To identify the particle's path and find the corresponding Cartesian equation, we start with the given parametric equations:

$x = 9t^2$ $y = 3t$

Step 1: Express $t$ in terms of $y$

From the equation $y = 3t$, we can solve for $t$:

$t = \frac{y}{3}$

Step 2: Substitute $t$ into the equation for $x$

Next, substitute $t = \frac{y}{3}$ into the equation $x = 9t^2$:

$x = 9\left(\frac{y}{3}\right)^2$

Simplify the expression:

$x = 9\left(\frac{y^2}{9}\right)$

$x = y^2$

Step 3: Identify the Cartesian Equation

The Cartesian equation of the particle's path is:

$x = y^2$

Step 4: Determine the Portion of the Graph Traced and the Direction of Motion

The parametric equations describe the path of a particle along the parabola $x = y^2$. Since $t$ ranges from $-\infty$ to $\infty$, and given that $x = 9t^2$ (which is always non-negative), the particle moves along the parabola in the positive $x$-direction.

• For $t > 0$: $y = 3t$ is positive, so the particle moves upward along the right branch of the parabola.
• For $t < 0$: $y = 3t$ is negative, so the particle moves downward along the left branch of the parabola.

Thus, the graph $x = y^2$ is a parabola opening to the right, and the particle moves from the lower left (downward) branch to the upper right (upward) branch as $t$ increases from negative infinity to positive infinity.

Step 5: Graph the Cartesian Equation

To graph $x = y^2$:

• The parabola opens to the right with its vertex at the origin $(0, 0)$.
• The portion traced by the particle includes the entire parabola since $t$ ranges over all real numbers.
• The direction of motion starts from the left branch of the parabola (as $t$ is negative) and moves towards the right branch as $t$ becomes positive.

Would you like to see a sketch of the graph, more details, or have any other questions? Here are some follow-up questions:

1. How would the graph change if $t$ was limited to a specific range?
2. What happens to the graph if the sign of $x$ is changed in the parametric equations?
3. How do you determine the direction of motion for other parametric curves?
4. Can you describe the motion of a particle for parametric equations with a trigonometric function?
5. What is the significance of parametric equations in modeling real-world motion?

Tip: When converting parametric equations to a Cartesian equation, always solve for one parameter in terms of the other to eliminate the parameter effectively.