To determine which function corresponds to the graph, let's analyze the graph's characteristics:

1. Amplitude: The maximum value is $4$, and the minimum value is $0$. The amplitude is: $A = \frac{\text{Max} - \text{Min}}{2} = \frac{4 - 0}{2} = 2$

2. Vertical shift: The graph oscillates between $4$ and $0$, so the midline is: $y = \frac{\text{Max} + \text{Min}}{2} = \frac{4 + 0}{2} = 2$ This means the graph has been shifted up by 2 units.

3. Phase shift and shape:

   • The graph starts at its maximum point at $x = 0$, which matches the behavior of a cosine function.
   • Therefore, we are looking for a cosine function with a phase shift.

4. Equation format: A cosine function has the general form: $f(x) = A \cos(B(x - C)) + D$

   • $A = -2$ flips the cosine wave vertically (since the graph opens down initially).
   • The midline shift $D = 2$.
   • The phase shift $C$ aligns the maximum with $x = 0$.

Matching Options

1. Option 2: $f(x) = -3\cos\left(x - \frac{3\pi}{2}\right) + 1$
   • Amplitude = $3$ (incorrect; the amplitude is $2$).
2. Option 4: $f(x) = 2\cos\left(x + \frac{\pi}{2}\right) - 1$
   • This does not match because of the amplitude and phase shift.
3. Option 3: $f(x) = 2\sin(x - \pi) - 1$
   • Incorrect as it’s sine, and the phase shift does not match.
4. Option 1: $f(x) = -2\sin(x + \pi) - 1$
   • This does not match either due to its sine behavior.

Final Answer:

The correct function is not listed. Do you need more clarification on matching graph behavior?