how to find the phase shift or horizontal shift of a sine eqaution

To find the phase shift (or horizontal shift) of a sine equation, you need to examine the general form of the sine function:

\[
y = a \sin(bx - c) + d
\]

In this equation:

- \(a\) is the amplitude (the vertical stretch or compression of the wave).
- \(b\) affects the period (how stretched or compressed the wave is horizontally).
- \(c\) affects the horizontal shift or phase shift.
- \(d\) is the vertical shift (up or down).

### Steps to Find the Phase Shift:
1. **Identify the value of \(c\):** Look at the term inside the sine function, specifically the expression \(bx - c\).
2. **Factor \(b\) out of the expression \(bx - c\):** This gives you:
   \[
   y = a \sin\left(b\left(x - \frac{c}{b}\right)\right) + d
   \]
3. **Determine the phase shift:** The phase shift (horizontal shift) is \(\frac{c}{b}\). If \(\frac{c}{b}\) is positive, the graph shifts to the right. If \(\frac{c}{b}\) is negative, the graph shifts to the left.

### Example:
Consider the equation:
\[
y = 2 \sin\left(3x - \frac{\pi}{2}\right) + 1
\]

1. Here, \(b = 3\) and \(c = \frac{\pi}{2}\).
2. Factor out the 3 inside the sine function:
   \[
   y = 2 \sin\left(3\left(x - \frac{\pi}{6}\right)\right) + 1
   \]
3. The phase shift is \(\frac{\pi}{6}\). Since it's positive, the graph shifts to the right by \(\frac{\pi}{6}\).

Would you like more details on this, or do you have any specific questions? Here are five related questions:

1. How does the phase shift affect the graph of a sine function?
2. Can you explain how to find the period of a sine function?
3. What is the difference between phase shift and vertical shift in trigonometric functions?
4. How would you find the phase shift of a cosine function?
5. Can phase shift be negative, and what does it mean?

**Tip:** Always remember to factor out \(b\) before determining the phase shift to avoid errors in calculation.

Learn how to find the phase shift (horizontal shift) of a sine equation step-by-step with examples. Explore the sine function's general form and how parameters affect the graph's horizontal translation. Suitable for students in Grades 10-12 studying trigonometry.

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