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Section9.2Parametric Equations

We are familiar with sketching shapes, such as parabolas, by following this basic procedure:

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The rectangular equation \(y=f(x)\) works well for some shapes like a parabola with a vertical axis of symmetry, but in the previous section we encountered several shapes that could not be sketched in this manner. (To plot an ellipse using the above procedure, we need to plot the “top” and “bottom” separately.)

In this section we introduce a new sketching procedure:

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Here, \(x\) and \(y\) are found separately but then plotted together. This leads us to a definition.

Definition9.2.1Parametric Equations and Curves

Let \(f\) and \(g\) be continuous functions on an interval \(I\text{.}\) The set of all points \(\big(x,y\big) = \big(f(t),g(t)\big)\) in the Cartesian plane, as \(t\) varies over \(I\text{,}\) is the graph of the parametric equations \(x=f(t)\) and \(y=g(t)\text{,}\) where \(t\) is the parameter. A curve is a graph along with the parametric equations that define it.

This is a formal definition of the word curve. When a curve lies in a plane (such as the Cartesian plane), it is often referred to as a plane curve. Examples will help us understand the concepts introduced in the definition.

Example9.2.2Plotting parametric functions

Plot the graph of the parametric equations \(x=t^2\text{,}\) \(y=t+1\) for \(t\) in \([-2,2]\text{.}\)

Solution

We often use the letter \(t\) as the parameter as we often regard \(t\) as representing time. Certainly there are many contexts in which the parameter is not time, but it can be helpful to think in terms of time as one makes sense of parametric plots and their orientation (for instance, “At time \(t=0\) the position is \((1,2)\) and at time \(t=3\) the position is \((5,1)\text{.}\)”).

Example9.2.4Plotting parametric functions

Sketch the graph of the parametric equations \(x=\cos^2(t)\text{,}\) \(y=\cos(t) +1\) for \(t\) in \([0,\pi]\text{.}\)

Solution

Technology Note: Most graphing utilities can graph functions given in parametric form. Often the word “parametric” is abbreviated as “PAR” or “PARAM” in the options. The user usually needs to determine the graphing window (i.e, the minimum and maximum \(x\)- and \(y\)-values), along with the values of \(t\) that are to be plotted. The user is often prompted to give a \(t\) minimum, a \(t\) maximum, and a “\(t\)-step” or “\(\Delta t\text{.}\)” Graphing utilities effectively plot parametric functions just as we've shown here: they plots lots of points. A smaller \(t\)-step plots more points, making for a smoother graph (but may take longer). In Figure 9.2.3, the \(t\)-step is 1; in Figure 9.2.5, the \(t\)-step is \(\pi/4\text{.}\)

One nice feature of parametric equations is that their graphs are easy to shift. While this is not too difficult in the “\(y=f(x)\)” context, the resulting function can look rather messy. (Plus, to shift to the right by two, we replace \(x\) with \(x-2\text{,}\) which is counter–intuitive.) The following example demonstrates this.

Example9.2.6Shifting the graph of parametric functions

Sketch the graph of the parametric equations \(x=t^2+t\text{,}\) \(y=t^2-t\text{.}\) Find new parametric equations that shift this graph to the right 3 places and down 2.

Solution

Because the \(x\)- and \(y\)-values of a graph are determined independently, the graphs of parametric functions often possess features not seen on “\(y=f(x)\)” type graphs. The next example demonstrates how such graphs can arrive at the same point more than once.

Example9.2.10Graphs that cross themselves

Plot the parametric functions \(x=t^3-5t^2+3t+11\) and \(y=t^2-2t+3\) and determine the \(t\)-values where the graph crosses itself.

Solution

Subsection9.2.1Converting between rectangular and parametric equations

It is sometimes useful to rewrite equations in rectangular form (i.e., \(y=f(x)\)) into parametric form, and vice–versa. Converting from rectangular to parametric can be very simple: given \(y=f(x)\text{,}\) the parametric equations \(x=t\text{,}\) \(y=f(t)\) produce the same graph. As an example, given \(y=x^2\text{,}\) the parametric equations \(x=t\text{,}\) \(y=t^2\) produce the familiar parabola. However, other parametrizations can be used. The following example demonstrates one possible alternative.

Example9.2.12Converting from rectangular to parametric

Consider \(y=x^2\text{.}\) Find parametric equations \(x=f(t)\text{,}\) \(y=g(t)\) for the parabola where \(t=\frac{dy}{dx}\text{.}\) That is, \(t=a\) corresponds to the point on the graph whose tangent line has slope \(a\text{.}\)

Solution

We sometimes chose the parameter to accurately model physical behavior.

Example9.2.13Converting from rectangular to parametric

An object is fired from a height of 0ft and lands 6 seconds later, 192ft away. Assuming ideal projectile motion, the height, in feet, of the object can be described by \(h(x) = -x^2/64+3x\text{,}\) where \(x\) is the distance in feet from the initial location. (Thus \(h(0) = h(192) = 0\)ft.) Find parametric equations \(x=f(t)\text{,}\) \(y=g(t)\) for the path of the projectile where \(x\) is the horizontal distance the object has traveled at time \(t\) (in seconds) and \(y\) is the height at time \(t\text{.}\)

Solution

It is sometimes necessary to convert given parametric equations into rectangular form. This can be decidedly more difficult, as some “simple” looking parametric equations can have very “complicated” rectangular equations. This conversion is often referred to as “eliminating the parameter,” as we are looking for a relationship between \(x\) and \(y\) that does not involve the parameter \(t\text{.}\)

Example9.2.15Eliminating the parameter

Find a rectangular equation for the curve described by \begin{equation*} x= \frac{1}{t^2+1} \text{ and } y=\frac{t^2}{t^2+1}. \end{equation*}

Solution
Example9.2.17Eliminating the parameter

Eliminate the parameter in \(x=4\cos(t) +3\text{,}\) \(y= 2\sin(t) +1\)

Solution

The Pythagorean Theorem can also be used to identify parametric equations for hyperbolas. We give the parametric equations for ellipses and hyperbolas in the following Key Idea.

