https://file.aiquickdraw.com/mathbot/file/1725883584418b03l12hy.jpg

Graph y = 3 sin 2(x − 45◦) + 4, and describe it’s relationship to y = sin x.

The graph shows two trigonometric functions:

1. **Red Curve**: \( y = \sin(x) \)
2. **Blue Curve**: \( y = 3 \sin(2(x - 45^\circ)) + 4 \)

Let's break down and compare these two functions:

### \( y = \sin(x) \) (Red Curve):
- This is the basic sine wave with:
  - **Amplitude**: 1
  - **Period**: \( 2\pi \) radians (or 360°)
  - **No phase shift** or vertical shift.

### \( y = 3 \sin(2(x - 45^\circ)) + 4 \) (Blue Curve):
- This is a transformed sine function, which undergoes several changes:
  1. **Amplitude**: The coefficient "3" increases the amplitude to 3. So, the wave peaks at 3 and valleys at -3 (before the vertical shift).
  2. **Frequency**: The coefficient "2" inside the sine argument affects the frequency. The period is now \( \frac{360^\circ}{2} = 180^\circ \). So, the wave completes one cycle in 180°.
  3. **Phase Shift**: The term \( (x - 45^\circ) \) shifts the graph **45° to the right**.
  4. **Vertical Shift**: The "+4" moves the entire wave **4 units upward**.

### Relationship:
- **Amplitude Change**: The blue curve is taller (amplitude of 3) compared to the red curve (amplitude of 1).
- **Frequency Change**: The blue curve has a shorter period of 180° (double the frequency of the red curve).
- **Phase Shift**: The blue curve is shifted 45° to the right.
- **Vertical Shift**: The blue curve is shifted 4 units up, so its centerline is at \( y = 4 \) instead of the x-axis.

### Summary:
- The blue curve represents a sine function that oscillates faster (double frequency), with higher peaks (amplitude 3), is shifted both vertically (up by 4 units) and horizontally (to the right by 45°).

Would you like further details on how these transformations affect the graph, or clarification on any of the terms used?

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### Related Questions:
1. How do you calculate the period of a trigonometric function with frequency modifications?
2. What would happen if you changed the phase shift to \( x + 45^\circ \)?
3. How does the vertical shift affect the maximum and minimum points of the sine wave?
4. What are the real-world applications of trigonometric transformations like these?
5. How do cosine and sine transformations compare visually?

### Tip:
To determine the period of a sine or cosine function \( y = A \sin(B(x - C)) + D \), use the formula \( \text{Period} = \frac{360^\circ}{|B|} \) or \( \frac{2\pi}{|B|} \) in radians.

Graph y = 3 sin 2(x − 45°) + 4, and describe its relationship to y = sin x.

Explore the graph of y = 3 sin 2(x − 45°) + 4 and its relationship to y = sin x. Learn about amplitude, frequency, phase shift, and vertical shift transformations of trigonometric functions. Suitable for Grades 10-12.

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