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Eliminating the parameter from parametric equations is a method to transition from a set of equations that define a curve in terms of a third variable, often labeled as \( t \), to a single equation in two variables, typically \( x \) and \( y \). For the given parametric equations \( X = \sqrt{t} \) and \( y = 1 - t \), eliminating the parameter involves solving one of these equations for \( t \) and substituting into the other equation.
To eliminate \( t \), first express \( t \) from \( X = \sqrt{t} \):

• Solve for \( t \) to get \( t = X^2 \).

Next, substitute \( X^2 \) for \( t \) in the second equation:

• The equation becomes \( y = 1 - X^2 \).

This conversion yields a singular relationship between \( x \) and \( y \), allowing the curve to be described without the parameter \( t \).

A rectangular coordinate equation, sometimes referred to as a Cartesian equation, is a traditional representation of curves in the coordinate plane, using two variables, usually \( x \) and \( y \). The goal is to write an equation that describes the same curve as represented by the given parametric equations without needing the third variable \( t \).
For the parametric equations \( X = \sqrt{t} \) and \( y = 1 - t \), the rectangular coordinate equation becomes \( y = 1 - X^2 \) after the elimination of the parameter was executed.

• This equation still describes the same curve but in a more unified form.
• It represents a clearer view of the relationship between \( x \) and \( y \), facilitating easier visualization and analysis of the curve.

In practical terms, this conversion is useful in analysis or sketching where a single equation format offers convenience.

Sketching curves from parametric equations is a foundational skill in understanding how different parameters influence a curve's shape and placement in the coordinate plane.
To sketch the curve:

• Calculate several points from the parametric equations by substituting values for \( t \). For instance, when \( t = 0, 1, \) and \( 4 \), you get points \((0, 1)\), \((1, 0)\), \((2, -3)\).
• Plot these points on a coordinate plane.
• Draw a smooth curve through the plotted points to illustrate the curve's path.

This graphical approach offers insights into the curve's behavior, allowing predictions on direction and shape based on underlying mathematical principles. Curve sketching helps in visualizing concepts that might otherwise be abstract.

A parabola is a specific type of curve that appears frequently in mathematics and physics, characterized by its "U" or inverted "U" shape. The general equation for a parabola found in the rectangular coordinate system is given by:

• Standard form: \( y = ax^2 + bx + c \).
• The parabola derived from the problem, \( y = 1 - X^2 \), is in the form \( y = -X^2 + 1 \).

This specific equation describes a parabola opening downward, with its vertex at the point \((0, 1)\). The vertex is where the maximum point of the parabola lies, which is critical information when analyzing the shape and orientation of the parabola.

• Understanding the vertex gives insight into the curve's "peak" or "trough" behavior.
• The negative sign before \( X^2 \) indicates the downward direction of the parabola.

These features provide a clear framework for predicting the curve's trajectory in the coordinate plane.