Question

The graph of an exponential function passes through (2,−32) and (3,−64). Find the exponential function that describes the graph.

Answer 1

Answer:
`y=(-8)2^x`

Explanation:

The exponential function will have this form:

`y=ab^x`

We know that the function passes through the points
`(2,-32)` and
`(3,-64)`. Then, we can substitute the coordinates of the point
`(2,-32)` into
`y=ab^x` and solve for "a":

`-32=ab^2\\\\a=(-32)/(b^2)`

Then, we know that:

`y=((-32)/(b^2))b^x`

Now, we neeed to substitute the coordinates of the second point
`(3,-64)` into
`y=((-32)/(b^2))b^x` and solve for "b":

`-64=((-32)/(b^2))b^3\\\\-64=-32b\\\\(-64)/(-32)=b\\\\b=2`

Substituting the value of "b" into
`a=(-32)/(b^2)` we can find "a":

`a(-32)/(2^2)\\\\a=-8`

Therefore, we get that the exponential function that describes the graph, is:

`y=(-8)2^x`