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