A parametric curve gives x and y each as a function of a third variable, usually time.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What this buys you
A curve can now double back or cross itself, which a function y = f(x) can never do.
Eliminating the parameter
Solve one equation for t and substitute into the other. The direction of travel is lost when you do.
The slope
dy/dx = (dy/dt) / (dx/dt). It is a ratio of two rates, not a single derivative.
Two functions of a third variable
x and y are each given as functions of a parameter t. As t runs, the point (x(t), y(t)) traces a curve. The parameter usually represents time, so the form describes motion, not merely shape.
It describes curves functions cannot
A circle fails the vertical line test and has no form y = f(x). Parametrically it is x = cos t, y = sin t. Parametric equations escape the one-output-per-input restriction entirely.
Direction and interval are part of the curve
Two parametrisations can trace the same path in opposite directions, at different speeds, or over different ranges of t. Sketches need arrows, and answers need the interval stated.
Eliminating t loses information
Solving for t and substituting gives a Cartesian equation identifying the shape while discarding the timing and direction. The conversion is genuinely lossy and should be noted as such.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Omar Eliminate the Parameter
Omar converts x = t + 1, y = t² to Cartesian form.
- Step 1
He solves the first for t: t = x − 1.
Step 4: Your Turn
Practice makes it stick.
At a Moment
Problem 1 of 2
x = 2t, y = t². What is x when t = 3?
The Slope
Problem 2 of 2
dx/dt = 2 and dy/dt = 6. What is dy/dx?
Parametrically
1 of 8
x = 3t, y = t. What is x when t = 4?
2 of 8
dx/dt = 4, dy/dt = 12. What is dy/dx?
3 of 8
dx/dt = 5, dy/dt = 0. What is dy/dx?
4 of 8
x = t, y = t². Eliminating t gives y = x to what power?
5 of 8
Select every true statement about parametric curves.
6 of 8
x = t + 2, y = 3t. What is y when x = 5?
7 of 8
dx/dt = 0 at a point. Is the tangent vertical there? 1 for yes, 0 for no.
8 of 8
Order the steps for eliminating the parameter.
- 1Substitute into the other
- 2Simplify to a relation in x and y
- 3Solve one equation for t
Step 5: Quick Check
Show what you know.
Question 1 of 1
dx/dt = 3, dy/dt = 15. What is dy/dx?
What You Learned
- A parametric curve gives x and y each as a function of a parameter.
- Eliminating the parameter recovers a Cartesian relation but loses direction.
- dy/dx is the ratio (dy/dt) / (dx/dt).