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Part 1

Parametric equations and a parameter interval for the motion of a particle in the​ xy-plane are given. Identify the​ particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.

xequals=99t squaredt2​, 

yequals=33​t, 

minus−infinity∞less than<tless than<infinity∞

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Part 1




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.

Learn how to identify a particle's path using parametric equations, derive its Cartesian equation, and graph the result. This problem covers fundamental concepts in parametric and Cartesian equations, suitable for advanced high school students.

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