In this lesson, our instructor Vincent Selhorst-Jones teaches about Parametric Equations. Youll review the plane curve and the key ideas, and then jump right into graphing. Youll learn how you can have the same graph but different equations. Vincent also shows you a metaphor for parametric equations to aid in understanding, Youll go over eliminating parameter, creating parametric graphs, and explore some very interesting ones. Before the five practice questions, Vincent reviews some graphing calculator information.
Parametric equations are a new way to look at graphing. Instead of graphing input versus output, we'll base both x and y on a third, new variable: a parameter.
Using this new idea for graphing, we can describe a set of points in the plane (a graph) as a plane curve. A plane curve is created by two functions f(t) and g(t) defined on some interval of the real numbers. The curve is the set of points (x,y)
= ( f(t), g(t) ). The equations
x = f(t) and y=g(t)
are called parametric equations and t is the parameter.
Graphing with parametric equations is very similar to "normal" graphing. You plug in a value, then see what point you get. Just instead of plugging in x to get y, you plug in t to get x and y.
If we want to show the direction of motion in the plane curve, we can draw arrows along the curve to show which way it moves.
Sometimes it's useful to turn a pair of parametric equations into an old fashioned rectangular equation (one using just x and y). To do that, we must eliminate the parameter t from the equations. We do this by solving for t in one equation, then
plugging it into the other. POOF! No more parameter. [Caution: Be careful when eliminating parameters. We will sometimes need to alter domains to keep the same graph. Furthermore, it is not always possible to solve for t directly, so occasionally
we'll have to be clever.]
If you have access to a graphing calculator, it's great to try graphing some parametric equations with it. It's a new way of looking at graphing, so it helps just to play around. For more information, check out the appendix on graphing calculators.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.