The equation you provided is $y = -2\sin(4x) + 3$. Let's break this down:

1. Amplitude:

The amplitude of a sine function is determined by the coefficient in front of the sine term. In this case, the coefficient is $-2$, so the amplitude is $| -2 | = 2$.

2. Period:

The period of the sine function is determined by the coefficient inside the sine function. For $y = \sin(kx)$, the period is given by $\frac{2\pi}{|k|}$. Here, $k = 4$, so the period is: $\text{Period} = \frac{2\pi}{4} = \frac{\pi}{2}$

3. Phase Shift:

There is no horizontal phase shift since there is no added or subtracted constant inside the sine function with the $x$-term.

4. Vertical Shift:

The sine function is vertically shifted by +3. This means that the midline of the sine function is at $y = 3$ instead of $y = 0$.

5. Reflection:

Because of the negative sign in front of the sine, the graph will reflect over the midline (flipping the usual upward sine wave into a downward one).

Graph Summary:

• Amplitude: 2
• Period: $\frac{\pi}{2}$
• Vertical Shift: 3 units up
• Reflection: Over the midline due to the negative sign.

Would you like more details on the graph, or have any questions?

Here are 5 relative questions:

1. How do you calculate the phase shift in sinusoidal functions?
2. How does the coefficient inside the sine function affect its frequency?
3. What are the key characteristics to identify in any sine function equation?
4. How would the graph change if the equation was $y = 2\sin(4x) + 3$?
5. Can you explain the effect of vertical shifts on the midline of trigonometric functions?

Tip: To quickly identify the period of any trigonometric function, use $\frac{2\pi}{|k|}$, where $k$ is the coefficient of $x$.