https://file.aiquickdraw.com/mathbot/file/173069899206772xssz37.jpg

Are you sure you look at the grahs and choose the right one?

To identify which diagram correctly represents the parametric curve given by:

\[
x = -\sin t, \quad y = \cos t, \quad 0 \leq t \leq \pi,
\]

we can analyze the behavior of \(x\) and \(y\) over the interval \(0 \leq t \leq \pi\) to understand the path of the curve.

1. **When \(t = 0\):**
   - \( x = -\sin(0) = 0 \)
   - \( y = \cos(0) = 1 \)
   - The curve starts at the point \((0, 1)\).

2. **When \(t = \pi\):**
   - \( x = -\sin(\pi) = 0 \)
   - \( y = \cos(\pi) = -1 \)
   - The curve ends at the point \((0, -1)\).

3. **Intermediate values of \(t\):**
   - For \(t\) between \(0\) and \(\pi\), \(\sin(t)\) varies from \(0\) to \(1\) (positive values) and then back to \(0\). Therefore, \(-\sin(t)\) will range from \(0\) to \(-1\) and back to \(0\).
   - \( \cos(t) \) varies from \(1\) to \(-1\).

From this information, we can deduce that the curve traces a semicircle, moving from \((0, 1)\) to \((0, -1)\), with \(x\) values remaining negative or zero. This matches a left-facing semicircle centered at the origin.

### Diagram Choice:
- The correct option is **(B)**, which shows a left-facing semicircle from \((0, 1)\) to \((0, -1)\).

Let me know if you'd like more details on any step.

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**Additional Questions:**
1. How can you determine the direction of a parametric curve?
2. Why does the value of \(t\) only go up to \(\pi\) here?
3. What would happen if the range of \(t\) extended beyond \(\pi\)?
4. Can parametric curves represent other shapes besides circles?
5. How would the graph change if \(x = \sin t\) instead of \(x = -\sin t\)?

**Tip:** When analyzing parametric curves, checking endpoint values and the general shape by looking at intermediate values can help quickly identify the curve’s form.

Which of the following diagrams represents the parametric curve x = -sin t, y = cos t, 0 ≤ t ≤ π?

Learn how to identify the correct diagram for the parametric curve defined by x = -sin(t) and y = cos(t) over the interval 0 ≤ t ≤ π. This guide covers parametric equations, trigonometric functions, and curve sketching concepts suitable for students in Grades 10-12.

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