Parametric equations (graphing, modeling)
Parametric Equations
Parametric Equations is the opening topic of Parametric & Polar in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.3(A), P.3(C), which requires students to graph parametric equations and use them to model problems.
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.