To analyze the function $y = -3 \sin\left(\frac{\pi}{4} x - \frac{\pi}{4}\right) + 2$, let's break down each part as follows:

1. Midline: The midline of a sinusoidal function $y = A \sin(Bx - C) + D$ is given by $y = D$.

   • Here, $D = 2$.
   • So, the midline is $y = 2$.

2. Amplitude: The amplitude is the absolute value of $A$ in $y = A \sin(Bx - C) + D$.

   • Here, $A = -3$, so the amplitude is $|A| = 3$.

3. Period: The period of a sine function $y = A \sin(Bx)$ is given by $\frac{2\pi}{|B|}$.

   • In this function, $B = \frac{\pi}{4}$.
   • Therefore, the period is $\frac{2\pi}{\frac{\pi}{4}} = 2\pi \cdot \frac{4}{\pi} = 8$.

4. Phase Shift: The phase shift is found by solving $\frac{C}{B}$ where $C$ and $B$ are the coefficients in $y = A \sin(Bx - C) + D$.

   • Here, $C = \frac{\pi}{4}$ and $B = \frac{\pi}{4}$.
   • The phase shift is $\frac{\frac{\pi}{4}}{\frac{\pi}{4}} = 1$.
   • Since the expression inside the sine function is $\left(\frac{\pi}{4} x - \frac{\pi}{4}\right)$, the phase shift is positive 1 unit to the right.

To summarize:

• Midline: $y = 2$
• Amplitude: 3
• Period: $8$
• Phase Shift: $1$ (to the right)

Would you like further details, or do you have any questions?

Related Questions:

1. How would these values change if the amplitude were doubled?
2. What would happen to the period if $B$ was doubled?
3. How does a negative amplitude affect the graph of the function?
4. How can we express this function in terms of a cosine function?
5. What are the x-intercepts of this function?

Tip:

Remember, a positive phase shift means a shift to the right, while a negative phase shift means a shift to the left.