1. **Problem Statement:** Determine the values of $a$, $b$, and $c$ in the function $f(x) = a \cos(bx) + c$ for $0^\circ \leq x \leq 360^\circ$, given the graph oscillates between $y=0.5$ and $y=1.5$ with three full cycles. 2. **Formula and Explanation:** The general form is $f(x) = a \cos(bx) + c$ where: - $a$ is the amplitude (half the distance between max and min values), - $b$ affects the period (number of cycles in $360^\circ$), - $c$ is the vertical shift (midline of the wave). 3. **Find $a$ (Amplitude):** Amplitude $a = \frac{\text{max} - \text{min}}{2} = \frac{1.5 - 0.5}{2} = 0.5$ 4. **Find $c$ (Vertical Shift):** Vertical shift $c = \frac{\text{max} + \text{min}}{2} = \frac{1.5 + 0.5}{2} = 1$ 5. **Find $b$ (Frequency):** The period $T$ is the length of one full cycle. Since there are 3 full cycles in $360^\circ$, $$T = \frac{360^\circ}{3} = 120^\circ$$ The relationship between $b$ and period $T$ is: $$b = \frac{360^\circ}{T} = \frac{360^\circ}{120^\circ} = 3$$ 6. **Final values:** $$a = 0.5, \quad b = 3, \quad c = 1$$ These values match the graph's amplitude, frequency, and vertical shift exactly.