Question

Function f is an exponential function. It predicts the value of a famous sculpture, in thousands of dollars, as a function of the number of years since it was last purchased. What equation models this function?

Answer 1

Remember that an exponential function takes the form
`b \cdot a^x`. We can see that the problem additionally gave us two points on the curve, which means that we can make two equations in attempt to find
`a`. (This is done by using the x-values and y-values that the points on the graph show, and substituting them into the general form.)

`b \cdot a^0 = b = 5`

`b \cdot a^1 = 7`

Substituting the value we found for
`b` that we found in the first equation into the second equation, we can create a better equation and solve for
`a`.

`5 \cdot a = 7`

`a = (7)/(5)`

Now, we have found
`b = 5` and
`a = (7)/(5)`, which means that we can create the final curve.

`\boxed{y = 5 \cdot \Big( (7)/(5) \Big)^x}`