Question 173127
In order to graph {{{f(x)=4^(x-3)}}}, we need to plot some points:



Note: I'll use "y" in place of f(x)



Let's find the y value when {{{x=0}}}



{{{y=4^(x-3)}}} Start with the given equation



{{{y=4^(0-3)}}} Plug in {{{x=0}}}



{{{y=4^(-3)}}} Subtract



{{{y=1/4^(3)}}} Flip the fraction to make the exponent positive



{{{y=1/64}}} Cube 4 to get 64



{{{y=0.015625}}} Divide.



So when {{{x=0}}}, then {{{y=0.015625}}}




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Now let's find the y value when {{{x=1}}}



{{{y=4^(x-3)}}} Start with the given equation



{{{y=4^(1-3)}}} Plug in {{{x=1}}}



{{{y=4^(-2)}}} Subtract



{{{y=1/4^(2)}}} Flip the fraction to make the exponent positive



{{{y=1/16}}} Square 4 to get 16



{{{y=0.0625}}} Divide.



So when {{{x=1}}}, then {{{y=0.0625}}}




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Now let's find the y value when {{{x=2}}}



{{{y=4^(x-3)}}} Start with the given equation



{{{y=4^(2-3)}}} Plug in {{{x=2}}}



{{{y=4^(-1)}}} Subtract



{{{y=1/4^(1)}}} Flip the fraction to make the exponent positive



{{{y=1/4}}} Raise 4 to the first power to get 4



{{{y=0.25}}} Divide.



So when {{{x=2}}}, then {{{y=0.25}}}




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Now let's find the y value when {{{x=3}}}



{{{y=4^(x-3)}}} Start with the given equation



{{{y=4^(3-3)}}} Plug in {{{x=3}}}



{{{y=4^(0)}}} Subtract



{{{y=1}}} Raise 4 to the zeroth power to get 1



So when {{{x=3}}}, then {{{y=1}}}




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Now let's find the y value when {{{x=4}}}



{{{y=4^(x-3)}}} Start with the given equation



{{{y=4^(4-3)}}} Plug in {{{x=4}}}



{{{y=4^(1)}}} Subtract



{{{y=4}}} Raise 4 to the first power to get 4



So when {{{x=4}}}, then {{{y=4}}}




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Now let's find the y value when {{{x=5}}}



{{{y=4^(x-3)}}} Start with the given equation



{{{y=4^(5-3)}}} Plug in {{{x=5}}}



{{{y=4^(2)}}} Subtract



{{{y=16}}} Square 4 to get 16



So when {{{x=5}}}, then {{{y=16}}}




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So we have the following values:



<table width="100" border="1"><th>x</th><th>y</th>
<tr><td>0</td><td>0.015625</td></tr>
<tr><td>1</td><td>0.0625</td></tr>
<tr><td>2</td><td>0.25</td></tr>
<tr><td>3</td><td>1</td></tr>
<tr><td>4</td><td>4</td></tr>
<tr><td>5</td><td>16</td></tr>
</table>




Now let's plot these points



{{{ drawing(500, 500, -10, 10, -3, 17,
 grid(1),
 graph( 500, 500, -10, 10, -3, 17,0),
 circle(0,0.015625,0.05),
 circle(0,0.015625,0.08),
 circle(0,0.015625,0.1),
 circle(1,0.0625,0.05),
 circle(1,0.0625,0.08),
 circle(1,0.0625,0.1),
 circle(2,0.25,0.05),
 circle(2,0.25,0.08),
 circle(2,0.25,0.1),
 circle(3,1,0.05),
 circle(3,1,0.08),
 circle(3,1,0.1),
 circle(4,4,0.05),
 circle(4,4,0.08),
 circle(4,4,0.1),
 circle(5,16,0.05),
 circle(5,16,0.08),
 circle(5,16,0.1)

)}}}



Now draw a curve through those points to graph {{{f(x)=4^(x-3)}}}



{{{ drawing(500, 500, -10, 10, -3, 17,
 grid(1),
 graph( 500, 500, -10, 10, -3, 17,4^(x-3)),
 circle(0,0.015625,0.05),
 circle(0,0.015625,0.08),
 circle(0,0.015625,0.1),
 circle(1,0.0625,0.05),
 circle(1,0.0625,0.08),
 circle(1,0.0625,0.1),
 circle(2,0.25,0.05),
 circle(2,0.25,0.08),
 circle(2,0.25,0.1),
 circle(3,1,0.05),
 circle(3,1,0.08),
 circle(3,1,0.1),
 circle(4,4,0.05),
 circle(4,4,0.08),
 circle(4,4,0.1),
 circle(5,16,0.05),
 circle(5,16,0.08),
 circle(5,16,0.1)

)}}}