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

2. 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.

3. 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.

4. Parametric Curve, Parametric Equations

5. Relations

6. Example Relations

7. 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.

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

9. 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.

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

11. Example Circles

12. Ellipses

13. 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.

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

15. 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.

16. Example Lines and Other Curves

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

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