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### Course: AP®︎/College Calculus BC>Unit 9

Lesson 1: Defining and differentiating parametric equations

# Parametric equations intro

In this video, we learn about parametric equations using the example of a car driving off a cliff. Parametric equations define x and y as functions of a third parameter, t (time). They help us find the path, direction, and position of an object at any given time. Created by Sal Khan.

## Want to join the conversation?

• Other than a moving object in space, what are some real-life applications for parametric equations?
• what is a parameter
• Basically a defined limit that you operate within.
• What is a parameter? What makes it useful?
• A parameter is some constant that relates two or more functions. In the example, the x-position and the y-position are not related to each other directly, but they are both defined in terms of time. Time is the parameter that allows us to see what the x and y functions are doing together. Try graphing the x and y functions separately. (In a calculator, you may have to call them y1 and y2, and change the t's to x's.) Can you look at them and know intuitively what the car is doing? Parametric equations allow us to break up a complicated problem, like motion in two or more dimensions, into simpler problems that can be solved separately and then recombined (if we want) through their shared parameter.
• when will there be exercises for parametric equations? are you considering making them at all? It'd be really nice, thanks!
• In "normal" functions (for the lack of a better word), like f(x)=x+2 for example, is "x" a parameter?
• Sort of. It's tempting to say so, but parameter has a special meaning in this context. Each of the functions in the example are 'normal,' separate functions. What makes them parametric is that they share a parameter. Parametric equations are used when x and y are not directly related to each other, but are both related through a third term. In the example, the car's position in the x-direction is changing linearly with time, i.e. the graph of its function is a straight line. In the y-direction, however, its position is changing exponentially with time. The unifying 'parameter' is time. The car is moving through time equally "in both directions." This allows us to graph (x, y) coordinates to show the position of the car, as Sal showed. This is much more useful and intuitive than looking at the graphs of y(t) and x(t) separately. You can also use the parameter to find a unifying function that does directly relate x to y, as Sal hinted at.
Wikipedia has a pretty good blurb about the math uses of "parameter." http://en.wikipedia.org/wiki/Parameter#Mathematical_models
• There is a topic on the "Precalculus" mission called "Parametric equations and polar coordinates" but it has no skill excercises. Did they took them awary?? Or are there non?? Anyone knows?? Thank you very much.
• I'm also looking for this badge, but I think there is no possibility to get it at the moment.
(1 vote)
• I'm pretty tired so I may be looking too far into this... Is weight a factor? For example we figured out if this car is falling off a cliff at 5 m/sec sqrd it will land in the area sal figured out. Let's say that we had a tank or a marble traveling at the same speed with the same gravity acting on it, and our object is falling from the same height. I can't imagine the results will be the same.
• The actual, real-world results are far more complicated because of the friction between the falling object and air as well as some minor issues with buoyancy. But, those kinds of computations are very advanced physics and engineering questions. They are just too complicated for this level of study.

I would anticipate, given its shape, that a marble would actually fall more quickly than a car. This would be because the car has an irregular shape and has lots of friction from the air slowing down its acceleration. A marble has a smooth, spherical shape, so it would have considerably less air friction.
• Did anyone else see that on the y-coordinate, as t went up, the amount that y decreased every time t went up increased by ten? (When t went from zero to 1, the difference in y was 5, when t went from 1 to 2, the difference in y was 15, when t went from two to three the difference in y was 25, and if there was no ground on the x-axis from t=3 to t=4 y would have decreased by 35.) Anyone know exactly why this happens the way it does?
Thanks!
• Yes, this is typical of quadratic functions. If you look at the successive differences between values, those differences form a linear pattern. This fact is sometimes used to fit a polynomial function to data, known as the "method of successive differences". In the quadratic case, you find that the "second difference" (the successive difference between the differences) is twice the quadratic coefficient. You observed that the second difference is -10, thus the quadratic coefficient is -5.