Example: Determine an exponential function of the form \(f(x) = C\cdot B^x\) which passes through the points \((0,1)\) and \(\left(-2,\frac{{1}}{{9}}\right)\).
Solution
When answering questions like this, whether they are for lines, parabolas, or in this case, exponential functions, the strategy is almost always the same: take the given points and plug them into the function. Once we do so, we may have a couple of equations, but we can use those equations to solve for one or more of the parameters of the model: \[\solve{ 1 &=&C\cdot B^0\\ 1&=&C\cdot (1)\\ 1&=&C } \] In this case, the first point we plug in actually solves for the variable \(C\), which is quite helpful. However, this should make sense because \(C\) should always be the vertical intercept of the exponential function. Using the fact that \(C=1\), we continue with the second point: \[ \solve{ \frac{1}{9}&=&B^{-2}\\ \frac{1}{9}&=&\frac{1}{B^2}\\ B^2&=&9\\ B&=&\pm3\\ B&=&3 } \]When solving for the Base of an exponential model, recall that the base must be positive, so we reject any negative solutions. Now that we have our parameters, we can write our final answer: \[ f(x)=3^x \]