Algebra 3--Transformation of Functions

Algebra 3--Transformation of Functions

by Mrs. Harvey

What are transformations?

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​Transforming a graph turns one function into another function.

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​Transformations can involve:

• ​shifting the graph vertically or horizontally

• ​stretching or shrinking the graph

• ​reflecting the graph across a line or axis

Vertical Shifts

This allows the function to move UP and DOWN or vertically. These are outside of the parentheses, bars, radicals, etc.

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​Positive numbers --> Shift up

​Negative numbers --> Shift down

Multiple Choice

y = |x| - 4

• UP 4

• DOWN 4

• LEFT 4

• RIGHT 4

Multiple Choice

y = x² + 2

• LEFT 2

• RIGHT 2

• UP 2

• DOWN 2

Horizontal Shifts

This allows the function to move LEFT and RIGHT or horizontally. These are inside of the parentheses, bars, radicals, etc.

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​Positive numbers --> Shift left

​Negative numbers --> Shift right

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​**This is opposite of what we think it would do!**

Multiple Choice

y = |x - 6|

• UP 6

• DOWN 6

• LEFT 6

• RIGHT 6

Multiple Choice

y = (x + 4)³

• UP 4

• DOWN 4

• LEFT 4

• RIGHT 4

Reflections

*When there is a negative sign in front of your function, it means it reflects over the X - AXIS (the y-value changes)

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​*When there is a negative sign in front of the x-value, it means it reflects over the Y - AXIS (the x-value changes)

Multiple Choice

$$f\left(x\right)=\ -\left(x-5\right)^2\$$  

• Reflect across the y-axis

• Reflect across the x-axis

Multiple Choice

$$f\left(x\right)=2^{-x}$$  

• Reflects across the y-axis

• Reflects across the x-axis

--> If |a| > 1 : There is a VERTICAL STRETCH (orange)

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--> If 0 < |a| < 1 : There is a VERTICAL COMPRESSION OR SHRINK (purple)

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​-->Multiply your y-values by a

​-->Vertical stretch/compression by factor of a

Multiple Choice

How does this function compare to its parent function?

$$f\left(x\right)=\frac{1}{4}\left|x\right|$$  

• Vertical compression by factor of 1/4

• Vertical stretch by a factor of 1/4

• No change

Multiple Choice

How does the function compare to the parent graph?

$$f\left(x\right)=4\ \sqrt[]{x}$$  

• Vertical stretch by factor of 4

• Vertical compression by factor of 4

• No change

--> If |b| > 1 : There is a HORIZONTAL COMPRESSION (orange)

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--> If 0 < |b| < 1 : There is a HORIZONTAL STRETCH (purple)

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​--> Divide your x-values by b

​--> Horizontal stretch/shrink by factor of 1/b

Multiple Choice

Describe the transformation:

$$g(x)=\left|\frac{1}{3}x\right|$$  

• Vertical stretch by factor of 3

• Vertical compression by factor of 1/3

• Horizontal stretch by factor of 3

• Horizontal compression by factor of 1/3

Multiple Choice

Describe the transformation:

$$g(x)=\left(4x\right)^2$$  

• Vertical stretch by factor of 4

• Vertical compression by factor of 1/4

• Horizontal stretch by factor of 4

• Horizontal compression by factor of 1/4

​Let's try multiple transformations

​1. Vertical shifts (up & down)

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​2. Horizontal shifts (left & right)

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​3. Reflections across x- and y-axis

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​4. Vertical stretch/shrink

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​5. Horizontal stretch/shrink

Multiple Choice

Write an equation for a square root function that has been shifted right 4 and reflected over the y-axis

• $$f\left(x\right)=\sqrt{-\left(x-4\right)}$$

• $$g\left(x\right)=\sqrt{-\left(x+4\right)}$$

• $$h\left(x\right)=-\sqrt{x-4}$$

• $$j\left(x\right)=-\sqrt{x}+4$$

Multiple Choice

Describe the translation from the blue dotted graph to the red graph.

• $$f\left(x-4\right)-3$$

• $$f\left(x-4\right)+3$$

• $$f\left(x+4\right)+3$$

• $$f\left(x+4\right)-3$$

Multiple Choice

Name the transformation from the function  $$y=\sqrt{x}$$   to the function   $$y=-\sqrt{x+3}$$  .

• Reflection over the y-axis, left 3.

• Reflection over the y-axis, right 3.

• Reflection over the x-axis, left 3.

• Reflection over the x-axis, right 3.

Multiple Choice

$$-f\left(x+3\right)+4$$  

Describe the transformations:

• Shifted left 3 units
  Shifted up 4 units
  

• Reflection over the y-axis

  Shifted right 3 units Shifted up 4 units

• Shifted right 3 units
  Shifted up 4 units

• Reflection over the x-axis

  Shifted left 3 units Shifted up 4 units

Multiple Choice

Describe the transformation: $$f\left(x\right)=\frac{1}{3}x^2+2$$  

• Vertical compression by a factor of of 1/3
  Up 2

• Vertical stretch by a factor of 1/3
  Up 2

• reflection over the x-axis
  Up 2

• Vertical compression by a factor of of 1/3

  Right 2

Multiple Choice

Describe the transformations from the parent function

$$y=\frac{1}{x}$$  

• Right 5  and  Up 7

• Left 7  and  Up 5

• Reflect over the x-axis  and  Up 7

• Left 5  and  Up 7

Multiple Choice

What is the transformation?

• Vertical Translation up 2 and Horizontal translation right 1
• Vertical Translation down 2 and Horizontal Translation left 1
• Horizontal Translation left 2 and Vertical Translation up 1
• Horizontal Translation right 2 and Vertical Translation down 1

Multiple Choice

Given the original exponential function defined by y=4^x, how would the graph change if the function were:

y=2(4)^x-3+1?

• Vertically stretched by a factor of 2, right 3, and up 1

• Vertically compressed by a factor of 1/2, left 3 and down 1

• Vertically compressed by a factor of 1/2, right 3, and down 1

• Vertically stretched by a factor of 2, left 3, and up 1

Multiple Select

CHECK ALL THAT APPLY!

Which transformations are involved in the equation $$y=\frac{1}{2}\sqrt{x+1}-1$$  ?

• Horizontal translation left

• Vertical shrink

• Vertical translation down

• Horizontal translation right

• Vertical stretch

Multiple Choice

What function has the transformations:

Vertical stretch by 7 and Reflected over x-axis

• $$f\left(x\right)\ =\ -7x^2$$

• $$f\left(x\right)\ =\ 7\left|-x\right|$$

• $$f\left(x\right)\ =\ \sqrt{-7x}$$

• $$f\left(x\right)\ =\ -\left|7x\right|$$

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

How many different transformations are occurring from the parent absolute value to the function:

$$y=-3\left|x-2\right|+7$$  

• 1

• 2

• 3

• 4

Poll

With which transformations do you feel MOST confident?

• Vertical shifts

• Horizontal shifts

• Reflections

• Vertical stretches & compressions

• Horizontal stretches & compressions

Poll

With which transformations do you feel LEAST confident?

• Vertical shifts

• Horizontal shifts

• Reflections

• Vertical stretches & compressions

• Horizontal stretches & compressions