Transformations of Square Root Functions

Transformations of Square Root Functions

By Monta Jeffley

This is the square root function in a more complex form, it will help you understand how the graph moves through the coordinate plane. ​

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​Square Root Function

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If the "a" is negative, you will reflect over the x-axis.

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If the "b" is negative, you will reflect over the y-axis. ​

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​Reflections

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Multiple Choice

Describe the transformation for:

$$-2\ \sqrt[]{x}$$  

• Reflection over the x-axis

• Reflection over the y-axis

Multiple Choice

Describe the transformation for:

$$\ \sqrt[]{-5\ x}$$  

• Reflection over the x-axis

• Reflection over the y-axis

Multiple Choice

Describe the transformation for:

$$\ -6\ \sqrt[]{x\ +\ 3}\ -\ 6$$  

• Reflection over the x-axis

• Reflection over the y-axis

Multiple Choice

Describe the transformation for:

$$\ \sqrt[]{-2x\ +\ 4}\ +\ 7$$  

• Reflection over the x-axis

• Reflection over the y-axis

Multiple Select

Describe the transformation for:

$$\ -\frac{2}{3}\sqrt[]{-x\ -\ 5}\ -\ 2$$  

• Reflection over the x-axis

• Reflection over the y-axis

The letter "a" controls the vertical compression and stretch of a graph.

-If a > 1, it is a vertical stretch

-If 0<a<1 (a is between 0 and 1), it is a vertical compression. ​

​Vertical Compressions and Stretches

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Multiple Choice

Describe the transformation for:

$$\frac{1}{3}\ \sqrt[]{x}$$  

• Vertical Stretch

• Vertical Compression

Multiple Choice

Describe the transformation for:

$$6\ \sqrt[]{x}$$  

• Vertical Stretch

• Vertical Compression

Multiple Choice

Describe the transformation for:

$$-\frac{5}{4}\ \sqrt[]{3x}\ +\ 3$$  

• Vertical Stretch

• Vertical Compression

Multiple Choice

Describe the transformation for:

$$\frac{2}{3}\ \sqrt[]{-2x}\ -\ 13$$  

• Vertical Stretch

• Vertical Compression

The letter "b" controls the horizontal compression and stretch of a graph.

-If b > 1, it is a horizontal compression

-If 0<b<1 (a is between 0 and 1), it is a horizontal stretch.

​Horizontal Compressions and Stretches

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Multiple Choice

Describe the transformation for:

$$\ \sqrt[]{2x}$$  

• Horizontal Stretch

• Horizontal Compression

Multiple Choice

Describe the transformation for:

$$\ \sqrt[]{\frac{3}{4}x}$$  

• Horizontal Stretch

• Horizontal Compression

Multiple Choice

Describe the transformation for:

$$-\ \sqrt[]{\frac{1}{8}x\ -4}\ +\ 7$$  

• Horizontal Stretch

• Horizontal Compression

Multiple Choice

Describe the transformation for:

$$5\ \sqrt[]{3x\ +\ 6}\ -\ 1$$  

• Horizontal Stretch

• Horizontal Compression

The letter "c" controls the horizontal shift of a graph.

Remember, to move the opposite direction of what you see. ​

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-If you see "+ c" move left.

-If you see "- c" move right.

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​Horizontal Shifts (moving left and right)

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​This number is underneath the square root and behind the x.

Multiple Choice

Describe the transformation for:

$$\sqrt[]{x\ +\ 6}$$  

• Shift right 6 units

• Shift left 6 units

Multiple Choice

Describe the transformation for:

$$\sqrt[]{x\ -\ 10}$$  

• Shift right 10 units

• Shift left 10 units

Multiple Choice

Describe the transformation for:

$$3\ \sqrt[]{2x\ -\ 1}\ +\ 5$$  

• Shift right 1 unit

• Shift left 1 unit

• Shift left 5 units

• Shift right 5 units

Multiple Choice

Describe the transformation for:

$$-\frac{3}{4}\ \sqrt[]{x\ +\ 9}\ -\ 4\$$  

• Shift right 4 units

• Shift left 9 units

• Shift left 4 units

• Shift right 9 units

The letter "d" controls the vertical shift of a graph.

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-If you see "+ d" move up.