Key Idea9.2.19Parametric Equations of Ellipses and Hyperbolas

  • The parametric equations \begin{equation*} x=a\cos(t) +h, y=b\sin(t) +k \end{equation*} define an ellipse with horizontal axis of length \(2a\) and vertical axis of length \(2b\text{,}\) centered at \((h,k)\text{.}\)

  • The parametric equations \begin{equation*} x= a\tan(t) +h, y=\pm b\sec(t) +k \end{equation*} define a hyperbola with vertical transverse axis centered at \((h,k)\text{,}\) and \begin{equation*} x=\pm a\sec(t) +h, y=b\tan(t) + k \end{equation*} defines a hyperbola with horizontal transverse axis. Each has asymptotes at \(y=\pm b/a(x-h)+k\text{.}\)

Subsection9.2.2Special Curves

Figure 9.2.20 gives a small gallery of “interesting” and “famous” curves along with parametric equations that produce them. Interested readers can begin learning more about these curves through internet searches.

One might note a feature shared by two of these graphs: “sharp corners,” or cusps. We have seen graphs with cusps before and determined that such functions are not differentiable at these points. This leads us to a definition.

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\text{ Astroid
\(x=\cos^3(t) \)
\(y=\sin^3(t) \) }

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\text{ Rose Curve
\(x=\cos(t)\sin(4t)\)
\(y=\sin(t)\sin(4t)\) }

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\text{ Hypotrochoid
\(x=2\cos(t)+5\cos(2t/3)\)
\(y=2\sin(t)-5\sin(2t/3)\) }

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\text{ Epicycloid
\(x=4\cos(t)-\cos(4t)\)
\(y=4\sin(t)-\sin(4t)\) }
Table9.2.20A gallery of interesting planar curves.
Definition9.2.21Smooth

A curve \(C\) defined by \(x=f(t)\text{,}\) \(y=g(t)\) is smooth on an interval \(I\) if \(\fp\) and \(g'\) are continuous on \(I\) and not simultaneously 0 (except possibly at the endpoints of \(I\)). A curve is piecewise smooth on \(I\) if \(I\) can be partitioned into subintervals where \(C\) is smooth on each subinterval.

Consider the astroid, given by \(x=\cos^3(t)\text{,}\) \(y=\sin^3(t)\text{.}\) Taking derivatives, we have: \begin{equation*} x' = -3\cos^2(t) \sin(t) \text{ and } y' = 3\sin^2(t) \cos(t) . \end{equation*}

It is clear that each is 0 when \(t=0,\ \pi/2,\ \pi,\ldots\text{.}\) Thus the astroid is not smooth at these points, corresponding to the cusps seen in the figure.

We demonstrate this once more.

Example9.2.22Determine where a curve is not smooth

Let a curve \(C\) be defined by the parametric equations \(x=t^3-12t+17\) and \(y=t^2-4t+8\text{.}\) Determine the points, if any, where it is not smooth.

Solution

If a curve is not smooth at \(t=t_0\text{,}\) it means that \(x'(t_0)=y'(t_0)=0\) as defined. This, in turn, means that rate of change of \(x\) (and \(y\)) is 0; that is, at that instant, neither \(x\) nor \(y\) is changing. If the parametric equations describe the path of some object, this means the object is at rest at \(t_0\text{.}\) An object at rest can make a “sharp” change in direction, whereas moving objects tend to change direction in a “smooth” fashion.

One should be careful to note that a “sharp corner” does not have to occur when a curve is not smooth. For instance, one can verify that \(x=t^3\text{,}\) \(y=t^6\) produce the familiar \(y=x^2\) parabola. However, in this parametrization, the curve is not smooth. A particle traveling along the parabola according to the given parametric equations comes to rest at \(t=0\text{,}\) though no sharp point is created.

Our previous experience with cusps taught us that a function was not differentiable at a cusp. This can lead us to wonder about derivatives in the context of parametric equations and the application of other calculus concepts. Given a curve defined parametrically, how do we find the slopes of tangent lines? Can we determine concavity? We explore these concepts and more in the next section.

Subsection9.2.3Exercises

In the following exercises, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

In the following exercises, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

In the following exercises, four sets of parametric equations are given. Describe how their graphs are similar and different. Be sure to discuss orientation and ranges.

In the following exercises, eliminate the parameter in the given parametric equations.

In the following exercises, eliminate the parameter in the given parametric equations. Describe the curve defined by the parametric equations based on its rectangular form.

In the following exercises, find parametric equations for the given rectangular equation using the parameter \(\ds t=\frac{dy}{dx}\text{.}\) Verify that at \(t=1\text{,}\) the point on the graph has a tangent line with slope of 1.

In the following exercises, find the values of \(t\) where the graph of the parametric equations crosses itself.

In the following exercises, find the value(s) of \(t\) where the curve defined by the parametric equations is not smooth.

In the following exercises, find parametric equations that describe the given situation.