Question 130638
In order to graph {{{f(x)=(2/3)^x}}}, we need to plot some points. To do that, we need to plug in some x values to get some y values



So let's find the first point:




{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(-5)=(2/3)^(-5)}}} Plug in {{{x=-5}}}



{{{f(-5)=(0.666666666666667)^(-5)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(-5)=(0.666666666666667)^(-5)}}} Raise 0.666666666666667 to the negative 5th power to get 7.59374999999998



So when {{{x=-5}}}, we have {{{y=7.59374999999998}}}



So our 1st point is (-5,7.59374999999998)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



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



{{{f(-4)=(0.666666666666667)^(-4)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(-4)=(0.666666666666667)^(-4)}}} Raise 0.666666666666667 to the negative 4th power to get 5.06249999999999



So when {{{x=-4}}}, we have {{{y=5.06249999999999}}}



So our 2nd point is (-4,5.06249999999999)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(-3)=(2/3)^(-3)}}} Plug in {{{x=-3}}}



{{{f(-3)=(0.666666666666667)^(-3)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(-3)=(0.666666666666667)^(-3)}}} Raise 0.666666666666667 to the negative 3rd power to get 3.375



So when {{{x=-3}}}, we have {{{y=3.375}}}



So our 3rd point is (-3,3.375)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(-2)=(2/3)^(-2)}}} Plug in {{{x=-2}}}



{{{f(-2)=(0.666666666666667)^(-2)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(-2)=(0.666666666666667)^(-2)}}} Raise 0.666666666666667 to the negative 2nd power to get 2.25



So when {{{x=-2}}}, we have {{{y=2.25}}}



So our 4th point is (-2,2.25)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(-1)=(2/3)^(-1)}}} Plug in {{{x=-1}}}



{{{f(-1)=(0.666666666666667)^(-1)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(-1)=(0.666666666666667)^(-1)}}} Raise 0.666666666666667 to the negative 1st power to get 1.5



So when {{{x=-1}}}, we have {{{y=1.5}}}



So our 5th point is (-1,1.5)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(0)=(2/3)^(0)}}} Plug in {{{x=0}}}



{{{f(0)=(0.666666666666667)^(0)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(0)=(0.666666666666667)^(0)}}} Raise 0.666666666666667 to the 0th power to get 1



So when {{{x=0}}}, we have {{{y=1}}}



So our 6th point is (0,1)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(1)=(2/3)^(1)}}} Plug in {{{x=1}}}



{{{f(1)=(0.666666666666667)^(1)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(1)=(0.666666666666667)^(1)}}} Raise 0.666666666666667 to the 1st power to get 0.666666666666667



So when {{{x=1}}}, we have {{{y=0.666666666666667}}}



So our 7th point is (1,0.666666666666667)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(2)=(2/3)^(2)}}} Plug in {{{x=2}}}



{{{f(2)=(0.666666666666667)^(2)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(2)=(0.666666666666667)^(2)}}} Raise 0.666666666666667 to the 2nd power to get 0.444444444444445



So when {{{x=2}}}, we have {{{y=0.444444444444445}}}



So our 8th point is (2,0.444444444444445)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(3)=(2/3)^(3)}}} Plug in {{{x=3}}}



{{{f(3)=(0.666666666666667)^(3)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(3)=(0.666666666666667)^(3)}}} Raise 0.666666666666667 to the 3rd power to get 0.296296296296297



So when {{{x=3}}}, we have {{{y=0.296296296296297}}}



So our 9th point is (3,0.296296296296297)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(4)=(2/3)^(4)}}} Plug in {{{x=4}}}



{{{f(4)=(0.666666666666667)^(4)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(4)=(0.666666666666667)^(4)}}} Raise 0.666666666666667 to the 4th power to get 0.197530864197531



So when {{{x=4}}}, we have {{{y=0.197530864197531}}}



So our 10th point is (4,0.197530864197531)



--------------  Let's find another point  --------------



{{{f(x)=(2/3)^x}}} Start with the given function



{{{f(5)=(2/3)^(5)}}} Plug in {{{x=5}}}



{{{f(5)=(0.666666666666667)^(5)}}} Divide 3 into 2 to get 0.666666666666667



{{{f(5)=(0.666666666666667)^(5)}}} Raise 0.666666666666667 to the 5th power to get 0.131687242798354



So when {{{x=5}}}, we have {{{y=0.131687242798354}}}



So our 11th point is (5,0.131687242798354)



Now lets make a table of the values we have calculated

<pre>
<TABLE width=500>

<TR><TD> x</TD><TD>y</TD></TR>

<TR><TD> -5</TD><TD>7.59374999999998</TD></TR> 
<TR><TD> -4</TD><TD>5.06249999999999</TD></TR> 
<TR><TD> -3</TD><TD>3.375</TD></TR> 
<TR><TD> -2</TD><TD>2.25</TD></TR> 
<TR><TD> -1</TD><TD>1.5</TD></TR> 
<TR><TD> 0</TD><TD>1</TD></TR> 
<TR><TD> 1</TD><TD>0.666666666666667</TD></TR> 
<TR><TD> 2</TD><TD>0.444444444444445</TD></TR> 
<TR><TD> 3</TD><TD>0.296296296296297</TD></TR> 
<TR><TD> 4</TD><TD>0.197530864197531</TD></TR> 
<TR><TD> 5</TD><TD>0.131687242798354</TD></TR> 
</TABLE>
</pre>Now plot the points

