Parametric Equations

The central idea

Parametric equations $x=x(t)$, $y=y(t)$ assign both coordinates as functions of a parameter over a stated interval. The same geometric curve may have different parametrizations, speeds, and orientations. Eliminating $t$ identifies a Cartesian relation but can lose direction or parameter restrictions.

Definitions, hypotheses, and notation

A parametrization is regular when its velocity vector does not vanish. Regularity helps guarantee a well-defined tangent direction, but a curve may still be meaningful at a singular parameter value. Reparametrizing by an increasing function preserves orientation; a decreasing change reverses it.

For $x=a+h t,y=b+k t$, the path is a line through $(a,b)$ with direction $\langle h,k\rangle$. For periodic trigonometric coordinates, compare the parameter interval length with the fundamental period to determine whether the path is partial, complete, or repeatedly traced.

Conceptual meaning

Rather than requiring one output $y$ for each $x$, a parameter traces a path. This represents circles, loops, and motion naturally. The ordered pair at each parameter value is the position of the moving point.

A dependable method and decision rule

1. Record the permitted parameter interval.
2. Make a small table of parameter values and corresponding points.
3. Eliminate the parameter when useful, retaining coordinate restrictions.
4. Determine orientation from increasing $t$.
5. Check whether different parameter values repeat points or the entire curve.

Fully worked example

Graphical or geometric meaning

The parameter interval acts like a playback timeline. Coordinate graphs $x(t)$ and $y(t)$ separately oscillate, while the plane plot combines them into circular motion. Arrows record information absent from $x^2+y^2=4$.

Common mistakes and why they fail

Verification and reasonableness checks

• Substitute several parameter values into the Cartesian relation.
• Mark the start, end, and direction.
• Count repeated tracing before using arc-length or area formulas.

A parameter records more than shape

The pair $(x(t),y(t))$ specifies position, direction, speed, and the part of a curve traced on the stated interval. Eliminating $t$ may reveal a Cartesian equation, but can erase orientation or add points that the parameter never reaches. Make a small table of parameter values, mark arrows, and check endpoint inclusion from the parameter interval. Different parameterizations can traverse the same geometry at different speeds or multiple times. When solving for $t$, check every inverse branch against its allowed interval. In a motion interpretation, $(x',y')$ gives tangent direction and $\sqrt{(x')^2+(y')^2}$ gives speed; neither is preserved by ordinary elimination. State both the geometric curve and its tracing information when relevant.

Practice

1. Eliminate $t$ from $x=t+1,y=t^2$.
2. What direction does $x=\cos t,y=\sin t$ trace?
3. What does $x=t^2,y=t^4$ satisfy?

Connections and next steps

• Calculus with Parametric Curves
• Polar Coordinates in Calculus
• Vector-Valued Functions