1. **State the problem:** We need to find the equation of the periodic function based on the given graph. 2. **Identify the form:** The function is given as $f(t) = a \sin(bt)$ or $f(t) = a \cos(bt)$. 3. **Analyze the graph:** - The graph oscillates between -4 and 4, so the amplitude $a = 4$. - The graph passes through the origin $(0,0)$ and reaches a maximum at $\left(\frac{\pi}{2}, 4\right)$. - The period $T$ is the distance between repeating points. From $0$ to $2\pi$, the pattern repeats, so $T = 2\pi$. 4. **Find $b$:** The period formula is $T = \frac{2\pi}{b}$, so $$b = \frac{2\pi}{T} = \frac{2\pi}{2\pi} = 1.$$ 5. **Determine sine or cosine:** Since the graph passes through the origin and goes up to a maximum, it matches the shape of $\sin(t)$ with positive amplitude. 6. **Write the function:** $$f(t) = 4 \sin(t).$$ 7. **Check options:** The closest option is $y = 4 \cos x$, but since the graph matches sine behavior, none of the cosine options fit. The correct function is $4 \sin t$, but since it's not listed, the best match is $y = 4 \sin x$ (not listed). Given the options, none exactly match, but the graph corresponds to $4 \sin t$. **Final answer:** $$f(t) = 4 \sin t.$$