# 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)

The goal is to use the two given points to find a and b. Then, we can replace a and b in the equation y = abx with the values we found.

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

## A more complicated example showing how to write an exponential function.

Example #2

Find y = abx for a graph that includes (1, 2) and (-1, 8)

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 (-1, 8)

8 = ab-1

Substitute
2 / b
for a in 8 = ab-1

8   =
2 / b
b-1

Multiply both sides of the equation by b2.

b2 × 8   =
2 / b
b-1 × b2

b2 × 8   =
2 / b
b

b2 × 8 = 2

Divide both sides of the equation by 8

(b2 × 8) / 8 = 2 / 8

b2 = 2 / 8

b2 = 1 / 4

b = square root of 1 / 4

b = 1 / 2

Now, let us find a using the equation a  = 2 / b

a = 2 ÷ 1 / 2   =   2 / 1 ÷ 1 / 2    =    2 / 1 × 2 / 1    =    4 / 1   =   4

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

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