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.

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.