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 $a$.

$5 \cdot a = 7$

$a = \dfrac{7}{5}$

Now, we have found $b = 5$ and $a = \dfrac{7}{5}$, which means that we can create the final curve.

$\boxed{y = 5 \cdot \Big( \dfrac{7}{5} \Big)^x}$