Question

Define an exponential function, h(x), which passes through the points (1,16) and(5, 1296). Enter your answer in the form a*b^xh(x) =

Answer 1

We want the answer in exponential form:

`h(x)=a\cdot b^x`

where a and b are the constants to be determined. We can find a and b by substituting the given points and construct a system of equatons, that is, by replacing point (1,16) we have

`\begin{gathered} 16=a\cdot b^1 \\ so \\ 16=a\cdot b \end{gathered}`

and by substituting point (5,1296), we get

`1296=a\cdot b^5`

Then, we have 2 equations in 2 unknonws:

`\begin{gathered} a\mathrm{}b=16 \\ a\mathrm{}b^5=1296 \end{gathered}`

By isolating a from the first equation and substituting this result into the second one, we get

`\begin{gathered} ((16)/(b))b^5=1296 \\ \end{gathered}`

which gives

`\begin{gathered} 16b^4=1296 \\ b^4=(1296)/(16) \\ b^4=81 \\ b=\sqrt[4]{81} \\ b=3 \end{gathered}`

Then, by substituting this result into the first equation of our system, we obtain

`\begin{gathered} a\cdot3=16 \\ \text{then} \\ a=(16)/(3) \end{gathered}`

Therefore, the answer is

`h(x)=(16)/(3)\cdot(3)^x`