1. The problem is to identify the function of the form $$y = A \cos(B(x - C)) + D$$ given the graph characteristics. 2. The general cosine function formula is $$y = A \cos(B(x - C)) + D$$ where: - $A$ is the amplitude (height from the center line to peak), - $B$ affects the period (frequency), - $C$ is the horizontal shift (phase shift), - $D$ is the vertical shift. 3. From the graph description: - The y-axis ranges from -5 to 5, so the amplitude $A$ is likely 5 for the larger wave and smaller for the smaller wave. - The x-axis ticks are at multiples of $\frac{\pi}{3}$. - The solid curve has smaller amplitude and lower frequency. - The dashed curve has larger amplitude and higher frequency. 4. For the solid curve (smaller amplitude): - Amplitude $A$ is about 1 (since it fits within -1 to 1 range). - The period $T$ can be estimated from the distance between peaks. If one peak to next is $2\pi$, then $B = \frac{2\pi}{T} = 1$. - No horizontal or vertical shift observed, so $C=0$, $D=0$. 5. So the solid curve function is: $$y = 1 \cos(1(x - 0)) + 0 = \cos x$$ 6. For the dashed curve (larger amplitude and frequency): - Amplitude $A$ is about 5. - The period is shorter, for example $\frac{2\pi}{3}$, so $B = \frac{2\pi}{T} = 3$. - No horizontal or vertical shift, so $C=0$, $D=0$. 7. The dashed curve function is: $$y = 5 \cos(3(x - 0)) + 0 = 5 \cos 3x$$ Final answer for the solid curve (smaller amplitude): $$y = \cos x$$