Question

Give a possible formula for the function below. Let A = (-1, 32) and B = (1, 8). (Note: Use the general exponential function.)

Answer 1

Explanation:

Let the equation is
`y = ab^x` ............ (1)

Putting the values of point A in equation (1) as follows.

`y = ab^x`

`32 = ab^(-1)` ........... (2)

Putting the values of point B in equation (1) as follows.

`y = ab^x`

`8 = ab^(1)` ................ (3)

Now, divide the equation (2) by equation (3) as follows.

`(32)/(8) = (ab^(-1))/(ab^(1))`

Cancelling out with common factors, the equation will be written as follows.

4 =
`(b^(-1))/(b^1)`

4 =
`(1)/(b^1 * b^(1))`

4 =
`(1)/(b^2)`

or,
`b^(2) = (1)/(4)`

Taking square root on both the side, we get

b = ±
`(1)/(2)`

Case 1. Place the value of b = +
`(1)/(2)` in equation (3) as follows.

`8 = ab^1`

`8 = a((1)/(2))^(1)`

`8 * 2 = a`

a = 16

Case 2. Place the value of b = -
`(1)/(2)` in equation (3) as follows.

`8 = ab^1`

8 =
`a * ((-1)/(2))^(1)`

`- 8 * 2 = a`

a = - 16