Parametric equations

A graph like ${y=x^2}$ shows the shape of a path, but not how something moves along it: where it is at each moment, or which way it is going. Parametric equations give both. They describe ${x}$ and ${y}$ separately, each as a function of a third variable, usually time.

Curves given by parametric equations

If ${x}$ and ${y}$ are both given as functions of a variable ${t}$,

${x=f(t)}$${y=g(t)}$

then as ${t}$ changes, the point ${(x,y)}$ moves and traces a curve. The variable ${t}$ is called the parameter. Often ${t}$ is time, and the equations say where an object is at time ${t}$.

Example 1: Sketch the curve ${x=t^2}$, ${y=t+1}$ for ${-2\le t\le 2}$.

Solution:

Make a table: for each ${t}$, find ${x}$ and ${y}$.

${t}$	${-2}$	${-1}$	${0}$	${1}$	${2}$
${x}$	${4}$	${1}$	${0}$	${1}$	${4}$
${y}$	${-1}$	${0}$	${1}$	${2}$	${3}$

Plot the points ${(x,y)}$ in order of ${t}$, and join them:

As ${t}$ goes from ${-2}$ to ${2}$, the point moves up along a parabola. The arrows show the direction.

Eliminating the parameter

To find an ordinary equation in ${x}$ and ${y}$, get rid of ${t}$. Usually, solve one equation for ${t}$ and substitute into the other.

Example 2: Eliminate the parameter in Example 1.

Solution:

From ${y=t+1}$, we get ${t=y-1}$. Substitute into ${x=t^2}$:

${x=(y-1)^2}$

This is a parabola opening to the right with vertex ${(0,1)}$, as the figure shows. The parametric form tells us more: the direction of motion, and that only the part from ${(4,-1)}$ to ${(4,3)}$ is traced.

With sines and cosines, use ${\sin^2 t+\cos^2 t=1}$ instead.

Example 3: Eliminate the parameter, and describe each curve for ${0\le t\le 2\pi}$.

(a) ${x=3\cos t}$, ${y=3\sin t}$ (b) ${x=4\cos t}$, ${y=2\sin t}$

Solution:

(a) ${x^2+y^2=9\cos^2 t+9\sin^2 t=9}$. This is the circle of radius ${3}$, traced once counterclockwise, starting at ${(3,0)}$.

(b) Solve for the cosine and sine: ${\cos t=\dfrac{x}{4}}$ and ${\sin t=\dfrac{y}{2}}$. Then ${\cos^2 t+\sin^2 t=1}$ gives

${\dfrac{x^2}{16}+\dfrac{y^2}{4}=1}$

This is an ellipse, again traced once counterclockwise.

${x=3\cos t}$, ${y=3\sin t}$

${x=4\cos t}$, ${y=2\sin t}$

A curve can have many parametric descriptions. For example, ${x=3\cos 2t}$, ${y=3\sin 2t}$ traces the same circle twice as fast, and ${x=3\sin t}$, ${y=3\cos t}$ traces it clockwise.

Projectile motion

A ball thrown with speed ${v_0}$ at angle ${\theta}$ above the ground moves sideways at a steady speed, while gravity pulls it down. If we ignore air resistance and measure in feet and seconds, its position at time ${t}$ is

${x=(v_0\cos\theta)\,t}$

${y=(v_0\sin\theta)\,t-16t^2}$

The numbers ${v_0\cos\theta}$ and ${v_0\sin\theta}$ are the horizontal and vertical parts of the starting velocity (see Vectors). The term ${-16t^2}$ is the effect of gravity.

Example 4: A ball is thrown at ${64}$ ft/s at ${45^\circ}$ above level ground. When and where does it land, and how high does it go?

Solution:

Here $\cos 45^\circ=\sin 45^\circ=\dfrac{\sqrt{2}}{2}$, so both parts of the velocity are

${64\cdot\dfrac{\sqrt{2}}{2}=32\sqrt{2}}$

${x=32\sqrt{2}\,t}$${y=32\sqrt{2}\,t-16t^2}$

Landing: Set ${y=0}$ and factor:

${16t\,(2\sqrt{2}-t)=0}$

So ${t=0}$ (the throw) or ${t=2\sqrt{2}\approx 2.83}$ s. At that time, ${x=32\sqrt{2}\cdot 2\sqrt{2}=128}$ ft.

Highest point: The path is symmetric, so the top is halfway through the flight, at ${t=\sqrt{2}}$:

$\begin{align*}y&=32\sqrt{2}\cdot\sqrt{2}-16\cdot 2\\&=64-32=32\end{align*}$

The ball goes ${32}$ ft high.

The path of the ball. The dots show where it is every half second; they spread out evenly across, because the horizontal speed is constant.

The ${128}$ ft agrees with the range found in Double-angle and half-angle formulas.

The cycloid

Paint a dot on the rim of a wheel, and roll the wheel along a straight road. The path of the dot is a cycloid. If the wheel has radius ${a}$ and has turned through angle ${t}$ (in radians), the dot is at

${x=a(t-\sin t)}$${y=a(1-\cos t)}$

A cycloid. As the green wheel rolls along the ground, the point ${P}$ on its rim traces the arches.

The center of the wheel has moved a distance ${at}$ (the arc length that has touched the road), and the dot has turned around the center by angle ${t}$. That gives the two terms in each equation. This curve is hard to describe with a single equation in ${x}$ and ${y}$, but easy with a parameter.

Summary

Parametric equations ${x=f(t)}$, ${y=g(t)}$ describe a curve and the direction it is traced.
To graph, make a table of ${t}$, ${x}$, ${y}$, and plot in order of ${t}$.
To eliminate the parameter, solve for ${t}$ and substitute, or use ${\sin^2 t+\cos^2 t=1}$.
A projectile moves on ${x=(v_0\cos\theta)t}$, ${y=(v_0\sin\theta)t-16t^2}$ (feet, seconds).
A cycloid is ${x=a(t-\sin t)}$, ${y=a(1-\cos t)}$.

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