A diagram containing mostly text, that outlines the process for plotting a function. The steps illustrated are as follows:
Choose
Use function to find ()
Plot point
This is not really indicative of how a person would sketch a graph by hand (better techniques for doing so were covered in Section 3.5), but it is not far off from how a computer might generate a plot, or how a student who is first learning about functions might use a table of values to create a plot.
The rectangular equation 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.)
This is another text-based diagram, which illustrates the method for plotting curves that will be covered in this section. In this method, the coordinates and are both defined as functions of a third variable, . Four phrases are arranged in a diamond shape, with text left, right, top, and bottom.
To the left of the diagram is the text, “Choose ”. An arrow points up and to the left from this text to the phrase “Use a function to find ()”, which is located at the top of the diagram. Another arrow points down and to the left from “Choose ” to the phrase “Use a function to find ()”, which is located at the bottom of the diagram.
From the phrases at the top and bottom of the diagram there are two more arrows pointing to the right, to a final phrase, which states, “Plot point ”.
Let and be continuous functions on an interval . The set of all points in the Cartesian plane, as varies over , is the graph of the parametric equations and , where 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.
We plot the graphs of parametric equations in much the same manner as we plotted graphs of functions like : we make a table of values, plot points, then connect these points with a “reasonable” looking curve. Figure 10.2.6.(a) shows such a table of values; note how we have 3 columns.
The points from the table are plotted in Figure 10.2.6.(b). The points have been connected with a smooth curve. Each point has been labeled with its corresponding -value. These values, along with the two arrows along the curve, are used to indicate the orientation of the graph. This information helps us determine the direction in which the graph is “moving.”
(a)
The parametric curve obtained from the table of values in Figure 10.2.6.(a) is shown. On a set of coordinate axes, the points corresponding to the values are plotted. These are, respectively, .
The curve joining these points is plotted; it appears to be a parabola opening to the right, with its vertex at the point . Arrow heads on the curve indicate a direction of travel corresponding to an increasing value of .
(b)
Figure10.2.6.A table of values of the parametric equations in Example 10.2.5 along with a sketch of their graph
We often use the letter as the parameter as we often regard 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 the position is and at time the position is .”).
The parametric curve obtained from the table of values in Figure 10.2.8.(a) is shown. The curve is very similar in appearance to the one in Figure 10.2.6.(b); again it appears to be a parabola opening to the right, with its vertex at the point .
The primary difference is that the marked points now correspond to parameter values . The direction of travel is opposite to that in Figure 10.2.6.(b).
(b)
Figure10.2.8.A table of values of the parametric equations in Example 10.2.7 along with a sketch of their graph
It is not difficult to show that the curves in Examples 10.2.5 and 10.2.7 are portions of the same parabola. While the parabola is the same, the curves are different. In Example 10.2.5, if we let vary over all real numbers, we’d obtain the entire parabola. In this example, letting vary over all real numbers would still produce the same graph; this portion of the parabola would be traced, and re-traced, infinitely many times. The orientation shown in Figure 10.2.8 shows the orientation on , but this orientation is reversed on .
These examples begin to illustrate the powerful nature of parametric equations. Their graphs are far more diverse than the graphs of functions produced by “” functions.
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 - and -values), along with the values of that are to be plotted. The user is often prompted to give a minimum, a maximum, and a “-step” or “.” Graphing utilities effectively plot parametric functions just as we’ve shown here: they plots lots of points. A smaller -step plots more points, making for a smoother graph (but may take longer). In Figure 10.2.6, the -step is 1; in Figure 10.2.8, the -step is .
One nice feature of parametric equations is that their graphs are easy to shift. While this is not too difficult in the “” context, the resulting function can look rather messy. (Plus, to shift to the right by two, we replace with , which is counter-intuitive.) The following example demonstrates this.
The graph of the parametric equations is given in Figure 10.2.10.(a). It is a parabola with a axis of symmetry along the line ; the vertex is at .
In order to shift the graph to the right 3 units, we need to increase the -value by 3 for every point. The straightforward way to accomplish this is simply to add 3 to the function defining :. To shift the graph down by 2 units, we wish to decrease each -value by 2, so we subtract 2 from the function defining :. Thus our parametric equations for the shifted graph are ,. This is graphed in Figure 10.2.10.(a). Notice how the vertex is now at .
A plot of the curve is shown. The shape of the curve is similar to that of a parabola, but slightly distorted.
The curve begins near the point , and then descends to a intercept at . It briefly enters the second quadrant before passing through the origin into the fourth quadrant. A small portion of the curve lies in the fourth quarant before it crosses the axis at ; after this point, it continues into the first quadrant, exiting the image near the point .
