Let's analyze the sine function graphs given in the problem and determine the functions in the form $y = a \sin(bx - c) + d$ for each:

1. Analysis of the First Graph:

• Amplitude ($a$): The height from the centerline (0) to the peak is about 2, so $a = 2$.
• Period ($\frac{2\pi}{b}$): The period is the distance between two peaks, which appears to be $\pi$, so $b = 2$.
• Phase shift ($c$): It looks like there is no horizontal shift, so $c = 0$.
• Vertical shift ($d$): The graph is centered around $y = 0$, so $d = 0$.

The function is: $y = 2 \sin(2x)$

2. Analysis of the Second Graph:

• Amplitude ($a$): The amplitude is 3.
• Period ($\frac{2\pi}{b}$): The distance between two peaks is about 6, which gives $b = \frac{\pi}{3}$.
• Phase shift ($c$): The graph starts at 0, indicating $c = 0$.
• Vertical shift ($d$): The graph is centered around $y = 0$.

The function is: $y = 3 \sin\left(\frac{\pi}{3}x\right)$

3. Analysis of the Third Graph:

• This graph is described to have a maximum at $\left(\frac{\pi}{2}, 7\right)$ and a minimum at $\left(\frac{3\pi}{2}, -1\right)$.
• Amplitude ($a$): The difference between the maximum and minimum is $7 - (-1) = 8$, so $a = 4$.
• Period ($\frac{2\pi}{b}$): The distance between $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ is $\pi$, so $b = 2$.
• Phase shift ($c$): The maximum occurs at $\frac{\pi}{2}$, suggesting a phase shift of $\frac{\pi}{2}$.
• Vertical shift ($d$): The midline is at $\frac{7 + (-1)}{2} = 3$.

The function is: $y = 4 \sin(2x - \frac{\pi}{2}) + 3$

Would you like more details or have any questions on these solutions?

Relative Questions:

1. How do you determine the period of a sine function from a graph?
2. What effect does changing the amplitude have on the graph of a sine function?
3. How do phase shifts impact the position of sine and cosine graphs?
4. Can you derive the formula for the period of a sine function from its general form?
5. What are the differences between horizontal and vertical shifts in trigonometric functions?

Tip:

For a sine function $y = a \sin(bx - c) + d$, remember that $a$ affects the amplitude, $b$ affects the period ($\frac{2\pi}{b}$), $c$ affects the horizontal shift, and $d$ adjusts the vertical shift.