The problem asks to determine the period of the function $y = 4 \sin(2x)$ and find the five key points on the x-axis.

Step 1: Find the period

For a sine function of the form $y = A \sin(Bx)$, the period is given by the formula: $\text{Period} = \frac{2\pi}{|B|}$ Here, $B = 2$, so the period is: $\text{Period} = \frac{2\pi}{2} = \pi$

Step 2: Determine the five key points

The key points for a sine function are typically at:

• $x_1$: Start of the period
• $x_2$: First quarter of the period (maximum point)
• $x_3$: Half of the period (zero point)
• $x_4$: Three-quarters of the period (minimum point)
• $x_5$: Full period

Since the period is $\pi$, these points occur as follows: $x_1 = 0, \quad x_2 = \frac{\pi}{4}, \quad x_3 = \frac{\pi}{2}, \quad x_4 = \frac{3\pi}{4}, \quad x_5 = \pi$

Final Answer:

• The period of $y = 4 \sin(2x)$ is $\pi$.
• The five key x-values are:
  • $x_1 = 0$
  • $x_2 = \frac{\pi}{4}$
  • $x_3 = \frac{\pi}{2}$
  • $x_4 = \frac{3\pi}{4}$
  • $x_5 = \pi$

Let me know if you need further details or have questions! Here are some related questions to deepen your understanding:

1. What effect does the amplitude $A = 4$ have on the graph of $y = 4 \sin(2x)$?
2. How does the value of $B$ in $y = A \sin(Bx)$ affect the graph's frequency?
3. What are the corresponding y-values at each of the key points for the function $y = 4 \sin(2x)$?
4. How would the key points change if the function was $y = \sin(2x)$ without the amplitude?
5. How can you calculate the key points if the sine function had a phase shift?

Tip:

Always remember that the period of $\sin(Bx)$ or $\cos(Bx)$ is calculated by $\frac{2\pi}{|B|}$, while vertical stretching or shrinking is controlled by the amplitude $A$.