https://file.aiquickdraw.com/mathbot/file/1726374923972wj54c8iy.jpg

Let's analyze the given parametric equations to determine the behavior of the curve and which diagram best represents it.

The curve is defined by the parametric equations:
- \( x = 2 + t \)
- \( y = 3 - 2t^2 \)
for \( -1 \leq t \leq 0 \).

### Step 1: Endpoints of the curve
Let's find the points on the curve at the boundary values of \( t \).

- When \( t = -1 \):
  - \( x = 2 + (-1) = 1 \)
  - \( y = 3 - 2(-1)^2 = 3 - 2 = 1 \)
  - So, one endpoint is \( (1, 1) \).

- When \( t = 0 \):
  - \( x = 2 + 0 = 2 \)
  - \( y = 3 - 2(0)^2 = 3 \)
  - So, the other endpoint is \( (2, 3) \).

### Step 2: Shape of the curve
We need to analyze how the curve behaves between \( t = -1 \) and \( t = 0 \).

- As \( t \) increases from \( -1 \) to \( 0 \), \( x = 2 + t \) increases from \( x = 1 \) to \( x = 2 \), meaning \( x \) is increasing.
- For \( y = 3 - 2t^2 \), since \( t^2 \) is always positive and increases as \( |t| \) increases, \( y \) is decreasing as \( t \) moves from \( t = 0 \) to \( t = -1 \).

This means:
- The curve moves from \( (1, 1) \) to \( (2, 3) \), with \( x \) increasing and \( y \) also increasing as \( t \) approaches 0.

### Step 3: Determine the direction
From this analysis, the curve should rise from the point \( (1, 1) \) to \( (2, 3) \) as \( t \) increases from \( -1 \) to \( 0 \), indicating that the curve is moving upwards and to the right.

### Conclusion:
The correct diagram should show the curve moving from \( (1, 1) \) to \( (2, 3) \) in an upward direction.

The best match is **diagram A**.

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Would you like more details or have any further questions?

Here are 5 related questions:
1. How do parametric equations differ from regular Cartesian equations?
2. What does the derivative tell us about the slope of a parametric curve?
3. How would the shape of the curve change if the bounds on \( t \) were different?
4. Can parametric curves be used to model real-world motion?
5. How can we reparameterize a curve to simplify its form?

**Tip:** When working with parametric curves, it's helpful to compute a few points for different parameter values to visualize the curve better.

A curve is defined in parametric form by x = 2 + t and y = 3 - 2t^2 for -1 ≤ t ≤ 0. Which diagram best represents this curve?

Explore how to graph a curve defined by the parametric equations x = 2 + t and y = 3 - 2t^2 for the interval -1 ≤ t ≤ 0. Learn how parametric equations behave and which diagram best represents this curve. Suitable for students in Grades 10-12.

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