To draw the graph of $y = \sin(x)$ for $0 \leq x \leq \pi$, here's a brief outline of how the function behaves:

1. Amplitude: The sine function oscillates between -1 and 1.
2. Periodicity: The sine function has a period of $2\pi$, but since we are only concerned with the interval $0 \leq x \leq \pi$, we will see the first half of one period, going from 0 to $\pi$.
3. Key Points:
   • $\sin(0) = 0$
   • $\sin\left(\frac{\pi}{2}\right) = 1$ (This is the peak.)
   • $\sin(\pi) = 0$ (At this point, the function returns to 0.)

Graph Details:

• The graph starts at the origin (0,0).
• It rises to a maximum at $x = \frac{\pi}{2}$ where $y = 1$.
• Then, it decreases back to 0 at $x = \pi$.

I'll generate the graph for you now. numpy as np import matplotlib.pyplot as plt

Define the x values (ranging from 0 to pi)

x = np.linspace(0, np.pi, 1000) y = np.sin(x)

Create the plot

plt.figure(figsize=(8, 4)) plt.plot(x, y, label=r'$y = \sin(x)$', color='blue')

Mark key points

plt.scatter([0, np.pi/2, np.pi], [0, 1, 0], color='red') # Key points: (0,0), (pi/2, 1), (pi, 0) plt.text(0, 0, '(0, 0)', fontsize=12, verticalalignment='bottom', horizontalalignment='right') plt.text(np.pi/2, 1, r'$(\frac{\pi}{2}, 1)$', fontsize=12, verticalalignment='bottom', horizontalalignment='right') plt.text(np.pi, 0, '(\u03C0, 0)', fontsize=12, verticalalignment='top', horizontalalignment='left')

Set labels and title

plt.title(r'Graph of $y = \sin(x)$ for $0 \leq x \leq \pi$', fontsize=14) plt.xlabel(r'$x$', fontsize=12) plt.ylabel(r'$y$', fontsize=12)

Set grid and axis limits

plt.grid(True) plt.xlim(0, np.pi) plt.ylim(-1.2, 1.2)

Show the plot

plt.legend() plt.show()Here is the graph of $y = \sin(x)$ for $0 \leq x \leq \pi$. As you can see:

• The graph starts at $(0, 0)$.
• It reaches its maximum at $\left(\frac{\pi}{2}, 1\right)$.
• It returns to $( \pi, 0)$.

Would you like further details on the graph, or do you have any questions?

Related Questions:

1. How does the sine function behave over other intervals like $\left[0, 2\pi\right]$?
2. What are the differences between sine, cosine, and tangent graphs?
3. How do phase shifts affect the graph of $y = \sin(x)$?
4. How can we derive the maximum and minimum values of the sine function?
5. What is the relationship between the unit circle and the sine function?

Tip:

To understand the behavior of trigonometric functions better, visualizing them on the unit circle is a powerful method!