10.6 Parametric Equations

Parametric equations let you describe curves by defining $x$ and $y$ separately as functions of a third variable (usually $t$). Instead of writing $y$ directly in terms of $x$, you let both coordinates depend on a parameter, which makes it much easier to represent motion, direction, and curves that loop or cross themselves.

This approach is especially useful for modeling projectile motion, orbits, and other situations where position changes over time. It also shows up in computer graphics and animation. For a College Algebra course, the focus is on understanding what parametric equations are, how to graph them, and how to convert between parametric and rectangular form.

Parametric Equations

Parametric vs rectangular equations

A rectangular equation relates $x$ and $y$ directly, like $y = x^2 + 3$ or $x^2 + y^2 = 25$. You get one equation connecting the two variables.

A parametric equation introduces a third variable, the parameter $t$, and defines each coordinate as its own function:

• $x = f(t)$
• $y = g(t)$

As $t$ changes, the pair $(x, y)$ traces out a curve. The parameter $t$ often represents time, but it can also represent an angle or any other quantity.

Why bother? Some curves are awkward or impossible to write as a single rectangular equation. A circle like $x^2 + y^2 = 9$ isn't a function of $x$ (it fails the vertical line test), but parametrically it's clean: $x = 3\cos(t)$, $y = 3\sin(t)$. Parametric form also naturally encodes direction, telling you which way the curve is traced as $t$ increases.

Graphing parametric curves

The most reliable way to graph a parametric curve by hand:

1. Build a table. Choose several values of $t$ (often evenly spaced). Plug each into the equations for $x$ and $y$ to get coordinate pairs.
2. Plot the points. Place each $(x, y)$ pair on the coordinate plane.
3. Connect in order. Draw the curve by connecting points in the order of increasing $t$. Add arrows to show the direction of travel.

For example, given $x = t + 1$ and $y = t^2$:

Plotting these gives a parabola opening upward, traced from left to right as $t$ increases.

A few things to watch for:

• Direction: The arrows matter. For $x = \cos(t)$, $y = \sin(t)$, the circle is traced counterclockwise.
• Self-intersections: The curve can cross itself if different $t$ values produce the same $(x, y)$ point.
• Restricted domain: Sometimes $t$ is limited to an interval, so you only get part of the curve.

Eliminating the parameter

Converting from parametric to rectangular form is called eliminating the parameter. This helps you recognize the shape of the curve.

1. Solve one equation for $t$. Pick whichever equation is simpler. For $x = t + 1$, you get $t = x - 1$.

2. Substitute into the other equation. Replace $t$ in the $y$ equation: $y = (x - 1)^2$.

3. Simplify. Now you have a rectangular equation you can analyze directly.

For trig parametrics, use identities instead of solving for $t$. If $x = 3\cos(t)$ and $y = 3\sin(t)$, you know $\cos^2(t) + \sin^2(t) = 1$. So:

$\left(\frac{x}{3}\right)^2 + \left(\frac{y}{3}\right)^2 = 1$

which simplifies to $x^2 + y^2 = 9$, a circle of radius 3.

Applications of parametric equations

Parametric equations are a natural fit for any situation where position changes over time.

Projectile motion is the classic example. An object launched at speed $v_0$ and angle $\theta$ from ground level has:

• $x(t) = v_0 \cos(\theta) \cdot t$
• $y(t) = v_0 \sin(\theta) \cdot t - \frac{1}{2}(32)t^2$

(using $32 \text{ ft/s}^2$ for gravity). At any time $t$, you can find the object's exact position. You can also find when it hits the ground by setting $y(t) = 0$ and solving for $t$.

Other applications include:

• Planetary orbits: Elliptical paths are cleanly described with parametric trig equations
• Lissajous figures: Curves formed by $x = A\sin(at)$, $y = B\sin(bt + \delta)$, which model oscillations
• Computer graphics: Curves in animation software and CAD programs are built from parametric equations (Bézier curves, for instance)

Motion analysis with parametrics

If your course covers derivatives with parametric equations, here's how motion analysis works.

Velocity comes from the first derivatives:

• Horizontal velocity: $v_x = \frac{dx}{dt}$
• Vertical velocity: $v_y = \frac{dy}{dt}$
• Speed (how fast, regardless of direction): $\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}$

Acceleration comes from the second derivatives:

• $a_x = \frac{d^2x}{dt^2}$
• $a_y = \frac{d^2y}{dt^2}$

To analyze motion at a specific moment, plug that $t$ value into the position, velocity, and acceleration expressions. For instance, if $x(t) = 2t$ and $y(t) = -16t^2 + 48t$, then at $t = 1$: position is $(2, 32)$, horizontal velocity is $2$, and vertical velocity is $-32(1) + 48 = 16$.

The slope of the tangent line to a parametric curve is:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

This is useful for finding the direction of motion at any point along the curve.