### About Graphing Exponential Functions:

In some cases, we will be asked to sketch the graph of an exponential function of the form f(x) = a^{x}, where a is greater than 0 and not equal to 1. For this type of function, the points (-1,1/a), (0,1), and (1,a) are on the graph. From there, we can plot as many additional points as needed to get a good sketch. This type of function can be hard to graph manually because it takes off very quickly. Additionally, we need to understand how to apply function transformations to the graph of an exponential function.

Test Objectives

- Demonstrate the ability to graph an exponential function
- Demonstrate an understanding of function transformations

#1:

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

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

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#2:

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

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

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#3:

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

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

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#4:

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

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

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#5:

Instructions: sketch the graph of f(x).

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

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Written Solutions:

#1:

Solutions:

a) reflected across the x-axis, shifted down by 5 units

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#2:

Solutions:

a) reflected across the y-axis, shifted up by 2 units

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#3:

Solutions:

a) vertically compressed by a factor of 2, reflected across the x-axis and y-axis, shifted down by 2 units

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#4:

Solutions:

a) shifted 1 unit right, vertically stretched by a factor of 3, shifted 2 units up

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#5:

Solutions:

a)