Given the equation $y = 5 \sin\left(\frac{\pi}{4}x - \frac{\pi}{2}\right) + 4$, let's break down each component of the function:

1. Amplitude: This is the coefficient in front of the sine function, which is $5$. So, the amplitude is $5$.

2. Period: The period of a sine function is given by $\frac{2\pi}{B}$, where $B$ is the coefficient of $x$ inside the sine function. Here, $B = \frac{\pi}{4}$. Therefore, the period is: $\frac{2\pi}{\frac{\pi}{4}} = 2\pi \times \frac{4}{\pi} = 8.$

3. Horizontal Shift: The horizontal shift (or phase shift) is found by solving $\frac{\pi}{4}x - \frac{\pi}{2} = 0$. To find the phase shift: $\frac{\pi}{4}x - \frac{\pi}{2} = 0,$ $\frac{\pi}{4}x = \frac{\pi}{2},$ $x = \frac{\pi}{2} \times \frac{4}{\pi} = 2.$ Since this is $-\frac{\pi}{2}$ inside the function, it means the graph is shifted to the right by 2 units.

4. Midline: The midline of the sine function is given by the constant added outside the sine function, which is $y = 4$.

So, the correct answers should be:

• Amplitude: $5$
• Period: $8$
• Horizontal Shift: $2$ units to the right
• Midline: $y = 4$

It seems the horizontal shift might be the answer you need to adjust to get full credit.

Would you like more details or have any questions?

Related Questions:

1. How do you find the period of a sine or cosine function?
2. What is the effect of the horizontal shift on the graph of a sine function?
3. How does changing the amplitude affect the sine graph?
4. How can you identify the midline of a trigonometric function?
5. How do vertical shifts change the properties of a trigonometric graph?

Tip:

When identifying the horizontal shift (phase shift) of a sine function, always remember it moves in the opposite direction of the sign inside the function.