Let's analyze the question and solve each part step-by-step based on the information in the image:

Part (a): Find the amplitude, period, and horizontal shift.

1. Amplitude:

   • The amplitude is the maximum absolute value of $y$ from the center line (horizontal axis). In this graph, the maximum value of $y$ is $8$ and the minimum is $-8$.
   • Therefore, the amplitude is $8$.

2. Period:

   • The period is the distance along the $x$-axis that it takes for the wave to complete one full cycle.
   • In the graph, the sine curve completes one cycle from $x = 0$ to $x = 2\pi$.
   • Thus, the period is $2\pi$.

3. Horizontal Shift:

   • The horizontal shift refers to any left or right displacement of the wave.
   • The graph appears to start at $x = 0$ without any horizontal shift.
   • Therefore, the horizontal shift is $0$.

Part (b): Write an equation that represents the curve in the form $y = a \sin(k(x - b))$.

• Given values:
  • Amplitude ($a$) is $8$.
  • Period ($2\pi / k$) is $2\pi$, which implies $k = 1$.
  • Horizontal Shift ($b$) is $0$.

Thus, the equation of the sine wave is: $y = 8 \sin(x)$

Summary:

• Amplitude: $8$
• Period: $2\pi$
• Horizontal Shift: $0$
• Equation: $y = 8 \sin(x)$

Would you like more details on any specific part of this solution?

Related Questions:

1. How would the equation change if the amplitude was halved?
2. What would happen to the period if $k$ was doubled?
3. How would a horizontal shift of $\pi/2$ affect the equation?
4. How can you rewrite this equation as a cosine function?
5. How would the graph look if the amplitude was negative?

Tip:

When identifying the period of a sine or cosine function, remember that it is given by $\frac{2\pi}{|k|}$ where $k$ is the coefficient of $x$ in the function $\sin(kx)$ or $\cos(kx)$.