Let's analyze the sine function graph you uploaded.

1. Amplitude (A):

Amplitude is the distance from the midline to the maximum (or minimum) of the wave. From the graph:

• Maximum value: 6
• Minimum value: -6
• Therefore, the amplitude is: $A = \frac{6 - (-6)}{2} = 6$

2. Midline (y):

The midline is the horizontal line that represents the average of the maximum and minimum values:

• $\text{Midline} = \frac{6 + (-6)}{2} = 0$

Thus, the midline is $y = 0$.

3. Period (P):

The period is the length of one full cycle. In the graph:

• The wave starts at $x = -7$ and completes one full cycle at $x = 1$.
• The distance between these points is: $P = 1 - (-7) = 8$

4. Equation of the sine function:

The general form of the sine function is: $y = A \cdot \sin\left(\frac{2\pi}{P} \cdot (x - D)\right) + C$ where:

• $A = 6$ (amplitude)
• $P = 8$ (period)
• $C = 0$ (midline)
• $D$ represents a phase shift, but since the sine wave starts at the origin, we don't need a shift ($D = 0$).

Thus, the equation is: $y = 6 \cdot \sin\left(\frac{2\pi}{8} \cdot x\right) = 6 \cdot \sin\left(\frac{\pi}{4} \cdot x\right)$

Would you like more details or have any questions?

Relative Questions:

1. What happens if we change the amplitude of a sine wave?
2. How would the graph shift if there was a phase shift $D$?
3. What is the relationship between the period and frequency of a sine wave?
4. How can we modify the sine equation to represent a cosine function instead?
5. What real-world phenomena can be modeled using sinusoidal functions?

Tip:

When identifying the period of a sine function from a graph, locate two consecutive peaks (or troughs) and measure the distance between them.