Let's work through each part of the question based on the provided graph.

a) What is the period?

The period of a sine function can be calculated by looking at the distance between two successive points where the function repeats. In this case, the function completes a full cycle between $\frac{-\pi}{6}$ and $\frac{5\pi}{6}$.

• Period = $\frac{5\pi}{6} - \frac{-\pi}{6}$
• Period = $\frac{5\pi}{6} + \frac{\pi}{6} = \frac{6\pi}{6} = \pi$

Thus, the period is $\pi$.

b) What is the amplitude?

The amplitude is the distance from the centerline (which is the x-axis here) to the peak of the sine wave. From the graph, the maximum value of the curve is 2, and the minimum value is -2. The amplitude is the absolute value of this maximum or minimum value.

So, the amplitude is 2.

c) What is the phase shift?

The phase shift is the horizontal shift from the standard sine function. A standard sine function $y = \sin(x)$ has its first peak at $\frac{\pi}{2}$. In this graph, the first peak occurs at $\frac{\pi}{12}$.

To find the phase shift:

• Standard peak point = $\frac{\pi}{2}$
• Peak on the graph = $\frac{\pi}{12}$

Phase shift = $\frac{\pi}{2} - \frac{\pi}{12} = \frac{6\pi}{12} - \frac{\pi}{12} = \frac{5\pi}{12}$

So, the phase shift is $\frac{5\pi}{12}$ to the right.

d) Write an equation for this sine curve

Using the general form of the sine function: $y = A \sin(B(x - C)) + D$ Where:

• $A$ is the amplitude
• $B$ is related to the period by $B = \frac{2\pi}{\text{Period}}$
• $C$ is the phase shift
• $D$ is the vertical shift (which is 0 here as the graph oscillates around the x-axis)

1. Amplitude $A = 2$.
2. Period $\pi$, so $B = \frac{2\pi}{\pi} = 2$.
3. Phase shift $C = \frac{5\pi}{12}$.
4. Vertical shift $D = 0$.

Thus, the equation is: $y = 2 \sin \left( 2 \left( x - \frac{5\pi}{12} \right) \right)$

Would you like to go into more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the equation change if the graph had a vertical shift?
2. What would the graph look like if the amplitude was 1 instead of 2?
3. How can we determine the phase shift from key points other than the peak?
4. What is the relationship between period and frequency in sinusoidal functions?
5. How would the graph change if the period doubled?

Tip: Always double-check whether the vertical shift ($D$) exists by observing if the graph oscillates around a shifted horizontal line.