Question

Suppose that the graph of f(x) = Ca^x passes through the points (2, 9.375) and (5, 18.311). Find a formula for f(x).

Answer 1

The given function is:

`f(x)=Ca^x`

It is given that its graph passes through the points (2,9.375) and (5,18.311).

Recall that if the graph of a function passes through a point, then the coordinates of the point satisfy the equation of the function.

Substitute x=2 and f(x)=9.375 into the equation of the function:

`9.375=Ca^2`

Substitute x=5 and f(x)=18.311 into the equation of the function:

`18.311=Ca^5`

Hence, the system of equation is:

`\begin{gathered} 9.375=Ca^2 \\ 18.311=Ca^5 \end{gathered}`

Solve the system of equations to find the constants C and a.

Divide the second equation by the first equation:

`\begin{gathered} (18.311)/(9.375)=(Ca^5)/(Ca^2) \\ \Rightarrow(18.311)/(9.375)=\frac{\cancel{C}a^5}{\cancel{C}a^2}\Rightarrow(18.311)/(9.375)=(a^5)/(a^2) \end{gathered}`

Solve for a in the equation:

`\begin{gathered} \text{Swap the sides of the equation:} \\ (a^5)/(a^2)=(18.311)/(9.375)\Rightarrow a^(5-2)=1.9532 \\ \Rightarrow a^3=1.9532 \\ \Rightarrow a=\sqrt[3]{1.9532}\approx1.25 \end{gathered}`

Substitute this value of a into the first equation to solve for C:

`\begin{gathered} 9.375=C(1.25)^2 \\ \Rightarrow C=(9.375)/(1.25^2)=6 \end{gathered}`

Substitute the values of a and C into the initial equation of the function:

`\begin{gathered} f(x)=Ca^x;a=1.25,c=6 \\ \Rightarrow f(x)=6(1.25)^x \end{gathered}`

The required formula for f(x) is:

`f(x)=6(1.25)^x`