Question

Function f is an exponential function. It predicts the value of a famous painting, in thousands of dollars, as a function of the number of years since it was last purchased. What equation models this function? Graph on a coordinate plane with axes labeled x and f of x. An exponential curve passes through 0 comma 8, 1 comma 10, and 2 comma 12.5. Enter your answer in the box. f(x)=

Answer 1

Answer:

$y=8 \cdot (\frac{5}{4})^x$

$f(x)=8 \cdot (\frac{5}{4})^x$

or

$f(x)=8 \cdot (1.25)^x$

Step-by-step explanation:

We are going to see if the exponential curve is of the form:

$y=a \cdot b^x$, ($b\neq 0$).

If you are given the $y-$intercept, then $a$ is easy to find.

It is just the $y-$coordinate of the $y-$intercept is your value for $a$.

(Why? The $y-$intercept happens when $x=0$. Replacing $x$ with 0 gives $y=a \cdot b^0=a \cdot 1=a$. This says when $x=0 \text{ that} y=a$.)

So $a=8$.

So our function so far looks like this:

$y=8 \cdot b^x$

Now to find $b$ we need another point. We have two more points. So we will find $b$ using one of them and verify for our resulting equation works for the other.

Let's do this.

We are given $(1,10)$ is a point on our curve.

So when $x=1$, $y=10$.

$10=8 \cdot b^1$

$10=8 \cdot b$

Divide both sides by 8:

$\frac{10}{8}=b$

Reduce the fraction:

$\frac{5}{4}=b$

So the equation if it works out for the other point given is:

$y=8 \cdot (\frac{5}{4})^x$

Let's try it.  So the last point given that we need to satisfy is $(2,12.5)$.

This says when $x=2$, $y=12.5$.

Let's replace $x$ with 2 and see what we get for $y$:

$y=8 \cdot (\frac{5}{4})^2$

$y=8 \cdot \frac{25}{16}$

$y=\frac{8}{16} \cdot 25$

$y=\frac{1}{2} \cdot 25}$

$y=\frac{25}{2}$

$y=12.5$

So we are good. We have found an equation satisfying all 3 points given.

The equation is $y=8 \cdot (\frac{5}{4})^x$.