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?

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.