{{{drawing(900,900,-15,15,-15,15,
  grid( 1 ),
circle(-5,7.59374999999998,0.05),
circle(-5,7.59374999999998,0.08),
circle(-5,7.59374999999998,0.05),
circle(-5,7.59374999999998,0.1),
circle(-5,7.59374999999998,0.05),
circle(-5,7.59374999999998,0.12),
circle(-4,5.06249999999999,0.05),
circle(-4,5.06249999999999,0.08),
circle(-4,5.06249999999999,0.05),
circle(-4,5.06249999999999,0.1),
circle(-4,5.06249999999999,0.05),
circle(-4,5.06249999999999,0.12),
circle(-3,3.375,0.05),
circle(-3,3.375,0.08),
circle(-3,3.375,0.05),
circle(-3,3.375,0.1),
circle(-3,3.375,0.05),
circle(-3,3.375,0.12),
circle(-2,2.25,0.05),
circle(-2,2.25,0.08),
circle(-2,2.25,0.05),
circle(-2,2.25,0.1),
circle(-2,2.25,0.05),
circle(-2,2.25,0.12),
circle(-1,1.5,0.05),
circle(-1,1.5,0.08),
circle(-1,1.5,0.05),
circle(-1,1.5,0.1),
circle(-1,1.5,0.05),
circle(-1,1.5,0.12),
circle(0,1,0.05),
circle(0,1,0.08),
circle(0,1,0.05),
circle(0,1,0.1),
circle(0,1,0.05),
circle(0,1,0.12),
circle(1,0.666666666666667,0.05),
circle(1,0.666666666666667,0.08),
circle(1,0.666666666666667,0.05),
circle(1,0.666666666666667,0.1),
circle(1,0.666666666666667,0.05),
circle(1,0.666666666666667,0.12),
circle(2,0.444444444444445,0.05),
circle(2,0.444444444444445,0.08),
circle(2,0.444444444444445,0.05),
circle(2,0.444444444444445,0.1),
circle(2,0.444444444444445,0.05),
circle(2,0.444444444444445,0.12),
circle(3,0.296296296296297,0.05),
circle(3,0.296296296296297,0.08),
circle(3,0.296296296296297,0.05),
circle(3,0.296296296296297,0.1),
circle(3,0.296296296296297,0.05),
circle(3,0.296296296296297,0.12),
circle(4,0.197530864197531,0.05),
circle(4,0.197530864197531,0.08),
circle(4,0.197530864197531,0.05),
circle(4,0.197530864197531,0.1),
circle(4,0.197530864197531,0.05),
circle(4,0.197530864197531,0.12),
circle(5,0.131687242798354,0.05),
circle(5,0.131687242798354,0.08),
circle(5,0.131687242798354,0.05),
circle(5,0.131687242798354,0.1),
circle(5,0.131687242798354,0.05),
circle(5,0.131687242798354,0.12)
)}}}



Now connect the points to graph {{{y=(2/3)^x}}}  (note: the more points you plot, the easier it is to draw the graph)

{{{drawing(900,900,-15,15,-15,15,
grid( 1 ),
graph(900,900,-15,15,-15,15, (2/3)^x),
circle(-5,7.59374999999998,0.05),
circle(-5,7.59374999999998,0.08),
circle(-5,7.59374999999998,0.05),
circle(-5,7.59374999999998,0.1),
circle(-5,7.59374999999998,0.05),
circle(-5,7.59374999999998,0.12),
circle(-4,5.06249999999999,0.05),
circle(-4,5.06249999999999,0.08),
circle(-4,5.06249999999999,0.05),
circle(-4,5.06249999999999,0.1),
circle(-4,5.06249999999999,0.05),
circle(-4,5.06249999999999,0.12),
circle(-3,3.375,0.05),
circle(-3,3.375,0.08),
circle(-3,3.375,0.05),
circle(-3,3.375,0.1),
circle(-3,3.375,0.05),
circle(-3,3.375,0.12),
circle(-2,2.25,0.05),
circle(-2,2.25,0.08),
circle(-2,2.25,0.05),
circle(-2,2.25,0.1),
circle(-2,2.25,0.05),
circle(-2,2.25,0.12),
circle(-1,1.5,0.05),
circle(-1,1.5,0.08),
circle(-1,1.5,0.05),
circle(-1,1.5,0.1),
circle(-1,1.5,0.05),
circle(-1,1.5,0.12),
circle(0,1,0.05),
circle(0,1,0.08),
circle(0,1,0.05),
circle(0,1,0.1),
circle(0,1,0.05),
circle(0,1,0.12),
circle(1,0.666666666666667,0.05),
circle(1,0.666666666666667,0.08),
circle(1,0.666666666666667,0.05),
circle(1,0.666666666666667,0.1),
circle(1,0.666666666666667,0.05),
circle(1,0.666666666666667,0.12),
circle(2,0.444444444444445,0.05),
circle(2,0.444444444444445,0.08),
circle(2,0.444444444444445,0.05),
circle(2,0.444444444444445,0.1),
circle(2,0.444444444444445,0.05),
circle(2,0.444444444444445,0.12),
circle(3,0.296296296296297,0.05),
circle(3,0.296296296296297,0.08),
circle(3,0.296296296296297,0.05),
circle(3,0.296296296296297,0.1),
circle(3,0.296296296296297,0.05),
circle(3,0.296296296296297,0.12),
circle(4,0.197530864197531,0.05),
circle(4,0.197530864197531,0.08),
circle(4,0.197530864197531,0.05),
circle(4,0.197530864197531,0.1),
circle(4,0.197530864197531,0.05),
circle(4,0.197530864197531,0.12),
circle(5,0.131687242798354,0.05),
circle(5,0.131687242798354,0.08),
circle(5,0.131687242798354,0.05),
circle(5,0.131687242798354,0.1),
circle(5,0.131687242798354,0.05),
circle(5,0.131687242798354,0.12)
)}}}