https://file.aiquickdraw.com/mathbot/file/17266778700543d315122.jpg

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!

Explore the function y = -3 sin(sin(2x + π/2)) with a detailed breakdown of its vertical translation, vertical stretch, and phase shift. Learn how to apply transformations to trigonometric functions and understand their effects on the graph. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation