10.7 Parametric Equations: Graphs

Parametric equations let you describe curves by defining $x$ and $y$ separately as functions of an independent parameter, usually $t$. This approach is especially useful when a curve can't be written as a single $y = f(x)$ equation, or when you need to track how a point moves along a path over time. You'll see parametric equations used to model projectile motion, orbital paths, and any situation where position changes with time.

Parametric Equations and Their Graphs

Plotting parametric equation points

A parametric equation is actually a pair of equations that define the $x$ and $y$ coordinates of a point using an independent parameter $t$:

• $x(t)$ gives the x-coordinate at a given value of $t$
• $y(t)$ gives the y-coordinate at that same value of $t$

Together, each value of $t$ produces a single point $(x, y)$ on the coordinate plane. As $t$ changes, the point traces out a curve.

How to plot a parametric curve:

1. Choose a range of $t$ values (for example, $t = -2, -1, 0, 1, 2, 3$).
2. Substitute each $t$ value into both $x(t)$ and $y(t)$ to get coordinate pairs.
3. Plot the resulting $(x, y)$ points on the Cartesian plane.
4. Connect the points smoothly in order of increasing $t$. The order matters because it shows the direction the curve is traced.

For example, take $x(t) = 2t + 1$, $y(t) = 3t - 2$. Plugging in $t = 0$ gives the point $(1, -2)$. Plugging in $t = 1$ gives $(3, 1)$. Plugging in $t = 2$ gives $(5, 4)$. These points fall on a straight line, which makes sense because both $x$ and $y$ are linear in $t$.

For a circle, try $x(t) = 3\cos(t)$, $y(t) = 3\sin(t)$ with $t$ from $0$ to $2\pi$. At $t = 0$ you get $(3, 0)$; at $t = \pi/2$ you get $(0, 3)$; at $t = \pi$ you get $(-3, 0)$. The curve traces a circle of radius 3, and the direction is counterclockwise.

Shape analysis of parametric curves

Once you've plotted a few points, you'll want to understand the overall shape without plotting every single value. Here's how to analyze a parametric curve:

Examine each function separately:

• Find the domain and range of $x(t)$ and $y(t)$. If both are bounded (like sine and cosine), the curve stays in a finite region.
• Check for periodicity. If both $x(t)$ and $y(t)$ are periodic with the same period, the curve is closed (it loops back to where it started).

Look at the relationship between the two functions:

• Can you eliminate $t$ to get a Cartesian equation? For example, with $x(t) = 3\cos(t)$ and $y(t) = 3\sin(t)$, squaring and adding gives $x^2 + y^2 = 9$, confirming it's a circle.
• Check whether the curve is open (extends indefinitely) or closed (forms a loop).

Track direction and behavior:

• As $t$ increases, which way does the curve move? This is called the orientation of the curve.
• As $t$ approaches very large or very small values, does the curve approach an asymptote, spiral outward, or do something else?

Common parametric curve types to recognize:

Real-world applications of parametric equations

Parametric equations are natural for modeling motion because the parameter $t$ usually represents time.

Projectile motion is the classic example. An object launched at angle $\theta$ with initial velocity $v_0$ follows:

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

Here $g$ is the acceleration due to gravity (about $9.8 \text{ m/s}^2$). The $x$-equation tracks horizontal distance (constant speed, no air resistance), while the $y$-equation tracks height (gravity pulls the object down over time). You can't easily write this path as $y = f(x)$ without first eliminating $t$, which is why parametric form is so convenient.

Circular and orbital motion also fits naturally. A Ferris wheel with radius 20 feet and center 30 feet off the ground can be modeled as:

• $x(t) = 20\cos(t)$
• $y(t) = 20\sin(t) + 30$

The $+30$ shifts the entire circle up so the lowest point is 10 feet above the ground, not underground. Planetary orbits work similarly but use ellipses instead of circles.

Spirals appear when the "radius" grows with time. The equations $x(t) = t\cos(t)$, $y(t) = t\sin(t)$ produce a spiral that winds outward because the factor of $t$ in front makes the distance from the origin increase as $t$ grows.

Alternative Coordinate Systems and Functions

Parametric equations aren't the only way to describe curves beyond $y = f(x)$. A few related tools you should be aware of:

• Polar coordinates describe a point by its distance from the origin ($r$) and angle from the positive x-axis ($\theta$), rather than by $(x, y)$. Curves like spirals and roses are often simpler in polar form.
• Vector-valued functions package the parametric equations into a single vector: $\vec{r}(t) = \langle x(t), y(t) \rangle$. This notation is especially useful when you extend to three dimensions.
• Phase planes graph one variable against another (rather than against time) to visualize how a system evolves. You'll encounter these more in differential equations and dynamical systems courses.