In Exercises 57–58, the parametric equations of four plane curves are given. Graph each plane curve and determine how they differ from each other.x = t and y = t² − 4

Parametric equations express the coordinates of points on a curve as functions of a variable, typically denoted as 't'. In this case, x = t and y = t² - 4 define a relationship where 't' serves as a parameter that generates points (x, y) on the curve. Understanding how to manipulate and graph these equations is essential for visualizing the resulting curve.

Graphing techniques involve plotting points on a coordinate system based on the values derived from the parametric equations. For the given equations, one would calculate y for various values of t, then plot the corresponding (x, y) points. Recognizing the shape and behavior of the graph, such as whether it opens upwards or downwards, is crucial for analyzing the curve's characteristics.

The equation y = t² - 4 represents a quadratic function, which is characterized by its parabolic shape. The term t² indicates that the graph will be a parabola that opens upwards, with its vertex at the point (0, -4). Understanding the properties of quadratic functions, including their vertex, axis of symmetry, and direction of opening, is vital for distinguishing this curve from others.