 1/2, left 1, down 3
• stretch by 1/2, right 1, down 3
• compression by 1/2, left 1, down 3
• compression by 1/2, right 1, down 3