Question 8

Find the exponential function $f(x)=C a^{x}$ whose graph goes through the points $(0 ; 2)$ and $(2,8)$.
\[
\begin{array}{l}
a= \\
C=
\end{array}
\]
Question Help:
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Step 1 ：Let's denote the exponential function as \(f(x)=C a^{x}\). We know that the graph of this function goes through the points (0 ; 2) and (2,8).

Step 2 ：Substituting the point (0 ; 2) into the function, we get \(f(0) = C a^{0} = C = 2\). So, \(C = 2\).

Step 3 ：Substituting the point (2,8) into the function, we get \(f(2) = C a^{2} = 2a^{2} = 8\).

Step 4 ：Solving the equation \(2a^{2} = 8\) for \(a\), we get two possible solutions, \(a = -2\) and \(a = 2\).

Step 5 ：However, in the context of an exponential function, \(a\) should be positive. Therefore, we discard -2 and take \(a = 2\).

Step 6 ：Final Answer: \(a = \boxed{2}, C = \boxed{2}\)