- Create and sketch graphs of curves given their parametric equations
- Convert parametric equations into a regular y = f(x) form by eliminating the parameter
- Recognize and describe the curve called a cycloid
Parametric Equations and Their Graphs
The Main Idea
Traditional functions [latex]y = f(x)[/latex] work great for many curves, but what about loops, vertical lines, or paths that double back on themselves? Parametric equations solve this by letting both [latex]x[/latex] and [latex]y[/latex] depend on a third variable called a parameter.
In parametric equations, we write [latex]x = x(t)[/latex] and [latex]y = y(t)[/latex], where [latex]t[/latex] is the parameter. Think of [latex]t[/latex] as time on a stopwatch—as [latex]t[/latex] changes, both coordinates change simultaneously, tracing out a path through the plane.
The parameter [latex]t[/latex] gives curves an orientation—a direction of travel. As [latex]t[/latex] increases, you can follow the path from start to finish with arrows showing which way you’re moving.
The Process: Make a table with [latex]t[/latex] values, calculate corresponding [latex]x[/latex] and [latex]y[/latex] coordinates, then plot and connect the points. The arrows show the direction as [latex]t[/latex] increases.
Sketch the curve described by the parametric equations
Eliminating the Parameter
The Main Idea
Sometimes you want to figure out what familiar curve your parametric equations represent. Eliminating the parameter means getting rid of [latex]t[/latex] to find a direct relationship between [latex]x[/latex] and [latex]y[/latex].
The Basic Strategy: Solve one parametric equation for [latex]t[/latex], then substitute that expression into the other equation. Choose whichever equation makes the algebra easier.
Standard Approach:
- From [latex]y = 2t + 1[/latex], solve for [latex]t[/latex]: [latex]t = \frac{y-1}{2}[/latex]
- Substitute into [latex]x = t^2 - 3[/latex]: [latex]x = \left(\frac{y-1}{2}\right)^2 - 3[/latex]
- Simplify to get the final [latex]x[/latex]–[latex]y[/latex] relationship
The parameter restrictions matter! If [latex]-2 \leq t \leq 3[/latex], your curve is only a piece of the full equation, not the entire parabola or circle.
You can also start with a regular equation like [latex]y = 2x^2 - 3[/latex] and create parametric equations. The simplest way is [latex]x = t, y = 2t^2 - 3[/latex], but you could also choose [latex]x = 3t - 2[/latex] and adjust [latex]y[/latex] accordingly.
Eliminate the parameter for the plane curve defined by the following parametric equations and describe the resulting graph.
Find two different sets of parametric equations to represent the graph of [latex]y={x}^{2}+2x[/latex].
Cycloids and Other Parametric Curves
The Main Idea
Imagine a piece of gum stuck to your bicycle tire. As you ride forward, that gum traces out a specific curve through space—and that curve has a name: a cycloid. This isn’t just abstract math; it’s the actual path traced by any point on a rolling wheel.
The Cycloid: For a wheel of radius [latex]a[/latex] rolling along a straight line, the parametric equations are:
- [latex]x(t) = a(t - \sin t)[/latex]
- [latex]y(t) = a(1 - \cos t)[/latex]
The motion combines two parts—the wheel’s center moving forward at constant height [latex]a[/latex], plus the point rotating around that moving center. The [latex]t - \sin t[/latex] and [latex]1 - \cos t[/latex] terms capture this combined motion perfectly.
Hypocycloids: Now imagine a smaller circle rolling inside a larger circle. The curves become even more fascinating, creating star-like shapes with sharp points called cusps.
The Magic Ratio: The ratio [latex]\frac{a}{b}[/latex] (big circle radius ÷ small circle radius) determines everything:
- [latex]\frac{a}{b} = 3[/latex] → 3-pointed star (deltoid)
- [latex]\frac{a}{b} = 4[/latex] → 4-pointed star (astroid)
- Rational ratios → finite cusps that eventually close
- Irrational ratios → infinite cusps that never repeat