-If you see "- d" move down.

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Vertical Shifts (moving up and down)

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​This number is added to or subtracted from the square root symbol. It is NOT underneath the square root.

Multiple Choice

Describe the transformation for:

$$\sqrt[]{x\ }\ -\ 4\$$  

• Shift down 4 units

• Shift up 4 units

Multiple Choice

Describe the transformation for:

$$\sqrt[]{x\ }+\ 3$$  

• Shift up 3 units

• Shift down 3 units

Multiple Choice

Describe the transformation for:

$$-4\ \sqrt[]{\frac{1}{2}x\ -4\ }+\ 6$$  

• Shift up 4 units

• Shift down 4 units

• Shift down 6 units

• Shift up 6 units

Multiple Choice

Describe the transformation for:

$$\frac{4}{5}\ \sqrt[]{8x\ +\ 5\ }-\ 2$$  

• Shift up 8 units

• Shift down 5 units

• Shift down 2 units

• Shift up 2 units

Mixed Practice

Read each question carefully, and select the correct answer. USE YOUR NOTES IF YOU NEED THEM!​

Multiple Select

SELECT ALL THAT APPLY for the transformations that best describe this graph:

$$f\left(x\right)\ =\ -2\ \sqrt[]{x}\ +\ 4\$$  

• Reflection over the x-axis

• Reflection over the y-axis

• Horizontal Compression

• Vertical Stretch

• Vertical Shift

  $$\uparrow\ or\ \downarrow$$  

Multiple Select

SELECT ALL THAT APPLY for the transformations that best describe this graph:

$$f\left(x\right)\ =\ \sqrt[]{-\frac{2}{5}x\ -\ 3}\$$  

• Reflection over the x-axis

• Reflection over the y-axis

• Horizontal Stretch

• Vertical Compression

• Horizontal Shift

  $$\longleftarrow\ or\ \longrightarrow$$  

Multiple Choice

Which equation best represents the graph?

 

• $$f\left(x\right)\ =\ \sqrt[]{x\ }+\ 1$$  

• $$f\left(x\right)\ =\ \sqrt[]{x}\ -\ 1$$  

• $$f\left(x\right)\ =\ \sqrt[]{x\ +\ 1}$$  

• $$f\left(x\right)\ =\ \sqrt[]{x\ -\ 1}$$  

Multiple Choice

Which equation best represents the graph?

 

• $$f\left(x\right)\ =\ \sqrt[]{x\ }+\ 3$$  

• $$f\left(x\right)\ =\ \sqrt[]{x}\ -\ 3$$  

• $$f\left(x\right)\ =\ \sqrt[]{x\ +\ 3}$$  

• $$f\left(x\right)\ =\ \sqrt[]{x\ -\ 3}$$  

Multiple Choice

Which equation would be:

-Reflected over the y-axis

-Compressed horizontally by 4

-Shifted left 10 units

-Shifted down 9 units

 

• $$f\left(x\right)\ =\ -4\ \sqrt[]{x\ +10\ }-\ 9$$  

• $$f\left(x\right)\ =\ \sqrt[]{-4x\ +\ 10}\ -\ 9$$  

• $$f\left(x\right)\ =\ -\ \sqrt[]{4x\ +\ 10}\ -\ 9$$  

• $$f\left(x\right)\ =\ \sqrt[]{-4x\ -\ 10}\ -\ 9$$  

Multiple Choice

Which equation would be:

-Compressed vertically by 1/2

-Stretched horizontally by 2/3

-Shifted right 7 units

-Shifted up 4 units

 

• $$f\left(x\right)\ =\ \frac{2}{3}\ \sqrt[]{\frac{1}{2}x\ -\ 7\ }+\ 4$$  

• $$f\left(x\right)\ =\ \frac{2}{3}\ \sqrt[]{\frac{1}{2}x\ +\ 7}\ +\ 4_{ }$$  

• $$f\left(x\right)\ =\ \frac{1}{2}\ \sqrt[\ ]{\frac{2}{3}x\ -\ 7}\ +\ 4$$  

• $$f\left(x\right)\ =\ \frac{1}{2}\ \sqrt[]{\frac{2}{3}x\ +\ 7}\ +\ 4$$