What is the $y$-intercept of $f(x)=3^{x+2}$ ?
To find the y-intercept of a function \(f(x)\), we must determine the point where the graph of the function crosses the y-axis. This occurs at the point where the x-coordinate is zero. Therefore, we need to evaluate the function \(f(x) = 3^{x+2}\) at \(x = 0\).
We substitute \(x = 0\) into the given function expression:
\[f(0) = 3^{0+2}\]
This simplifies to:
\[f(0) = 3^2\]
Calculating the power, we find:
\[f(0) = 9\]
Thus, the function value at \(x = 0\) is 9.
The y-intercept is expressed as a coordinate pair \((x, y)\), where \(x\) is 0 and \(y\) is the value of the function at \(x = 0\). Since we found \(f(0) = 9\), the y-intercept is the point \((0, 9)\). Comparing this result with the provided options, we conclude that the correct option is \((0, 9)\).
The concept of a y-intercept is crucial in real-world scenarios involving linear and exponential growth. For example, if you are tracking the initial balance of a savings account where interest is added over time, the starting balance at time \(t = 0\) represents the y-intercept of your growth function. In our case, \(f(x) = 3^{x+2}\) could model the growth of a population where 9 represents the starting size at time \(x = 0\). Understanding these intercepts helps in predicting initial conditions before any growth or decay takes place.