Write an exponential function

You can write an exponential function from two points on the function's graph. For example, write an exponential function y = abx for a graph that includes (1,1) and (2, 4)

Use the general form of the exponential function y = abx and substitute for x and y using (1, 1)

1 = ab1

1 = ab

Divide both sides by b to solve for a

a  =  
1 / b

Use the general form of the exponential function y = abx again and substitute for x and y using (2, 4)

4 = ab2

Substitute    
1 / b
   for a in 4 = ab2

4   =  
1 / b
b2

4   =  
b2 / b


4 = b.

a  =  
1 / b

a  =  
1 / 4

The exponential function is y = (1/4)(4)x


Example #2

Find y = abx for a graph that includes (1,-2) and (2, -1)

Use the general form of the exponential function y = abx and substitute for x and y using (1, -2)

-2 = ab1

-2 = ab

Divide both sides by b to solve for a

a  =  
-2 / b

Use the general form of the exponential function y = abx again and substitute for x and y using (2, -1)

-1 = ab2

Substitute    
-2 / b
   for a in -1 = ab2

-1   =  
-2 / b
b2

-1   =  
2b2 / b

-1 = -2b

1  = 2b

b   =  
1 / 2

a   =  
-2 / b

a   =  
-2 / 1/2
= -2 × 2 = -4

The exponential function is y = -4(1/2)x




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