Question

Find the equation of an exponential function in the form y = ab^x, given the points (0, 3) and (2, 108/25). Please simplify your answer.

Answer 1

We have the equation:

`y=a\cdot b^x`

We know two points and we will use them to calculate the parameters a and b.

The point (0,3) will let us know a, as b^0=1.

`\begin{gathered} y=a\cdot b^x \\ 3=a\cdot b^0=a \\ a=3 \end{gathered}`

Now, we use the point (2, 108/25) to calcualte b:

`\begin{gathered} y=3\cdot b^x \\ (108)/(25)=3\cdot b^2 \\ 3\cdot b^2=(108)/(25) \\ b^2=(108)/(25\cdot3)=(108)/(3)\cdot(1)/(25)=(36)/(25) \\ b=\sqrt[]{(36)/(25)} \\ b=\frac{\sqrt[]{36}}{\sqrt[]{25}} \\ b=(6)/(5) \end{gathered}`

Then, we can write the equation as:

`y=3\cdot((6)/(5))^x`