Given the function $y = 2 + \frac{1}{2} \cos(\pi x)$, we need to determine the amplitude and period of the cosine term.

1. Amplitude:

The general form of a cosine function is: $y = A \cos(Bx) + C$ Where:

• $A$ represents the amplitude.
• $B$ affects the period.
• $C$ shifts the graph vertically.

In your function: $y = 2 + \frac{1}{2} \cos(\pi x)$ Here, the coefficient of cosine is $\frac{1}{2}$, so the amplitude is: $\text{Amplitude} = \left| \frac{1}{2} \right| = \frac{1}{2}$

2. Period:

The period of a cosine function is given by: $\text{Period} = \frac{2\pi}{B}$ Where $B$ is the coefficient of $x$ inside the cosine function. In your case, $B = \pi$. Therefore, the period is: $\text{Period} = \frac{2\pi}{\pi} = 2$

Summary:

• Amplitude: $\frac{1}{2}$
• Period: 2

Would you like more details or clarifications?

Related Questions:

1. How do vertical shifts (like $C = 2$ in this case) affect the graph of the function?
2. What would happen to the amplitude if the coefficient of cosine were negative?
3. How would you find the phase shift of a cosine function?
4. How does changing the period affect the spacing of peaks in the cosine graph?
5. Can you explain how the period changes if the argument of cosine is $\pi/2 \cdot x$?

Tip: When determining the amplitude of a trigonometric function, always take the absolute value of the coefficient of the sine or cosine function.