Prerequisites for Calculus AP CALCULUS AB Chapter 1: Prerequisites for Calculus Section 1.4: Parametric Equations

What you’ll learn about… Relations Circles Ellipses Lines and Other Curves …and why Parametric equations can be used to obtain graphs of relations and functions.

Relations A relation is a set of ordered pairs (x, y) of real numbers. The graph of a relation is the set of points in a plane that correspond to the ordered pairs of the relation. If x and y are functions of a third variable t, called a parameter, then we can use the parametric mode of a grapher to obtain a graph of the relation.

Parametric Curve, Parametric Equations

Relations

Example Relations

Section 1.4 – Parametric Equations To convert from a set of parametric equations to the Cartesian equation of the curve, solve either equation for t and plug into the other equation.

Section 1.4 – Parametric Equations You try: Describe the graph of the relation determined by

Section 1.4 – Parametric Equations In applications, t often denotes time, an angle or the distance a particle has traveled along its path from a starting point. Parametric graphing can be used to simulate the motion of a particle.

Section 1.4 – Parametric Equations Circles can be parameterized as where a is the radius Ellipses can be parameterized as

Example Circles

Ellipses

Section 1.4 – Parametric Equations You try: Graph the parametric curve Find the Cartesian equation for a curve that presents the same parametric curve. What portion of the graph of the Cartesian equation is traced by the parametric curve? Indicate the direction in which the curve is traced and the initial and terminal points, if any.

Lines and Other Curves Lines, line segments and many other curves can be defined parametrically.

Section 1.4 – Parametric Equations To parameterize a line segment with endpoints let This line goes through the point when t= 0. To find a and b, make the line reach when t=1.

Example Lines and Other Curves

Section 1.4 – Parametric Equations You try: Draw and identify the graph of the parametric curve determined by

Section 1.4 – Parametric Equations You try: Find a parameterization for the line segment with endpoints (2, -3) and (1, 4).