Let's address each part of the problem one by one.

Given Parametrization:

• $x = 2t$
• $y = 3t - 5$
• $0 \leq t \leq 1$

Part (a): Graph the curve

Initial and Terminal Points:

1. Initial Point: This occurs at $t = 0$.

   • When $t = 0$, we get: $x = 2(0) = 0, \quad y = 3(0) - 5 = -5$ So, the initial point is $(0, -5)$.

2. Terminal Point: This occurs at $t = 1$.

   • When $t = 1$, we get: $x = 2(1) = 2, \quad y = 3(1) - 5 = -2$ So, the terminal point is $(2, -2)$.

Direction of the Curve:

The direction of the curve is traced from the initial point $(0, -5)$ to the terminal point $(2, -2)$ as $t$ increases from 0 to 1.

The curve is a straight line because both $x$ and $y$ are linear in $t$.

Part (b): Find a Cartesian Equation

To eliminate $t$ and find the Cartesian equation, solve for $t$ in terms of $x$ from the equation $x = 2t$: $t = \frac{x}{2}$ Now substitute this expression for $t$ into the equation for $y$: $y = 3t - 5 = 3\left(\frac{x}{2}\right) - 5$ Simplifying: $y = \frac{3x}{2} - 5$ Thus, the Cartesian equation is: $y = \frac{3x}{2} - 5$

Portion of the Graph Traced:

The parameter $t$ is restricted to $0 \leq t \leq 1$, which corresponds to the $x$-values between 0 and 2 (since $x = 2t$).

• For $t = 0$, we get $x = 0$.
• For $t = 1$, we get $x = 2$.

Therefore, the portion of the graph of the line $y = \frac{3x}{2} - 5$ that is traced by the parametrized curve is from $(0, -5)$ to $(2, -2)$.

Conclusion:

• The initial point is $(0, -5)$.
• The terminal point is $(2, -2)$.
• The direction of the curve is from the initial to the terminal point.
• The Cartesian equation of the curve is $y = \frac{3x}{2} - 5$.
• The portion of the graph traced is the segment between $x = 0$ and $x = 2$.

Would you like a graph of this curve or further explanation on any part of the solution?

5 Related Questions:

1. How do you parametrize curves in 3D space?
2. What are the differences between parametric and Cartesian equations?
3. How can you determine the length of a parametric curve?
4. How do you compute the slope of a curve from its parametric equations?
5. How can you reparametrize a curve to trace it in reverse?

Tip:

When converting from parametric to Cartesian form, always check if the relationship is linear or non-linear to predict the type of curve (line, parabola, etc.).