4.1: Parametric Equations | AP Precalculus

How it works

Parametric equations express both coordinates in terms of a shared parameter. Eliminating the parameter gives a relation, but the parameter interval and direction must be retained. Different parameterizations can trace the same curve at different speeds.

x=x(t),\quad y=y(t)

Worked examples

Example 1

Eliminate t from $\displaystyle x=2t+1$ , $\displaystyle y=t-3$ , for all real t.

Since $\displaystyle t=(x-1)/2$ , $\displaystyle y=(x-1)/2-3=(x-7)/2$ .

Example 2

For $\displaystyle x=\cos t$ , $\displaystyle y=\sin t$ , $\displaystyle 0\le t\le2\pi$ , identify the curve and direction.

The unit circle is traced once counterclockwise from $\displaystyle (1,0)$ .

Problems

Problem 4.1.1

For $\displaystyle x=2t$ , $\displaystyle y=t^2$ , $\displaystyle 0\le t\le2$ , eliminate t and state the domain and direction.

Parabola segment from 0,0 to 4,4. — Elimination gives y = x squared / 4, with x restricted to [0,4].

Show worked solution

$\displaystyle y=x^2/4$ for $\displaystyle 0\le x\le4$ , traced from $\displaystyle (0,0)$ to $\displaystyle (4,4)$ as t increases.

Parabola segment from 0,0 to 4,4. The solution adds labeled relationships. — As t increases, the point moves from (0,0) toward (4,4).

Problem 4.1.2

For $\displaystyle x=3\cos t$ , $\displaystyle y=2\sin t$ , eliminate t.

Show worked solution

The identity gives $\displaystyle x^2/9+y^2/4=1$ , an ellipse.

Problem 4.1.3

Why does eliminating t from $\displaystyle x=t^2$ , $\displaystyle y=t^2$ not give the entire line $\displaystyle y=x$ ?

Show worked solution

The parameter forces $\displaystyle x\ge0$ , so only the ray with $\displaystyle y=x\ge0$ is traced.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.1.1

Problem 4.1.2

Problem 4.1.3

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