The graph provided appears to represent a sinusoidal function, and you are asked to find its amplitude, period, and equation. Let's analyze it step by step:

1. Amplitude:

The amplitude of a sinusoidal function is the distance from the midline of the graph to the peak (or trough).

• From the graph, the midline seems to be at $y = 3$.
• The maximum value is at $y = 4$ and the minimum is at $y = 2$.
• The amplitude is calculated as the difference between the maximum and the midline (or midline and minimum): $\text{Amplitude} = 4 - 3 = 1$ Amplitude: 1

2. Period:

The period of a sinusoidal function is the horizontal length of one full cycle.

• From the graph, one full cycle appears to start at $x = -150^\circ$ and end at $x = 90^\circ$, giving a period of: $\text{Period} = 90^\circ - (-150^\circ) = 240^\circ$ Period: $240^\circ$

3. Equation:

The general form for a sinusoidal function is: $y = A \sin(Bx + C) + D$ Where:

• $A$ is the amplitude,
• $B$ is related to the period by $B = \frac{360^\circ}{\text{Period}}$,
• $D$ is the vertical shift (the midline),
• $C$ is the horizontal shift (phase shift).

For this function:

• The amplitude $A = 1$.
• The period is $240^\circ$, so $B = \frac{360^\circ}{240^\circ} = 1.5$.
• The vertical shift $D = 3$ (since the midline is at $y = 3$).
• It looks like the function has been shifted horizontally, starting at $x = -150^\circ$, so the phase shift needs to account for that. The horizontal shift is $-150^\circ$.

Thus, the equation can be written as: $y = \sin\left(1.5x + 225^\circ\right) + 3$

Would you like further clarification on any part of this explanation?

Related Questions:

1. How can we derive the period from the general form of a trigonometric function?
2. What effect does changing the amplitude have on the graph of the function?
3. How would the equation change if the function were a cosine function instead of sine?
4. Can you explain how phase shift affects the starting point of the sine wave?
5. How would you modify the equation if the sine function were reflected across the x-axis?

Tip:

To quickly find the period of a sine or cosine function from its equation, remember the formula: $\text{Period} = \frac{360^\circ}{B}$, where $B$ is the coefficient of $x$.