[tex]\huge{\bf{{\underline{\colorbox{red} {\color{black} {Answer}}}}}}[/tex]
You're right, let's determine the equation of the given trigonometric graph based on its key characteristics:
1. **Amplitude**: The amplitude is the maximum value of the cosine function from its centerline. Observing the graph, the maximum value is 3 and the minimum value is -3. Hence, the amplitude $A$ is 3.
2. **Period**: The period of the cosine function is the length of one complete cycle. In the graph, one cycle is completed from $-\pi$ to $\pi$, so the period $P$ is $2\pi$.
3. **Phase Shift**: The graph does not appear to be horizontally shifted, as it starts at its maximum at $x = 0$. Therefore, the phase shift $C$ is 0.
4. **Vertical Shift**: The centerline of the graph appears to be at $y = 0$, indicating no vertical shift. Therefore, the vertical shift $D$ is 0.
Given these characteristics, the equation of the cosine function can be written as:
$y = A \cos \left( \frac{2\pi}{P} x \right) + D$
Substituting the values, we get:
$y = 3 \cos(x) + 0$
$y = 3 \cos(x)$
Therefore, the equation of the given trigonometric graph is $y = 3 \cos(x)$.
