It seems like the function you provided is incomplete. Based on what you've written, I assume you meant:

$y = 3 \sin\left(-2x - \frac{\pi}{2}\right)$

To find the transformations of this sine function, let's break it down:

The general form of a sine function is:

$y = A \sin(Bx + C) + D$

Where:

• $A$ is the amplitude (vertical stretch or shrink),
• $B$ affects the period of the function (horizontal stretch or shrink),
• $C$ causes a horizontal shift (phase shift),
• $D$ is a vertical shift.

For the given function $y = 3 \sin\left(-2x - \frac{\pi}{2}\right)$:

Step 1: Amplitude

• The coefficient $3$ indicates a vertical stretch by a factor of 3.
  • This means the graph will oscillate between 3 and -3 instead of 1 and -1.

Step 2: Horizontal Stretch/Shrink

• The coefficient $-2$ inside the sine function affects the period of the graph.
  • The formula for the period of a sine function is $\frac{2\pi}{|B|}$.
  • In this case, $B = -2$, so the period is: $\text{Period} = \frac{2\pi}{2} = \pi$
  • This means the sine wave will complete one full cycle in $\pi$ units, which is a horizontal shrink by a factor of 2 (because normally, it would complete one cycle in $2\pi$).

Step 3: Phase Shift (Horizontal Shift)

• The phase shift is determined by the formula $\frac{-C}{B}$.
  • In the equation $-2x - \frac{\pi}{2}$, $C = -\frac{\pi}{2}$ and $B = -2$.
  • The phase shift is: $\frac{-(-\frac{\pi}{2})}{-2} = \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4}$
  • Since the phase shift is positive, the graph shifts to the left by $\frac{\pi}{4}$ units.

Step 4: Reflection

• The negative sign in front of the $2x$ indicates a reflection across the y-axis.
  • This means the graph is flipped horizontally.

Summary of Transformations:

1. Vertical Stretch by a factor of 3 (amplitude is 3).
2. Horizontal Shrink by a factor of 2 (period is $\pi$).
3. Phase Shift to the left by $\frac{\pi}{4}$ units.
4. Reflection across the y-axis.

Would you like a more detailed explanation or a graph of the function?

Here are some related questions:

1. How does the period of a sine function change when the coefficient inside the function changes?
2. What is the effect of a negative coefficient on the amplitude of a sine function?
3. How do you calculate the phase shift when the function is in the form $y = A \sin(Bx + C) + D$?
4. How does the reflection across the x-axis differ from a reflection across the y-axis in trigonometric functions?
5. How can you use the transformations to graph the function step by step?

Tip: To quickly find the phase shift, always use the formula $\frac{-C}{B}$, but remember to consider the signs of $C$ and $B$ carefully.