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
Traditional functions like [latex]y = f(x)[/latex] work well for many curves, but some curves can’t be described this way. What if a curve loops back on itself or is vertical in places? This is where parametric equations become essential.
In parametric equations, both [latex]x[/latex] and [latex]y[/latex] are defined as functions of a third variable called a parameter. This parameter, often denoted as [latex]t[/latex], acts as an independent variable that controls both coordinates simultaneously.
parametric equations
If [latex]x[/latex] and [latex]y[/latex] are continuous functions of [latex]t[/latex] on an interval [latex]I[/latex], then the equations
are called parametric equations and [latex]t[/latex] is called the parameter.
[latex]\\[/latex]
The set of points [latex](x,y)[/latex] obtained as [latex]t[/latex] varies over the interval [latex]I[/latex] is called the graph of the parametric equations. The graph of parametric equations is called a parametric curve or plane curve, and is denoted by [latex]C[/latex].
Consider Earth’s orbit around the Sun. Earth’s position changes continuously throughout the year, tracing an elliptical path. We can use the day of the year as our parameter [latex]t[/latex].
Day 1 = January 1
Day 31 = January 31
Day 59 = February 28
And so on…
Figure 1. Earth’s orbit around the Sun during one year. The point labeled [latex]{F}_{2}[/latex] is one of the foci of the ellipse; the other focus is occupied by the Sun.
As [latex]t[/latex] (the day) increases from 1 to 365, Earth’s position [latex](x(t), y(t))[/latex] traces out its complete orbital path. After 365 days, we return to the starting position and begin a new cycle.
Think of the parameter [latex]t[/latex] as time on a stopwatch. As time progresses, both the [latex]x[/latex] and [latex]y[/latex] coordinates change according to their respective functions, creating a path through the coordinate plane.
It’s crucial to understand that [latex]x[/latex] and [latex]y[/latex] serve dual roles in parametric equations:
As functions: [latex]x(t)[/latex] and [latex]y(t)[/latex] are functions of the parameter [latex]t[/latex]
As variables: [latex]x[/latex] and [latex]y[/latex] represent the coordinates of points on the curve
When [latex]t[/latex] varies over an interval, the functions [latex]x(t)[/latex] and [latex]y(t)[/latex] generate ordered pairs [latex](x,y)[/latex] that form the parametric curve.
Sketch the curves described by the following parametric equations:
To create a graph of this curve, first set up a table of values. Since the independent variable in both [latex]x\left(t\right)[/latex] and [latex]y\left(t\right)[/latex] is t, let t appear in the first column. Then [latex]x\left(t\right)[/latex] and [latex]y\left(t\right)[/latex] will appear in the second and third columns of the table.
t
[latex]x\left(t\right)[/latex]
[latex]y\left(t\right)[/latex]
−3
−4
−2
−2
−3
0
−1
−2
2
0
−1
4
1
0
6
2
1
8
The second and third columns in this table provide a set of points to be plotted. The graph of these points appears in Figure 3. The arrows on the graph indicate the orientation of the graph, that is, the direction that a point moves on the graph as t varies from −3 to 2.
Figure 3. Graph of the plane curve described by the parametric equations in part a.
To create a graph of this curve, again set up a table of values.
t
[latex]x\left(t\right)[/latex]
[latex]y\left(t\right)[/latex]
−2
1
−3
−1
−2
−1
0
−3
1
1
−2
3
2
1
5
3
6
7
The second and third columns in this table give a set of points to be plotted (Figure 4). The first point on the graph (corresponding to [latex]t=-2[/latex]) has coordinates [latex]\left(1,-3\right)[/latex], and the last point (corresponding to [latex]t=3[/latex]) has coordinates [latex]\left(6,7\right)[/latex]. As [latex]t[/latex] progresses from −2 to 3, the point on the curve travels along a parabola. The direction the point moves is again called the orientation and is indicated on the graph.
Figure 4. Graph of the plane curve described by the parametric equations in part b.
In this case, use multiples of [latex]\frac{\pi}{6}[/latex] for t and create another table of values:
t
[latex]x\left(t\right)[/latex]
[latex]y\left(t\right)[/latex]
t
[latex]x\left(t\right)[/latex]
[latex]y\left(t\right)[/latex]
0
4
0
[latex]\frac{7\pi }{6}[/latex]
[latex]-2\sqrt{3}\approx -3.5[/latex]
2
[latex]\frac{\pi }{6}[/latex]
[latex]2\sqrt{3}\approx 3.5[/latex]
[latex]2[/latex]
[latex]\frac{4\pi }{3}[/latex]
−2
[latex]-2\sqrt{3}\approx -3.5[/latex]
[latex]\frac{\pi }{3}[/latex]
[latex]2[/latex]
[latex]2\sqrt{3}\approx 3.5[/latex]
[latex]\frac{3\pi }{2}[/latex]
0
−4
[latex]\frac{\pi }{2}[/latex]
0
4
[latex]\frac{5\pi }{3}[/latex]
2
[latex]-2\sqrt{3}\approx -3.5[/latex]
[latex]\frac{2\pi }{3}[/latex]
−2
[latex]2\sqrt{3}\approx 3.5[/latex]
[latex]\frac{11\pi }{6}[/latex]
[latex]2\sqrt{3}\approx 3.5[/latex]
2
[latex]\frac{5\pi }{6}[/latex]
[latex]-2\sqrt{3}\approx -3.5[/latex]
2
[latex]2\pi[/latex]
4
0
[latex]\pi[/latex]
−4
0
The graph of this plane curve appears in the following graph.
Figure 5. Graph of the plane curve described by the parametric equations in part c.
This is the graph of a circle with radius 4 centered at the origin, with a counterclockwise orientation. The starting point and ending points of the curve both have coordinates [latex]\left(4,0\right)[/latex].
Watch the following video to see the worked solution to the example above.
For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.
