Parametric Functions

Beyond the exam. The AP® Precalculus exam does not test this unit. Many classes teach it, and it prepares you for calculus.

Most graphs you have drawn so far show one quantity against another: $y$ against $x$ . A parametric function adds a third quantity that is not drawn at all. It is the clock that tells each point when to appear. This page shows how to read and draw such curves, and why they can do things an ordinary graph cannot.

A parameter runs the show

A parametric function is a pair of functions of one input, written $f(t) = (x(t), y(t))$ . The input $t$ is called the parameter. For each allowed $t$ , the function $x(t)$ gives a horizontal position and $y(t)$ gives a vertical position, so the pair is one point of the plane.

Think of $t$ as time. A drone's log might say that after $t$ seconds it is $x(t)$ meters east and $y(t)$ meters north of where it took off. The flight path is the set of all points $(x(t), y(t))$ . Time does not show on the picture, but it decides the order of the points.

From a table to a curve

The surest way to sketch a parametric curve is a table. Choose several values of $t$ , compute $x(t)$ and $y(t)$ , plot the points, and join them in order of increasing $t$ .

Worked example 1. Sketch $x(t) = t^2 - 2t$ , $y(t) = 2t - 1$ for $-1 \le t \le 3$ .

$t$	$-1$	$0$	$1$	$2$	$3$
$x(t) = t^2 - 2t$	$3$	$0$	$-1$	$0$	$3$
$y(t) = 2t - 1$	$-3$	$-1$	$1$	$3$	$5$

For example, $x(-1)$ $= (-1)^2 - 2(-1)$ $= 1 + 2$ $= 3$ . Plot $(3, -3)$ , $(0, -1)$ , $(-1, 1)$ , $(0, 3)$ , $(3, 5)$ and join them in that order. The curve moves left until $t = 1$ , then turns and moves right, climbing the whole time.

The curve x(t) = t squared minus 2t, y(t) = 2t minus 1 for t from -1 to 3, a parabola opening to the right. Points are marked at t = -1 (3, -3), t = 0 (0, -1), t = 1 (-1, 1), t = 2 (0, 3) and t = 3 (3, 5). Arrows show the point moves left and up until t = 1, then right and up. — **Figure 1.** The curve of Worked example 1. Arrows show the direction of travel as $t$ increases. LevlPrep original diagram.

Direction of travel

A parametric curve comes with a direction: the order in which the points are traced as $t$ increases. Mark it with arrows. You can read the direction from the two functions separately:

where $x(t)$ is increasing, the point moves right; where it is decreasing, the point moves left;
where $y(t)$ is increasing, the point moves up; where it is decreasing, the point moves down.

In Worked example 1, $y(t) = 2t - 1$ increases for every $t$ , so the point climbs the whole time, while $x(t)$ falls and then rises.

Curves that are not graphs of functions

Both $x(t)$ and $y(t)$ are functions of $t$ : one input, one output. The curve they make is a different matter. In Worked example 1, $x = 0$ happens at $t = 0$ and at $t = 2$ , with $y = -1$ and $y = 3$ . One $x$ -value has two $y$ -values, so the curve is not the graph of a function of $x$ . A parametric curve may also cross itself, when two different values of $t$ land on the same point.

Same path, different parameter

Different parametric functions can trace the same set of points. The functions $(t, t^2)$ for $0 \le t \le 2$ and $(2s, 4s^2)$ for $0 \le s \le 1$ both trace the parabola $y = x^2$ from $(0, 0)$ to $(2, 4)$ . The second one just gets there with smaller parameter values. Replacing $t$ with $-t$ (and adjusting the interval) traces the same path backward.

Writing $y$ as a function of $x$

When $x(t)$ is simple to solve for $t$ , you can remove the parameter and get an equation in $x$ and $y$ alone.

Worked example 2. Write $y$ as a function of $x$ for $x(t) = 2t + 1$ , $y(t) = t^2$ .

Solve $x = 2t + 1$ for $t$ : subtract 1, then divide by 2, so $t = \frac{x - 1}{2}$ .

Substitute into $y = t^2$ : $y = \left(\frac{x - 1}{2}\right)^2$ . The curve is a parabola with vertex at $(1, 0)$ , the point where $t = 0$ .

Graph against the parameter, or curve in the plane?

Picture	Horizontal axis	Vertical axis	What it shows
Graph of $x(t)$	$t$	$x$	when the point moves right or left
Graph of $y(t)$	$t$	$y$	when the point moves up or down
Parametric curve	$x$	$y$	the path itself, with arrows for the order

Keep these apart. A common slip is to draw $y(t)$ against $t$ and call it the curve.