In Exercises 53–56, find two different sets of parametric equations for each rectangular equation. y = x² + 4

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, typically denoted as 't'. Instead of defining y directly in terms of x, both x and y are defined in terms of 't', allowing for a more flexible representation of curves. For example, for the equation y = x² + 4, one could set x = t and y = t² + 4, or use a different parameterization such as x = 2t and y = (2t)² + 4.

Rectangular Equations

A rectangular equation relates the x and y coordinates of points in a Cartesian plane without involving a third variable. In this case, the equation y = x² + 4 describes a parabola that opens upwards, with its vertex at the point (0, 4). Understanding how to convert this rectangular form into parametric equations is essential for exploring the curve's properties and behavior.

Graphing Parabolas

Graphing parabolas involves understanding their shape, direction, and key features such as the vertex and axis of symmetry. The equation y = x² + 4 indicates that the parabola opens upwards, with the vertex located at (0, 4). Recognizing these characteristics helps in creating accurate parametric equations that reflect the same geometric properties as the original rectangular equation.