To determine the function whose graph has a period of 2, we need to evaluate and find the periods of each given function.
### Step-by-Step Solution:
1. Function A: [tex]\( y = 3 \cos \pi x \)[/tex]
- The general form of a cosine function is [tex]\( y = a \cos(bx + c) + d \)[/tex].
- The period of [tex]\( \cos(bx) \)[/tex] is [tex]\( \frac{2\pi}{b} \)[/tex].
- For [tex]\( y = 3 \cos \pi x \)[/tex], the coefficient [tex]\( b \)[/tex] of [tex]\( x \)[/tex] is [tex]\( \pi \)[/tex].
- Period = [tex]\( \frac{2\pi}{\pi} = 2 \)[/tex].
2. Function B: [tex]\( y = \cos \left( x - \frac{\pi}{2} \right) \)[/tex]
- This is in the form [tex]\( \cos(bx + c) \)[/tex].
- The coefficient [tex]\( b \)[/tex] of [tex]\( x \)[/tex] is 1.
- Period = [tex]\( \frac{2\pi}{1} = 2\pi \)[/tex].
3. Function C: [tex]\( y = 2 \sin 3x \)[/tex]
- The general form of a sine function is [tex]\( y = a \sin(bx + c) + d \)[/tex].
- The period of [tex]\( \sin(bx) \)[/tex] is [tex]\( \frac{2\pi}{b} \)[/tex].
- For [tex]\( y = 2 \sin 3x \)[/tex], the coefficient [tex]\( b \)[/tex] of [tex]\( x \)[/tex] is 3.
- Period = [tex]\( \frac{2\pi}{3} \)[/tex].
4. Function D: [tex]\( y = -4 \sin 2x \)[/tex]
- This is in the form [tex]\( \sin(bx + c) \)[/tex].
- The coefficient [tex]\( b \)[/tex] of [tex]\( x \)[/tex] is 2.
- Period = [tex]\( \frac{2\pi}{2} = \pi \)[/tex].
Given these results, we see that:
- The period of [tex]\( y = 3 \cos \pi x \)[/tex] (Function A) is 2.
- The periods of the other functions do not match 2.
Therefore, the correct function whose graph has a period of 2 is:
A. [tex]\( y = 3 \cos \pi x \)[/tex]
