The equation given is:

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

The text next to it mentions a vertical translation down 2 units and a vertical stretch or shrink by the product of 1/2 and the existing function coefficient.

Let's break this down:

1. Base Function

The core of the function involves the nested sine functions:

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

• Outer Sine: The term \sin(something) indicates the sine wave behavior, with the amplitude controlled by the coefficient in front of it, which is -3 in this case.
• Inner Sine: The inner sine, \sin(2x + \frac{\pi}{2}), is itself a sine wave, with a phase shift of $\frac{\pi}{2}$ to the left and a horizontal compression due to the factor of 2 multiplying $x$.

2. Vertical Stretch/Shrink

• The coefficient of -3 causes a vertical stretch by a factor of 3, as compared to the base sine function.
• The negative sign causes a reflection over the x-axis.

The note refers to modifying this coefficient. If the vertical stretch or shrink is further multiplied by $\frac{1}{2}$, it means that the overall amplitude will be reduced to half its current value, or:

$-3 \times \frac{1}{2} = -\frac{3}{2}$

This would mean the new function would be:

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

3. Vertical Translation

The "vertical translation down 2 units" suggests the entire graph is shifted downward by 2 units. Thus, the final modified equation after the vertical translation would be:

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

This is the final equation that represents the function after both the stretch/shrink and translation.

Do you have any further questions, or would you like additional details? Here are some related questions:

1. How does the inner sine function affect the overall shape of the graph?
2. What is the significance of the phase shift $\frac{\pi}{2}$ in this function?
3. How does the vertical translation affect the maximum and minimum values of the sine wave?
4. How would you find the period of the inner sine function $\sin(2x + \frac{\pi}{2})$?
5. How would the graph look if the coefficient of -3 was positive instead?

Tip: When working with nested trigonometric functions, analyze the inner function's behavior first, then apply the modifications caused by the outer function!