### About Sequence of Graphing Transformations:

In this section, we will continue to explore the topic of function transformations. Here, we will be challenged with finding the correct sequence of function transformations to go from a parent function f(x) to a given function g(x).

Test Objectives

- Demonstrate a general understanding of function transformations
- Demonstrate the ability to find the correct sequence of function transformations

#1:

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

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

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

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#2:

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

$$a)\hspace{.2em}$$ $$f(x) = x^3$$ $$g(x) = (2x - 6)^3$$

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

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#3:

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

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

$$b)\hspace{.2em}$$ $$f(x) = \sqrt[3]{x}$$ $$g(x) = -2\sqrt[3]{2x - 2} - 1$$

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#4:

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

$$a)\hspace{.2em}$$ $$f(x) = x^2$$ $$g(x) = -5\left(\frac{1}{3}x + 1\right)^2 - 2$$

$$b)\hspace{.2em}$$ $$f(x) = x^3$$ $$g(x) = -\frac{1}{4}(2x + 6)^3 + 2$$

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#5:

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

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

$$b)\hspace{.2em}$$ $$f(x) = x^3 - x$$ $$g(x) = \frac{1}{3}(2x - 6)^3 - \frac{1}{3}(2x - 6) - 1$$

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}$$ Vertically compressed by a factor of 2 and shifted down 1 unit. Desmos Link for More Detail

$$b)\hspace{.2em}$$ Reflected across the x-axis, vertically stretched by a factor of 2, and shifted up 3 units. Desmos Link for More Detail

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#2:

Solutions:

$$a)\hspace{.2em}$$ Horizontally compressed by a factor of 2 and shifted right 3 units. Desmos Link for More Detail

$$b)\hspace{.2em}$$ Reflected across the y-axis, horizontally stretched by a factor of 2, and shifted right 2 units. Desmos Link for More Detail

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#3:

Solutions:

$$a)\hspace{.2em}$$ Vertically stretched by a factor of 3, horizontally stretched by a factor of 3, shifted 3 units right, and shifted 1 unit up. Desmos Link for More Detail

$$b)\hspace{.2em}$$Reflected across the x-axis, vertically stretched by a factor of 2, horizontally compressed by a factor of 2, shifted 1 unit right, and shifted 1 unit down. Desmos Link for More Detail

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#4:

Solutions:

$$a)\hspace{.2em}$$Reflected across the x-axis, vertically stretched by a factor of 5 horizontally stretched by a factor of 3, shifted left 3 units, and shifted down 2 units. Desmos Link for More Detail

$$b)\hspace{.2em}$$ Reflected across the x-axis, vertically compressed by a factor of 4, horizontally compressed by a factor of 2, shifted left 3 units, and shifted up 2 units. Desmos Link for More Detail

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#5:

Solutions:

$$a)\hspace{.2em}$$ Vertically compressed by a factor of 2, horizontally stretched by a factor of 3/2, shifted right 3/2 units, and shifted up 1 unit. Desmos Link for More Detail

$$b)\hspace{.2em}$$ Vertically compressed by a factor of 3, horizontally compressed by a factor of 2, shifted right 3 units, and shifted down 1 unit. Desmos Link for More Detail