SOLUTION: Sketch the graph of the function {{{f(x)=(2/3)^x}}}

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Question 130638: Sketch the graph of the function f%28x%29=%282%2F3%29%5Ex
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
In order to graph f%28x%29=%282%2F3%29%5Ex, 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%28-5%29=%282%2F3%29%5E%28-5%29 Plug in x=-5


f%28-5%29=%280.666666666666667%29%5E%28-5%29 Divide 3 into 2 to get 0.666666666666667


f%28-5%29=%280.666666666666667%29%5E%28-5%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%28-4%29=%282%2F3%29%5E%28-4%29 Plug in x=-4


f%28-4%29=%280.666666666666667%29%5E%28-4%29 Divide 3 into 2 to get 0.666666666666667


f%28-4%29=%280.666666666666667%29%5E%28-4%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%28-3%29=%282%2F3%29%5E%28-3%29 Plug in x=-3


f%28-3%29=%280.666666666666667%29%5E%28-3%29 Divide 3 into 2 to get 0.666666666666667


f%28-3%29=%280.666666666666667%29%5E%28-3%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%28-2%29=%282%2F3%29%5E%28-2%29 Plug in x=-2


f%28-2%29=%280.666666666666667%29%5E%28-2%29 Divide 3 into 2 to get 0.666666666666667


f%28-2%29=%280.666666666666667%29%5E%28-2%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%28-1%29=%282%2F3%29%5E%28-1%29 Plug in x=-1


f%28-1%29=%280.666666666666667%29%5E%28-1%29 Divide 3 into 2 to get 0.666666666666667


f%28-1%29=%280.666666666666667%29%5E%28-1%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%280%29=%282%2F3%29%5E%280%29 Plug in x=0


f%280%29=%280.666666666666667%29%5E%280%29 Divide 3 into 2 to get 0.666666666666667


f%280%29=%280.666666666666667%29%5E%280%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%281%29=%282%2F3%29%5E%281%29 Plug in x=1


f%281%29=%280.666666666666667%29%5E%281%29 Divide 3 into 2 to get 0.666666666666667


f%281%29=%280.666666666666667%29%5E%281%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%282%29=%282%2F3%29%5E%282%29 Plug in x=2


f%282%29=%280.666666666666667%29%5E%282%29 Divide 3 into 2 to get 0.666666666666667


f%282%29=%280.666666666666667%29%5E%282%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%283%29=%282%2F3%29%5E%283%29 Plug in x=3


f%283%29=%280.666666666666667%29%5E%283%29 Divide 3 into 2 to get 0.666666666666667


f%283%29=%280.666666666666667%29%5E%283%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%284%29=%282%2F3%29%5E%284%29 Plug in x=4


f%284%29=%280.666666666666667%29%5E%284%29 Divide 3 into 2 to get 0.666666666666667


f%284%29=%280.666666666666667%29%5E%284%29 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%28x%29=%282%2F3%29%5Ex Start with the given function


f%285%29=%282%2F3%29%5E%285%29 Plug in x=5


f%285%29=%280.666666666666667%29%5E%285%29 Divide 3 into 2 to get 0.666666666666667


f%285%29=%280.666666666666667%29%5E%285%29 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 



 xy



 -57.59374999999998
 

 -45.06249999999999
 

 -33.375
 

 -22.25
 

 -11.5
 

 01
 

 10.666666666666667
 

 20.444444444444445
 

 30.296296296296297
 

 40.197530864197531
 

 50.131687242798354
Now plot the points



Now connect the points to graph y=%282%2F3%29%5Ex (note: the more points you plot, the easier it is to draw the graph)