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Graphing curves from parametric equations involves creating a visual representation of the equations for given parameter ranges. A parametric equation describes both the x and y coordinates in terms of a third variable, commonly denoted as \( r \). This allows you to see the curve as \( r \) varies.
To graph the curve from the parametric equations \( x = 2\cos^2 r \) and \( y = 3\sin^2 r \), follow these steps:

• Identify the parameter range, here it is \( 0 \leq r \leq 2\pi \).
• Incrementally choose values of \( r \) within this range, like \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \).
• Calculate the corresponding \( x \) and \( y \) for each \( r \).
• Plot these points on the Cartesian coordinate plane to visualize the curve.

By selecting several points, you can connect these points to form the curve, often resembling an ellipse in our given example.

A curve is considered closed if it forms a complete loop, meaning the curve ends where it begins after covering its path. In the context of parametric equations, we check if as \( r \) traverses its entire range, the ending and starting points coincide.
For our parametric representation \( x = 2\cos^2r \) and \( y = 3\sin^2r \), as \( r \) moves from \( 0 \) to \( 2\pi \), the path traversed returns to the starting point. Hence, this curve is closed.
A simple curve is one that does not intersect itself at any point other than potentially at its starting/ending point. In our example, each \( r \) corresponds to a unique point. This uniqueness implies no self-intersections, categorizing it as a simple curve.

To derive the Cartesian equation from parametric ones, you aim to remove the parameter \( r \) from the equations. By doing this, you establish a direct relationship between \( x \) and \( y \).
In this example, we use the identity \( \sin^2 r + \cos^2 r = 1 \). From the parametric equations given:

• Express \( \cos^2 r \) from \( x = 2\cos^2 r \) as \( \cos^2 r = \frac{x}{2} \).
• Substitute \( \cos^2 r \) into \( \sin^2 r = 1 - \cos^2 r \), giving \( \sin^2 r = 1 - \frac{x}{2} \).
• This results in \( y = 3(1 - \frac{x}{2}) = 3 - \frac{3}{2}x \).

Thus, the Cartesian equation \( y = 3 - \frac{3}{2}x \) accurately represents the curve without referring back to the parameter \( r \).

Eliminating parameters transforms a parametric set of equations into a single equation in terms of \( x \) and \( y \). This is often useful for easier manipulations, especially when comparing with known equations.
To eliminate the parameter \( r \), we utilized trigonometric identities. Here's how it works:

• Start with the substitution \( \cos^2 r = \frac{x}{2} \) from the equation \( x = 2\cos^2 r \).
• Apply this to the identity \( \sin^2 r = 1 - \cos^2 r \), leading to \( \sin^2 r = 1 - \frac{x}{2} \).
• Add \( y = 3\sin^2 r \), substituting \( \sin^2 r \) yields \( y = 3 - \frac{3x}{2} \).

By removing \( r \), the resulting Cartesian equation \( y = 3 - \frac{3x}{2} \) is derived, showcasing the relationship directly between \( x \) and \( y \). This process solidifies the connection of trigonometric identities in simplifying parametric equations.