91Ó°ÊÓ

Coordinate geometry, also known as analytic geometry, involves plotting points, lines, and curves on a coordinate plane. It links algebraic equations to geometric shapes. Parametric equations pair algebraic expressions with geometry, giving a set of equations to describe the coordinates of points on a curve. In this exercise, we explore how geometric shapes, such as parabolas, can be expressed with parametric equations in terms of a parameter \( t \).

• The parametric equations \( x = t^2 + 2 \) and \( y = 4t \) represent a set of coordinates on a plane for various \( t \) values.
• By plotting points with selected \( t \) values, as shown in the table, a series of coordinates illustrate how the shape of the curve emerges as the parameter changes.

Creating a table of these points facilitates easier visualization and understanding of the curve's path, helping to link the algebraic form to the geometric shape.

Eliminating parameters involves converting parametric equations into a single equation to describe the relationship between \( x \) and \( y \) directly, without the parameter \( t \). This process can reveal the underlying geometric shape.

In this specific exercise, we started with:

• \( x = t^2 + 2 \)
• \( y = 4t \)

To eliminate \( t \), solve the equation with \( y \) for \( t \):

• \( t = \frac{y}{4} \)

Substitute this expression into the \( x \)-equation:

• \( x = \left(\frac{y}{4}\right)^2 + 2 \)

Simplifying gives the rectangular equation \( x = \frac{y^2}{16} + 2 \), providing a clearer and more direct depiction of the curve's shape independent of \( t \). This reveals the shape is a parabola primarily associated with coordinate geometry.

Curve sketching is the process of drawing a curve by understanding its equation and using key properties like orientation, direction, and curvature. With parametric equations, sketching a curve involves plotting points and recognizing the patterns formed as the parameter changes.

In this exercise, the range of \( t \) from \(-4\) to \(4\) gives a direction to the curve. By plotting these points and indicating the increasing direction of \( t \), the curve's orientation from bottom to top becomes clear. This informs how the shape is traced on the coordinate plane.

When sketching, keep in mind:

• The orientation of the curve. For our problem, the direction from \(-4\) to \(4\) shows a parabola opening to the right.
• The curve's nature. It follows a smooth path as \( t \) moves incrementally.

Understanding these aspects of the curve aids in visualizing and sketching the path accurately, further deepening comprehension of the geometric structure of parametric equations.