Write an exponential equation given 2 points: (3,12) and (4,24).

We are asked
to find the exponential function that contains the points (3,12) and (4,24).

Exponential functions can be written in the form `y=a*b^x` . b is the base of the
exponential function. b determines if the function is an exponential growth function (x>0,
b>1) or an exponential decay function (x>0, 0 a<0 the graph is reflected over a horizontal axis.)

We need to find the
coefficient a and the base b. We are given two coordinate pairs so we can substitute those into
the general equation to get two equations in two unknowns.

(3,12) ==>
`12=a*b^3`

(4,24) ==> `24=a*b^4`

A good approach to
solving simultaneous equations is to solve one of the equations for one of the variables and
then substitute the resulting expression into the other equation.

Solve the
first equation for a to get `a=12/(b^3)` Now substitute this in place of a in the second
equation to get:

`24=(12/(b^3))b^4`

Using properties of
exponents we get 24=12b or b=2.

Take b=2 and substitute into one of the
original equations (to avoid errors.) Then `12=a*(2)^3` or 12=8a so that a=12/8=3/2.

We now know the base b=2 and the leading coefficient a=3/2 so we can
write the function:

`y=3/2 *(2)^x`

Since
2>1 this is an exponential growth function. We should check to see that the original (given)
points lie on the graph of this function.

12=3/2(2)^3 true and
24=3/2(2)^4=3/2(16) true.

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