### About Horizontal and Vertical Shifts:

A vertical shift is very straightforward and is simply a movement up or down based on the original graph. For a given function f(x), the graph of f(x) + k is shifted up k units if k is positive and down by the absolute value of k units if k is negative. When we come across horizontal shifts, the process is not as straightforward. These movements are left or right based on the original graph. When we have a function such as f(x), then f(x + h), will shift the absolute value of h units right if h is negative and h units left if h is positive.

Test Objectives

- Demonstrate the ability to find a horizontal shift based on a parent function
- Demonstrate the ability to find a vertical shift based on a parent function
- Demonstrate the ability to find a horizontal and/or vertical shift based on a graph

#1:

Instructions: describe the shift of g(x) based on f(x).

$$a)\hspace{.2em}$$ $$f(x)=x^2$$ $$g(x)=(x-7)^2 + 5$$

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

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#2:

Instructions: describe the shift of g(x) based on f(x).

$$a)\hspace{.2em}$$ $$f(x)=[\![x]\!]$$ $$g(x)=[\![x+2]\!] + 1$$

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

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#3:

Instructions: describe the shift of g(x) based on f(x).

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

$$b)\hspace{.2em}$$ $$f(x)=|x|$$ $$g(x)=|x - 4| - 5$$

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#4:

Instructions: describe the shift of g(x) based on f(x).

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

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

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#5:

Instructions: give the parent function f(x), the transformed function g(x), and describe the transformation.

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

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

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Written Solutions:

#1:

Solutions:

a) Shifts 7 units right and 5 units up

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b) Shifts 1 unit left and 2 units down

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#2:

Solutions:

a) Shifts 2 units left and 1 unit up

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b) Shifts 2 units right and 1 unit up

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#3:

Solutions:

a) Shifts 3 units right and 2 units up

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b) Shifts 4 units right and 5 units down

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#4:

Solutions:

a) Shifts 2 units right and 3 units up

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b) Shifts 1 unit right and 2 units up

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#5:

Solutions:

a) Shifts 1 unit left and 3 units up

$$f(x)=\sqrt{x}$$ $$g(x)=\sqrt{x + 1}+ 3$$b) Shifts 6 units left and 1 unit down

$$f(x)=x^2$$ $$g(x)=(x + 6)^2 - 1$$