To graph $y = 4\sin x$, it's important to understand the key characteristics of the sine function and how it changes with a coefficient.

Characteristics of $y = 4\sin x$:

1. Amplitude: The amplitude of the function $y = 4\sin x$ is 4. This means the graph will oscillate between $-4$ and $4$.

2. Period: The period of $\sin x$ is $2\pi$. This remains unchanged since there is no coefficient affecting the $x$-variable.

3. Frequency: The frequency is also the same as $\sin x$ because the coefficient inside the function (which affects frequency) is 1.

4. Vertical Shift: There is no vertical shift since no constant is added or subtracted from $y$.

5. Horizontal Shift: There is no horizontal shift as there is no constant added or subtracted within the $x$-variable.

Key Points of the Graph:

• At $x = 0$: $y = 4\sin(0) = 0$
• At $x = \frac{\pi}{2}$: $y = 4\sin\left(\frac{\pi}{2}\right) = 4$
• At $x = \pi$: $y = 4\sin(\pi) = 0$
• At $x = \frac{3\pi}{2}$: $y = 4\sin\left(\frac{3\pi}{2}\right) = -4$
• At $x = 2\pi$: $y = 4\sin(2\pi) = 0$

The graph oscillates between 4 and -4 with a period of $2\pi$.

Graph Representation:

If you want to visualize the graph, it will be a standard sine curve but with a stretched amplitude reaching a maximum of 4 and a minimum of -4.

Would you like me to provide a visual representation of this graph?

Related Questions:

1. How does the graph of $y = 4\sin x$ compare to $y = \sin x$?
2. What happens to the graph if we change $y = 4\sin x$ to $y = 4\sin(2x)$?
3. What is the period of $y = 4\sin\left(\frac{1}{2}x\right)$?
4. How would you find the maximum and minimum values of $y = 4\sin x$?
5. How does changing the amplitude affect the shape of a sine wave?

Tip: When sketching trigonometric graphs, always identify key features like amplitude, period, and shifts first for a more accurate plot.