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Section 6-1 : Exponential Functions

3. Sketch each of the following.

1. $$f\left( x \right) = {6^x}$$
2. $$g\left( x \right) = {6^x} - 9$$
3. $$g\left( x \right) = {6^{x + 1}}$$

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a Show Solution

We can build up a quick table of values that we can plot for the graph of this function.

$$x$$ $$f\left( x \right)$$
-2 $$f\left( { - 2} \right) = {6^{ - 2}} = \frac{1}{{{6^{\,2}}}} = \frac{1}{{36}}$$
-1 $$f\left( { - 1} \right) = {6^{ - 1}} = \frac{1}{6}$$
0 $$f\left( 0 \right) = {6^0} = 1$$
1 $$f\left( 1 \right) = {6^1} = 6$$
2 $$f\left( 2 \right) = {6^2} = 36$$

Here is a quick sketch of the graph of the function. b Show Solution

For this part all we need to do is recall the Transformations section from a couple of chapters ago. Using the “base” function of $$f\left( x \right) = {6^x}$$ the function for this part can be written as,

$g\left( x \right) = {6^x} - 9 = f\left( x \right) - 9$

Therefore, the graph for this part is just the graph of $$f\left( x \right)$$ shifted down by 9.

The graph of this function is shown below. The blue dashed line is the “base” function, $$f\left( x \right)$$, and the red solid line is the graph for this part, $$g\left( x \right)$$. c Show Solution

For this part all we need to do is recall the Transformations section from a couple of chapters ago. Using the “base” function of $$f\left( x \right) = {6^x}$$ the function for this part can be written as,

$g\left( x \right) = {6^{x + 1}} = f\left( {x + 1} \right)$

Therefore, the graph for this part is just the graph of $$f\left( x \right)$$ shifted left by 1.

The graph of this function is shown below. The blue dashed line is the “base” function, $$f\left( x \right)$$, and the red solid line is the graph for this part, $$g\left( x \right)$$. 