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A Cartesian equation is an equation that describes a curve or a surface in terms of coordinates such as x and y in a two-dimensional space. It eliminates the parameter, allowing for the expression of a relationship within the familiar Cartesian plane where each point has a specific location on the X and Y axes.
When dealing with parametric equations, the process of transforming them into a Cartesian equation involves removing the parameter, usually denoted as \( t \), by finding an expression for \( t \) in terms of one variable and substituting it into the equation of the second variable.
For example, given \( x = t^2 - 1 \) and \( y = t + 1 \), you would express \( t \) in terms of \( y \) (as \( t = y - 1 \)) and substitute it back into the equation \( x = t^2 - 1 \) to eliminate \( t \).

• The result is a new equation, \( x = y^2 - 2y \), which is now dependent only on \( x \) and \( y \), common for use in Cartesian coordinates.
• This approach converts the parametric form into a single equation that defines the curve explicitly in the Cartesian plane.

In mathematics, a parabola is a U-shaped curve that can open in any direction. In the context of Cartesian equations, it is often expressed in the form \( y = ax^2 + bx + c \) if the parabola opens upward or downward. Or correspondingly, \( x = ay^2 + by + c \) if it opens to the left or right.
The key characteristic of a parabola is its symmetry and its single peak or vertex. In this exercise, we have the equation \( x = y^2 - 2y \), which describes a parabola that opens in the horizontal direction.

• The vertex of a parabola described by \( x = y^2 - 2y \) can be calculated by completing the square or using calculus to find the turning point.
• The direction in which the parabola opens depends on the coefficient of \( y^2 \). If positive, it opens to the right, and if negative, to the left.
• In this scenario, the parabola opens to the right as \( y \) increases, allowing \( x \) to change positively.

Eliminating parameters is a technique used to convert parametric equations into Cartesian equations. This process simplifies the representation of curves by removing parameters such as \( t \). It is a helpful skill that allows for exploring the properties of a curve without reference to the parameter itself.
Here is how parameters are eliminated:

• Express the parameter in terms of one variable: This step involves rewriting the equation for one variable to isolate and solve for \( t \).
• Substitute back into the other equation: Use the expression derived from the first equation to replace \( t \) in the second equation, effectively removing the parameter.
• Result in a single equation: The refined equation, which no longer includes \( t \), provides a direct relationship between the variables \( x \) and \( y \).

In our example, with \( x = t^2 - 1 \) and \( y = t + 1 \), solving for \( t \) from \( y = t + 1 \) gives \( t = y - 1 \). Substituting this into \( x = t^2 - 1 \) yields the Cartesian equation \( x = y^2 - 2y \). This process is essential for simplifying and analyzing complex curves in easy-to-understand forms.