Question

How do you write an exponential function of the form y = a b^x the graph of which passes through (-1, 4/5) and (2, 100)?

Answer 1

`y=4(5)^x`

Explanation:

Plug in `(x,y)=(-1,4/5)` and `(x,y)=(2,100)` to see the two equations that we're left with in terms of `a` and `b`.

For `(-1,4/5)`, we get

`4/5=a(b)^-1" "" "bb((1))`

and for `(2,100`) we get

`100=a(b)^2" "" "bb((2))`

Note that `bb((1))` can be rewritten as

`4/5=a/b" "" "bb((3))`

We can solve for `a` or `b`. Here, I'll solve for `a`.

`4/5b=a`

We can use `a=4/5b` to replace `a` with `4/5b` in `bb((2))`.

`100=a(b)^2" "=>" "100=4/5b(b)^2`

We can now solve this for `b`.

`100=4/5b^3`

Multiplying both sides by `5/4`:

`125=b^3`

`b=5`

Returning to `bb((3))` with our newfound value of `b`, we see that

`4/5=a/b" "=>" "4/5=a/5" "=>" "a=4`