1. **State the problem:** We are given the graph of the function $y = a \cos(bx) + c$. We need to find the values of $a$, $b$, and $c$ based on the graph's characteristics. 2. **Identify key features from the graph:** - The graph starts at $y=10$ when $x=0$. - The graph oscillates between a maximum of $10$ and a minimum of $-20$. - The period of the cosine wave is $\pi$. 3. **Recall the general form and properties:** - The amplitude $a$ is half the distance between the maximum and minimum values. - The vertical shift $c$ is the midpoint between the maximum and minimum values. - The period $T$ is related to $b$ by the formula $$T = \frac{2\pi}{b}$$ 4. **Calculate amplitude $a$:** $$a = \frac{\text{max} - \text{min}}{2} = \frac{10 - (-20)}{2} = \frac{30}{2} = 15$$ 5. **Calculate vertical shift $c$:** $$c = \frac{\text{max} + \text{min}}{2} = \frac{10 + (-20)}{2} = \frac{-10}{2} = -5$$ 6. **Calculate $b$ using the period:** Given period $T = \pi$, use $$T = \frac{2\pi}{b}$$ Rearranged: $$b = \frac{2\pi}{T} = \frac{2\pi}{\pi} = 2$$ 7. **Verify the starting value at $x=0$:** $$y(0) = a \cos(b \cdot 0) + c = a \cos(0) + c = a \cdot 1 + c = 15 + (-5) = 10$$ This matches the graph's starting point. **Final answer:** $$a = 15, \quad b = 2, \quad c = -5$$