Definition: Parametric equations describe a curve by giving x and y each as a function of a third variable, the parameter, often thought of as time.
What are parametric equations?
Instead of one equation relating x and y, you write x = f(t) and y = g(t). As t increases, the point (x, y) traces the curve and also shows direction and speed of travel.
How do you eliminate the parameter?
Solve one equation for t and substitute into the other.
Worked example: x = 2t + 1 and y = t2 – 4. From the first, t = (x – 1)/2. Then y = (x – 1)2/4 – 4, a parabola.
At t = 3, the point is (7, 5). Check: (7 – 1)2/4 – 4 = 9 – 4 = 5.
A thrown ball is a classic use: the horizontal position grows steadily with time while the height follows a quadratic in time.
What is a common mistake?
Forgetting the limits on t. If t only runs from 0 to 3, you get just part of the parabola, not all of it. Mathbot keeps the t-interval in the answer.