For the equation y=-1/2sin(x+3)-8 identify all transformations

For the equation \( y = -\frac{1}{2} \sin(x + 3) - 8 \), we can identify the following transformations applied to the basic sine function, \( y = \sin(x) \):

1. **Vertical Reflection**: The negative sign in front of the sine function (\( -\frac{1}{2} \)) reflects the graph across the x-axis. This means the sine wave will now be upside down.

2. **Vertical Compression**: The coefficient \( \frac{1}{2} \) compresses the graph vertically by a factor of \( \frac{1}{2} \), making the peaks and valleys only half as tall as those of the standard sine curve.

3. **Horizontal Shift (Phase Shift)**: The term \( x + 3 \) inside the sine function represents a horizontal shift. Specifically, the graph is shifted to the left by 3 units because \( x + 3 \) means the graph starts earlier (the opposite of the sign).

4. **Vertical Shift**: The term \( -8 \) at the end of the equation shifts the entire graph downward by 8 units.

In summary, the transformations are:

- Reflection across the x-axis (due to the negative sign),
- Vertical compression by a factor of \( \frac{1}{2} \),
- Horizontal shift to the left by 3 units,
- Vertical shift downward by 8 units.

Would you like further details or have any questions?

Here are 5 related questions:

1. What would the graph look like if the coefficient of sine was positive?
2. How does the period of the function change if the equation had a term like \( 2x \) inside the sine function?
3. What happens to the graph if we change the vertical shift to \( +8 \)?
4. How would the equation change if the sine function was horizontally stretched?
5. What if we used \( \cos(x) \) instead of \( \sin(x) \)?

**Tip**: Horizontal shifts in trigonometric functions always work in the opposite direction of the sign in the equation.

Explore how to identify transformations in the equation y = -1/2sin(x+3)-8. Learn about vertical reflection, vertical compression, horizontal shift, and vertical shift, and how they affect the graph of the sine function. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation