Question 173133
In order to graph {{{f(x)=(1/5)^x}}}, we need to plot some points:



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




Let's find the y value when {{{x=-1}}}



{{{y=(1/5)^x}}} Start with the given equation



{{{y=(1/5)^(-1)}}} Plug in {{{x=-1}}}



{{{y=(5/1)^1}}} Flip the fraction to make the exponent positive.



{{{y=(5)^1}}} Reduce



{{{y=5}}} Raise {{{5}}} to the first power to get 5



So when {{{x=-1}}}, then {{{y=5}}}




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



{{{y=(1/5)^x}}} Start with the given equation



{{{y=(1/5)^0}}} Plug in {{{x=0}}}



{{{y=1}}} Raise {{{1/5}}} to the zeroth power to get 1



So when {{{x=0}}}, then {{{y=1}}}




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



{{{y=(1/5)^x}}} Start with the given equation



{{{y=(1/5)^1}}} Plug in {{{x=1}}}



{{{y=1/5}}} Raise {{{1/5}}} to the first power to get {{{1/5}}}



{{{y=0.2}}} Divide



So when {{{x=1}}}, then {{{y=0.2}}}




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



{{{y=(1/5)^x}}} Start with the given equation



{{{y=(1/5)^2}}} Plug in {{{x=2}}}



{{{y=1/25}}} Square {{{1/5}}} to get {{{(1/5)^2=(1/5)(1/5)=1/25}}}



{{{y=0.04}}} Divide



So when {{{x=2}}}, then {{{y=0.04}}}




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



{{{y=(1/5)^x}}} Start with the given equation



{{{y=(1/5)^3}}} Plug in {{{x=3}}}



{{{y=1/125}}} Cube {{{1/5}}} to get {{{(1/5)^3=(1/5)(1/5)(1/5)=1/125}}}



{{{y=0.008}}} Divide



So when {{{x=3}}}, then {{{y=0.008}}}




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



{{{y=(1/5)^x}}} Start with the given equation



{{{y=(1/5)^4}}} Plug in {{{x=4}}}



{{{y=1/625}}} Raise {{{1/5}}} to the fourth power to get {{{(1/5)^4=(1/5)(1/5)(1/5)(1/5)=1/625}}}



{{{y=0.0016}}} Divide



So when {{{x=4}}}, then {{{y=0.0016}}}




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So we have the following values:



<table width="100" border="1"><th>x</th><th>y</th>
<tr><td>-1</td><td>5</td></tr>
<tr><td>0</td><td>1</td></tr>
<tr><td>1</td><td>0.2</td></tr>
<tr><td>2</td><td>0.04</td></tr>
<tr><td>3</td><td>0.08</td></tr>
<tr><td>4</td><td>0.0016</td></tr>
</table>




Now let's plot these points



{{{ drawing(500, 500, -2, 5, -2, 7,
 grid(1),
 graph( 500, 500, -2, 5, -2, 7,0),
 circle(-1,5,0.02),
 circle(-1,5,0.04),
 circle(-1,5,0.06),
 circle(0,1,0.02),
 circle(0,1,0.04),
 circle(0,1,0.06),
 circle(1,0.2,0.02),
 circle(1,0.2,0.04),
 circle(1,0.2,0.06),
 circle(2,0.04,0.02),
 circle(2,0.04,0.04),
 circle(2,0.04,0.06),
 circle(3,0.008,0.02),
 circle(3,0.008,0.04),
 circle(3,0.008,0.06),
 circle(4,0.0016,0.02),
 circle(4,0.0016,0.04),
 circle(4,0.0016,0.06)

)}}}



Now draw a curve through those points to graph {{{f(x)=(1/5)^x}}}



{{{ drawing(500, 500, -2, 5, -2, 7,
 grid(1),
 graph( 500, 500, -2, 5, -2, 7,(1/5)^x),
 circle(-1,5,0.02),
 circle(-1,5,0.04),
 circle(-1,5,0.06),
 circle(0,1,0.02),
 circle(0,1,0.04),
 circle(0,1,0.06),
 circle(1,0.2,0.02),
 circle(1,0.2,0.04),
 circle(1,0.2,0.06),
 circle(2,0.04,0.02),
 circle(2,0.04,0.04),
 circle(2,0.04,0.06),
 circle(3,0.008,0.02),
 circle(3,0.008,0.04),
 circle(3,0.008,0.06),
 circle(4,0.0016,0.02),
 circle(4,0.0016,0.04),
 circle(4,0.0016,0.06)

)}}}



Note: the graph does NOT cross or touch the x-axis. It just looks like that since the y values become very small.