Let's analyze the given functions and match each with the corresponding graph.

We are working with five trigonometric functions and five graphs (A-E). The key features of trigonometric functions we’ll focus on are amplitude, frequency (period), vertical shifts, and reflections.

Analyzing the functions:

1. $f(x) = 2\sin(2x) + 1$

   • Amplitude: 2
   • Period: $\frac{2\pi}{2} = \pi$ (since the coefficient of $x$ inside sine is 2)
   • Vertical shift: +1
   • Type: Sine function (starts from the middle of the wave at $y = 1$)
   • Graph: We expect it to start at $y = 1$, oscillate between 3 and -1 with a period of $\pi$.

2. $f(x) = -2\sin\left(\frac{\pi}{2}x\right) + 1$

   • Amplitude: 2 (but reflected due to the negative sign)
   • Period: $\frac{2\pi}{\frac{\pi}{2}} = 4$ (since $\frac{\pi}{2}$ is the coefficient of $x$)
   • Vertical shift: +1
   • Type: Reflected sine function (inverted wave)
   • Graph: This will oscillate between -1 and 3, with a period of 4 units.

3. $f(x) = -3\cos\left(\frac{\pi}{3}x\right) + 2$

   • Amplitude: 3 (with a reflection)
   • Period: $\frac{2\pi}{\frac{\pi}{3}} = 6$
   • Vertical shift: +2
   • Type: Reflected cosine function (starting from the peak at $y = -1$, shifted by +2)
   • Graph: This oscillates between -1 and 5.

4. $f(x) = 3\cos\left(\frac{x}{2}\right) + 2$

   • Amplitude: 3
   • Period: $\frac{2\pi}{\frac{1}{2}} = 4\pi$ (longer period)
   • Vertical shift: +2
   • Type: Cosine function
   • Graph: Oscillates between -1 and 5, starting at the peak at $y = 5$.

5. $f(x) = -3\sin\left(\frac{\pi}{3}x\right) - 1$

   • Amplitude: 3 (with reflection)
   • Period: $\frac{2\pi}{\frac{\pi}{3}} = 6$
   • Vertical shift: -1
   • Type: Reflected sine function
   • Graph: Oscillates between -4 and 2, starting from the middle of the wave at $y = -1$.

Matching the graphs:

• Graph A: This graph shows a sine wave, with a peak at $y = 3$, a trough at $y = -1$, and a vertical shift of +1. It corresponds to $f(x) = 2\sin(2x) + 1$. Thus, A matches with 1.

• Graph B: This shows a sine wave that oscillates between -1 and 3, with a period of 4. This corresponds to $f(x) = -2\sin\left(\frac{\pi}{2}x\right) + 1$. Thus, B matches with 2.

• Graph C: This graph oscillates between -1 and 5, with a period of 6 units and a reflection (inverted). This matches $f(x) = -3\cos\left(\frac{\pi}{3}x\right) + 2$. Thus, C matches with 3.

• Graph D: Oscillates between -1 and 5, with a vertical shift of +2, and is a cosine wave. This matches $f(x) = 3\cos\left(\frac{x}{2}\right) + 2$. Thus, D matches with 4.

• Graph E: This graph oscillates between -4 and 2, with a reflection and a vertical shift of -1. This matches $f(x) = -3\sin\left(\frac{\pi}{3}x\right) - 1$. Thus, E matches with 5.

Final answer:

1. A
2. B
3. C
4. D
5. E

Would you like further details on any of the steps?

Here are 5 related questions you might consider:

1. How do you determine the period of a trigonometric function?
2. What effect does a negative sign in front of a trigonometric function have on its graph?
3. How do vertical shifts affect the graph of a sine or cosine function?
4. What are some real-world applications of sine and cosine functions?
5. How can phase shifts (horizontal shifts) be introduced in these trigonometric functions?

Tip: Always remember that the coefficient inside the trigonometric function affects the period, while the coefficient outside affects the amplitude.