Let's break down the function step by step:

The given function is: $y = -4\sin\left(\frac{5\pi}{2}x + \frac{15\pi}{8}\right) + 3$

1. Amplitude:

The amplitude is the coefficient in front of the sine function. Here, the amplitude is 4, but the negative sign indicates that the graph is reflected across the horizontal axis. Therefore, the amplitude is 4.

2. Phase Shift:

The phase shift is determined by the equation inside the sine function: $y = \sin(bx + c)$ where the phase shift is given by: $\text{Phase Shift} = -\frac{c}{b}$ For our function: $b = \frac{5\pi}{2}, \quad c = \frac{15\pi}{8}$ So, the phase shift is: $\text{Phase Shift} = -\frac{\frac{15\pi}{8}}{\frac{5\pi}{2}} = -\frac{15\pi}{8} \times \frac{2}{5\pi} = -\frac{30\pi}{40\pi} = -\frac{3}{4}$ Thus, the phase shift is $-\frac{3}{4}$ units to the left.

3. Equation of Midline:

The midline is the horizontal line around which the sine wave oscillates. It is determined by the constant added or subtracted outside the sine function. In this case, the midline is: $y = 3$ So, the midline is the line $y = 3$.

4. Period:

The period of a sine function is given by: $\text{Period} = \frac{2\pi}{|b|}$ For our function, $b = \frac{5\pi}{2}$, so the period is: $\text{Period} = \frac{2\pi}{\frac{5\pi}{2}} = \frac{2\pi \times 2}{5\pi} = \frac{4}{5}$ Thus, the period is $\frac{4}{5}$ units.

Summary:

• Amplitude: 4
• Phase Shift: $-\frac{3}{4}$ units to the left
• Equation of Midline: $y = 3$
• Period: $\frac{4}{5}$

Would you like any further details or clarification? Here are some related questions:

1. How do you graph a sine function given its amplitude, phase shift, and period?
2. What happens to the graph of a sine function if the amplitude is negative?
3. How can you find the period and amplitude of a sine function from a graph?
4. What is the effect of changing the value of $c$ in the equation $y = \sin(bx + c)$?
5. How do the transformations affect the graph of a sine function?

Tip: To graph sine functions accurately, always start by identifying key features like amplitude, midline, and period, and then plot points based on these transformations.