### About Stretching or Shrinking a Graph:

In some cases, we will see a function transformation known as function dilation or simply stretching or shrinking of a graph. We will explore what happens when a function g(x) is defined by multiplying a parent function f(x) by some positive real number a. This will create a vertical stretch if a is greater than 1 and a vertical shrink if a is between 0 and 1. Additionally, we will explore horizontal compressions and stretches. These occur when a function such as g(x) is defined by plugging in ax in for x in the function f(x), so essentially this becomes g(x) = f(ax). Here if a is larger than 1, we have a horizontal compression and if a is between 0 and 1, we will have a horizontal stretch.

Test Objectives

- Demonstrate an understanding of function transformations
- Demonstrate the ability to determine a transformation that involves a horizontal stretch or compression
- Demonstrate the ability to determine a transformation that involves a vertical stretch or compression

#1:

Instructions: Find the transformation from f(x) to g(x).

$$a)\hspace{.2em}$$ $$f(x)=|x|$$ $$g(x)=\left|\frac{1}{2}x\right|$$

$$b)\hspace{.2em}$$ $$f(x)=\sqrt{x}$$ $$g(x)=\frac{1}{2}\sqrt{x}$$

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#2:

Instructions: Find the transformation from f(x) to g(x).

$$a)\hspace{.2em}$$ $$f(x)=x^3$$ $$g(x)=\left(\frac{1}{3}x\right)^3$$

$$b)\hspace{.2em}$$ $$f(x)=[x]$$ $$g(x)=3[x]$$

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#3:

Instructions: Find the transformation from f(x) to g(x).

$$a)\hspace{.2em}$$ $$f(x)=\frac{1}{x}$$ $$g(x)=\frac{1}{3x}$$

$$b)\hspace{.2em}$$ $$f(x)=x^3$$ $$g(x)=\frac{1}{3}x^3$$

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#4:

Instructions: Find the transformation from f(x) to g(x).

$$a)\hspace{.2em}$$ $$f(x)=x^2$$ $$g(x)=(3x)^2$$

$$b)\hspace{.2em}$$ $$f(x)=\frac{1}{x}$$ $$g(x)=\frac{1}{2x}$$

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#5:

Instructions: Find the transformation from f(x) (blue graph) to g(x) (orange graph).

$$a)\hspace{.2em}$$

$$b)\hspace{.2em}$$

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Written Solutions:

#1:

Solutions:

a) horizontally stretched by a factor of 2

b) vertically compressed by a factor of 2

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#2:

Solutions:

a) horizontally stretched by a factor of 3

b) vertically stretched by a factor of 3

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#3:

Solutions:

a) horizontally compressed by a factor of 3

b) vertically compressed by a factor of 3

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#4:

Solutions:

a) horizontally compressed by a factor of 3

b) horizontally compressed by a factor of 2

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#5:

Solutions:

a) vertically stretched by a factor of 3

b) horizontally stretched by a factor of 3