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find the cartesian equation for this parametric equation

The parametric equations given are:

\[
x = t^3, \quad y = 2t^2
\]

We want to eliminate the parameter \( t \) to find the Cartesian equation relating \( x \) and \( y \).

### Steps to Find the Cartesian Equation:

1. **Solve for \( t \) in terms of \( x \):**
   \[
   x = t^3 \implies t = \sqrt[3]{x}
   \]

2. **Substitute \( t = \sqrt[3]{x} \) into the equation for \( y \):**
   \[
   y = 2t^2 = 2\left(\sqrt[3]{x}\right)^2 = 2x^{\frac{2}{3}}
   \]

Thus, the Cartesian equation is:

\[
y = 2x^{\frac{2}{3}}
\]

Would you like any further explanation on this? Here are some related questions for practice:

1. How do you derive the Cartesian equation from other parametric equations?
2. What does the Cartesian equation \( y = 2x^{2/3} \) tell you about the shape of the curve?
3. How can you graph the equation \( y = 2x^{2/3} \)?
4. What is the domain of the function \( y = 2x^{2/3} \)?
5. How does the parameter \( t \) affect the curve described by the parametric equations?
6. How can you find the tangent line to the curve \( y = 2x^{2/3} \) at a given point?
7. What is the inverse function of the Cartesian equation \( y = 2x^{2/3} \)?
8. How can you determine if a parametric curve has any symmetries?

**Tip:** When converting parametric equations to Cartesian form, carefully isolate the parameter in one equation to substitute into the other. This method helps in simplifying the equations accurately.

Learn how to find the Cartesian equation from parametric equations step-by-step. This guide covers the elimination of parameters method with an example involving parametric equations x = t^3 and y = 2t^2. Suitable for advanced high school students and anyone preparing for calculus or algebra exams.

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