Select the correct answer.
What is the $y$-intercept of $f(x)=3^{x+2}$?
The function given is \(f(x) = 3^{x+2}\). A y-intercept is the point where the graph of a function intersects the y-axis. By definition, at any point on the y-axis, the x-coordinate must be \(0\). Therefore, to find the y-intercept of the function, we evaluate the function at \(x = 0\).
To find the y-intercept, we substitute \(x = 0\) into the function \(f(x) = 3^{x+2}\).
This gives:
\[f(0) = 3^{0+2}\]
\[f(0) = 3^2\]
Calculating the power, we have:
\[3^2 = 9\]
Thus, the value of the function at \(x=0\) is \(9\).
Since the y-intercept is the point where the graph crosses the y-axis, its coordinates are given by \((0, f(0))\). Substituting our calculated value, we find the coordinates to be \((0, 9)\). Comparing this to the given options:
The correct point is \((0, 9)\).
The y-intercept is a fundamental concept in modeling growth processes. For instance, if you are tracking the population growth of a certain bacteria culture over time, where the population \(P(t)\) at time \(t\) is represented by an exponential function like \(P(t) = P_0 \cdot 2^t\), the y-intercept represents the initial population of the bacteria at time \(t = 0\). Identifying this point allows scientists and researchers to establish a starting baseline from which to analyze future growth or decay.