Question

Which of the following exponential functions goes through the points (1, 6) and (2, 12)?

f(x) = 3(2)x
f(x) = 2(3)x
f(x) = 3(2)−x
f(x) = 2(3)−x

Answer 1

`f(x)=3(2)^x`

Explanation

To solve this, we re using the standard exponential equation:
`f(x)=ab^x`

Also, remember that
`f(x)=y`

We know that our exponential function goes through the points (1, 6) and (2, 12), so we are going to replace the points in our stander exponential equation and solve for
`a` and
`b`:

For (1, 6)

`x=1` and
`y/f(x)=6`, so:

`f(x)=ab^x`

`6=ab^1`

`6=ab`

`a=(6)/(b)` equation (1)

For (2, 12)

`x=2` and
`y/f(x)=12`, so:

`f(x)=ab^x`

`12=ab^2` equation (2)

Replace equation (1) in equation (2) and solve for b

`12=ab^2`

`12=((6)/(b)) b^2`

`12=6b`

`b=(12)/(6)`

`b=2` equation (3)

Replace equation (3) in equation (1) to find a

`a=(6)/(b)`

`a=(6)/(2)`

`a=3`

Now we can complete our exponential function:

`f(x)=ab^x`

`f(x)=3(2)^x`

We can conclude that the exponential function that goes through the points (1, 6) and (2, 12) is
`f(x)=3(2)^x`