Algebra - Exponential Functions

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Section 6.1 : Exponential Functions

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3. Sketch each of the following.

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

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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

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

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, $g\left( x \right)$ .