Parametric equations (graphing, modeling)
Parametric Equations
Parametric Equations is the opening topic of Parametric & Polar in the California Common Core State Standards. It is aligned to Standard F-IF, which requires students to represent motion and paths with parametric equations.
Parametric equations give \(x\) and \(y\) each as a function of a parameter \(t\), tracing a path with a direction — ideal for modeling motion.
Theory
As \(t\) varies, the point \((x,y)\) traces a path — and, unlike an ordinary graph, the path has a direction and a timing. The parameter often represents time, making this ideal for motion.
The parametric form:
How to work with parametric equations
- Make a table of \(t\), \(x\), and \(y\).
- Plot the \((x,y)\) points in order of increasing \(t\).
- Mark the direction the curve is traced.
- Interpret \(t\) (often time) in context.
Substitute each \(t\).
| \(t=0\) | \(\to\) | \((0,0)\) |
| \(t=1\) | \(\to\) | \((1,1)\) |
| \(t=2\) | \(\to\) | \((2,4)\) |
Since \(\cos^2 t+\sin^2 t=1\), the point stays on the unit circle.
| \(x^2+y^2\) | \(=\) | \(1\) |
It traces the unit circle counterclockwise.
As \(t\) grows, \(x=t\) increases, so the curve is traced left to right.
| \(t\uparrow\) | \(\Rightarrow\) | \(x\uparrow\) |
Substitute \(t=1\).
| \(x\) | \(=\) | \(20(1)=20\) |
| \(y\) | \(=\) | \(15-4.9=10.1\) |
At \((20,\ 10.1)\) meters.
Common pitfalls
Frequently asked questions
What are parametric equations?
Equations giving \(x\) and \(y\) each as a function of a parameter \(t\), tracing a path as \(t\) varies.
What is the parameter?
The independent variable \(t\) that drives both coordinates; it often represents time.
Why use parametric equations instead of y = f(x)?
They capture direction, timing, and curves that fail the vertical line test (like a full circle).
How do you graph parametric equations?
Build a table of \(t\), \(x\), \(y\), plot the points in order, and note the direction of travel.