To find the phase shift from the graph of a sine function, follow these steps:

Steps to Find the Phase Shift from a Graph:

1. Identify the basic properties of the sine graph:

   • Amplitude: Measure the vertical distance from the middle of the wave to its peak.
   • Period: Measure the horizontal distance between two consecutive peaks (or troughs).
   • Midline: The horizontal line that runs through the middle of the wave (i.e., the average value of the maximum and minimum points).

2. Determine the original sine wave:

   • The standard sine wave $y = \sin(x)$ starts at $x = 0$ (without any phase shift), crosses the midline, reaches its maximum at $x = \frac{\pi}{2}$, returns to the midline at $x = \pi$, reaches its minimum at $x = \frac{3\pi}{2}$, and completes one full cycle back at $x = 2\pi$.

3. Locate a key point on the graph:

   • Identify a characteristic point on the graph, such as where the sine wave crosses the midline going upward (zero crossing), reaches its maximum, or reaches its minimum.

4. Compare the identified point to the standard sine wave:

   • If the point occurs later than it would on the standard sine wave, the phase shift is to the right (positive).
   • If the point occurs earlier, the phase shift is to the left (negative).

5. Calculate the phase shift:

   • Measure the horizontal distance between the point you identified and where it would occur on the standard sine wave.
   • This distance is the phase shift. If the point occurs to the right of where it would be on the standard sine wave, the phase shift is positive; if to the left, it's negative.

Example:

Suppose you have a sine wave that reaches its first peak at $x = \frac{\pi}{4}$ instead of $x = \frac{\pi}{2}$.

1. Standard sine wave: The first peak normally occurs at $x = \frac{\pi}{2}$.
2. Observed graph: The peak occurs at $x = \frac{\pi}{4}$.
3. Calculate the phase shift:
   • The graph has shifted to the left by $\frac{\pi}{4}$.
   • Therefore, the phase shift is $-\frac{\pi}{4}$.

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

1. How can you determine the period of a sine function from its graph?
2. What is the relationship between phase shift and the starting point of a sine graph?
3. How do you find the vertical shift of a sine function from its graph?
4. Can you explain how to determine the amplitude of a sine function from its graph?
5. How would you find the phase shift of a cosine function from its graph?

Tip: Always look for the starting point of the sine curve (either at a peak, trough, or midline crossing) to determine the phase shift accurately.