Question

Find an exponential function that passes through each pair of points.

(2,1.75) & (-2,28)

Answer 1

`f(x) = 7\, ((1/2)^(x))`.

Explanation:

An exponential function is typically in the form
`f(x) = a\, (b^(x))`, where
`a` and
`b` (
`b > 0`) are constants to be found.

In this question:

`f(2) = 1.75` means that
`a\, (b^(2)) = 1.75`.

`f(-2) = 28` means that
`a\, (b^(-2)) = 28`.

Divide one of the two equations by the other to eliminate
`a` and solve for
`b`.

The number of the right-hand side of the second equation is larger than that of the first equation. Hence, divide the second equation with the first:

`\displaystyle (a\, (b^(-2)))/(a\, (b^(2))) = (28)/(1.75)`.

`\displaystyle b^(-4) = 16`.

`b^(-1) = 2`.

`\displaystyle b = (1)/(2)`.

Substitute
`b = (1/2)` back into either equation (for example, the first equation) and solve for
`a`:

`a\, ((1/2)^(2)) = 1.75`.

`a = 7`.

Substitute
`a = 7` and
`b = (1/2)` into the other equation. That equation should also be satisfied.

Therefore, this function would be:

`f = 7\, ((1 / 2)^(x))`.