1. The problem is to graph the function $y=4^x$ and identify the correct graph. 2. The function $y=4^x$ is an exponential function where the base $4$ is greater than $1$, so it represents exponential growth. 3. Important properties of exponential growth functions: - The graph passes through the point $(0,1)$ because any number to the zero power is $1$. - The graph is always positive, so it never touches or crosses the $x$-axis. - As $x$ increases, $y$ increases rapidly. - As $x$ decreases, $y$ approaches $0$ but never reaches it. 4. Among the given options, the graph that matches these properties is option B: an increasing exponential curve starting near $y=0$ for negative $x$ and increasing rapidly for positive $x$. 5. Therefore, the correct graph for $y=4^x$ is option B. Final answer: Option B is the correct graph for $y=4^x$.