Question

Find an equation for an exponential function which passes through the points (-3,4) and (2,3). You may round any numbers in your formula to three decimal places. Is this an exponential growth or decay function?

Answer 1

Because we don’t have the initial value, we substitute both points into an equation of the form

`f(x)=ab^x`

and then solve the system for a and b.

- Substituting (-3,4) gives:

`4=ab^(-3)`

- Substituting (2,3) gives:

`3=ab^2`

Use the first equation to solve for a in terms of b:

`\begin{gathered} 4=ab^(-3) \\ (4)/(b^(-3))=a \\ (4)/((1)/(b^3))=a \\ a=4b^3 \end{gathered}`

Substitute a in the second equation, and solve for b:

`\begin{gathered} 3=ab^2 \\ 3=4b^3b^2 \\ 3=4b^(3+2) \\ 3=4b^5 \\ (3)/(4)=b^5 \\ b=\sqrt[5]{(3)/(4)}=((3)/(4))^{(1)/(5)}=0.944 \end{gathered}`

Use the value of b in the first equation to solve for the value of a:

`a=4b^3=4(0.944)^3=3.365`

Thus, the equation is:

`f(x)=3.365(0.944)^x`

Next, we can graph the function:

Notice that the graph below passes through the initial points given in the problem (-3,4) and (2,3). So, the graph is an exponential decay function.

Answer:

- Equation

`f(x)=3.365(0.944)^x`

- This is an exponential decay function.