The graph provided represents a transformed trigonometric function. Let's analyze the options and match the equation to the graph step by step.

Observations:

1. Vertical Shift: The midline of the function appears to be at $y = 2$, indicating a vertical shift upwards by 2.
2. Amplitude: The graph oscillates between $y = -3$ and $y = 5$, confirming an amplitude of $1$ (i.e., standard sine or cosine function).
3. Horizontal Shift: The graph does not start at $(0, 2)$ as a standard sine or cosine wave would. Instead, there is a phase shift.
4. Correct Function Type: Observing the wave's starting point, the graph matches a sine wave shifted horizontally.

Option Matching:

• Correct Equation: The answer is $y = \sin\left(x - \frac{\pi}{4}\right) + 2$, as the graph reflects:
  • A sine wave ($\sin(x)$),
  • A horizontal shift of $\frac{\pi}{4}$ to the right,
  • A vertical shift of $+2$.

Would you like a detailed breakdown of how phase shifts and vertical shifts impact trigonometric graphs?

Here are 5 related questions to expand your understanding:

1. How do you calculate the amplitude and period of a sine or cosine function?
2. What are the key differences between horizontal and vertical shifts in trigonometric functions?
3. How do you determine if a graph is sine or cosine based on its starting point?
4. What does a negative sign in front of $\sin(x)$ or $\cos(x)$ do to the graph?
5. How would the equation change if the midline were $y = -2$ instead of $y = 2$?

Tip: Always compare key features (amplitude, period, phase shift, vertical shift) to match graphs to equations systematically.