### About Stretching or Compressing a Graph:

In some cases, we observe a function transformation called function dilation, involving the stretching or compressing of a graph. We'll examine what occurs when a function g(x) is formed by multiplying a related function f(x) by a positive real number a. This results in a vertical stretch if a > 1 and a vertical compression if 0 < a < 1. Additionally, we'll explore horizontal compressions and stretches. These occur when the function g(x) is created by substituting ax for x in the function f(x), essentially making it g(x) = f(ax). If a > 1, this leads to a horizontal compression, and if 0 < a < 1, it results in 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).

$$a)\hspace{.2em}$$ $$f(x)=|x|$$ $$g(x)=\left|\frac{1}{3}x\right|$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ $$f(x)=\sqrt{x}$$ $$g(x)=\frac{1}{2}\sqrt{x}$$ Desmos Link for More Detail

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#2:

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

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

$$b)\hspace{.2em}$$ $$f(x)=[x]$$ $$g(x)=3[x]$$ Desmos Link for More Detail

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#3:

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

$$a)\hspace{.2em}$$ $$f(x)=\frac{1}{x}$$ $$g(x)=\frac{1}{2x}$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ $$f(x)=x^3$$ $$g(x)=\frac{1}{4}x^3$$ Desmos Link for More Detail

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#4:

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

$$a)\hspace{.2em}$$ $$f(x)=x^2$$ $$g(x)=(3x)^2$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ $$f(x)=\sqrt[3]{x}$$ $$g(x)=\sqrt[3]{2x}$$ Desmos Link for More Detail

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#5:

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

$$a)\hspace{.2em}$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ Desmos Link for More Detail

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Written Solutions:

#1:

Solutions:

a) horizontally stretched by a factor of 3

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 2

b) vertically compressed by a factor of 4

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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