1. **Stating the problem:** We want to find a function that models the number of students participating in sports after $x$ years, given that the initial number is 317 and it increases by 4% each year. 2. **Understanding the problem:** The number of students grows by a percentage each year, which suggests exponential growth. The general formula for exponential growth is: $$f(x) = a(1 + r)^x$$ where $a$ is the initial amount, $r$ is the growth rate (as a decimal), and $x$ is the number of years. 3. **Applying the formula:** Here, $a = 317$ and $r = 0.04$ (4%). So the function is: $$f(x) = 317(1.04)^x$$ 4. **Checking the options:** - a) $317(4)^x$ is incorrect because 4 is not the growth factor. - b) $4x + 317$ is linear, not exponential. - c) $317(1.04)^x$ matches our formula. - d) $1.04x + 317$ is linear, not exponential. **Answer:** c) $f(x) = 317(1.04)^x$ --- **Regarding the graph:** - A vertical asymptote at $x=0$ suggests the function is undefined or tends to infinity there. - A horizontal asymptote at $y=30$ means as $x \to \infty$, $f(x) \to 30$. This behavior is typical of a function like: $$f(x) = 30 + \frac{k}{x}$$ or a rational function with vertical asymptote at $x=0$ and horizontal asymptote at $y=30$. Since the problem does not ask for a specific function here, this is just an observation.