Question

How do you find an exponential function given the points are (-1,8) and (1,2)?

Answer 1

`y=4(1/2)^x`

Explanation:

An exponential function is in the general form

`y=a(b)^x`

We know the points `(-1,8)` and `(1,2)`, so the following are true:

`8=a(b^-1)=a/b`

`2=a(b^1)=ab`

Multiply both sides of the first equation by `b` to find that

`8b=a`

Plug this into the second equation and solve for `b`:

`2=(8b)b`

`2=8b^2`

`b^2=1/4`

`b=+-1/2`

Two equations seem to be possible here. Plug both values of `b` into the either equation to find `a`. I'll use the second equation for simpler algebra.

If `b=1/2`:

`2=a(1/2)`

`a=4`

Giving us the equation: `color(green)(y=4(1/2)^x`

If `b=-1/2`:

`2=a(-1/2)`

`a=-4`

Giving us the equation: `y=-4(-1/2)^x`

However! In an exponential function, `b>0`, otherwise many issues arise when trying to graph the function.

The only valid function is

`color(green)(y=4(1/2)^x`