(a)
The curve in this image is identical to the one beside it in Figure 10.2.10.(a); the only difference is that it has been shifted 3 units to the right, and 2 units down.
(b)
Figure10.2.10.Illustrating how to shift graphs in Example 10.2.9
Because the - and -values of a graph are determined independently, the graphs of parametric functions often possess features not seen on “” type graphs. The next example demonstrates how such graphs can arrive at the same point more than once.
Using the methods developed in this section, we again plot points and graph the parametric equations as shown in Figure 10.2.12. It appears that the graph crosses itself at the point , but we’ll need to analytically determine this.
The curve in this image could describe the path of a fly that turns sharply in a loop, before heading off in another direction. It begins in the second quadrant, traveling down and to the right, and crossing the axis at approximately .
The curve continues in this direction until it turns sharply near the point . The curve then travels to the left and bends upwards, crossing itself at the point . After this point of self-intersection, the curve continues in the first quadrant, traveling up and to the right.
Figure10.2.12.A graph of the parametric equations from Example 10.2.11
We are looking for two different values, say, and , where and . That is, the -values are the same precisely when the -values are the same. This gives us a system of 2 equations with 2 unknowns:
Solving this system is not trivial but involves only algebra. Using the quadratic formula, one can solve for in the second equation and find that . This can be substituted into the first equation, revealing that the graph crosses itself at and . We confirm our result by computing and .
Subsection10.2.2Converting between rectangular and parametric equations
It is sometimes useful to rewrite equations in rectangular form (i.e., ) into parametric form, and vice-versa. Converting from rectangular to parametric can be very simple: given , the parametric equations , produce the same graph. As an example, given , the parametric equations , produce the familiar parabola. However, other parametrizations can be used. The following example demonstrates one possible alternative.
We start by computing :. Thus we set . We can solve for and find . Knowing that , we have . Thus parametric equations for the parabola are
.
To find the point where the tangent line has a slope of , we set . This gives the point . We can verify that the slope of the line tangent to the curve at this point indeed has a slope of .
Example10.2.14.Converting from rectangular to parametric.
An object is fired from a height of 0 feet and lands 6 seconds later, 192 feet away. Assuming ideal projectile motion, the height, in feet, of the object can be described by , where is the distance in feet from the initial location. (Thus feet.) Find parametric equations , for the path of the projectile where is the horizontal distance the object has traveled at time (in seconds) and is the height at time .
Physics tells us that the horizontal motion of the projectile is linear; that is, the horizontal speed of the projectile is constant. Since the object travels 192 ft in 6 s, we deduce that the object is moving horizontally at a rate of 32 ft⁄s, giving the equation . As , we find . We can quickly verify that ft⁄ft2, the acceleration due to gravity, and that the projectile reaches its maximum at , halfway along its path.
These parametric equations make certain determinations about the object’s location easy: 2 seconds into the flight the object is at the point . That is, it has traveled horizontally 64 ft and is at a height of 128 ft, as shown in Figure 10.2.15.
The curve is a parabola that opens downward from a vertex at . It illustrates the path of a projectile that is fired from the origin and travels up and to the right before reaching its maximum height at the vertex of the parabola. It then travels back down, returning to the axis at the point .
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 and that does not involve the parameter .
There is not a set way to eliminate a parameter. One method is to solve for in one equation and then substitute that value in the second. We use that technique here, then show a second, simpler method.
Starting with , solve for :. Substitute this value for in the equation for :
.
Two lines are plotted. The first is the line , which is drawn from the point to the point .
A second, thicker line is drawn over the first line, from to . At there is a “hollow” (white-filled) dot, indicating that the second line approaches, but does not reach, this point. At there is a solid dot, indicating the fact that the parametric version of the line reaches this point, but does not go past it.
Figure10.2.17.Graphing parametric and rectangular equations for a graph in Example 10.2.16
Thus . One may have recognized this earlier by manipulating the equation for :
.
This is a shortcut that is very specific to this problem; sometimes shortcuts exist and are worth looking for.
We should be careful to limit the domain of the function . The parametric equations limit to values in , thus to produce the same graph we should limit the domain of to the same.
The graphs of these functions is given in Figure 10.2.17. The portion of the graph defined by the parametric equations is given in a thick line; the graph defined by with unrestricted domain is given in a thin line.
We should not try to solve for in this situation as the resulting algebra/trig would be messy. Rather, we solve for and in each equation, respectively. This gives
and .
The Pythagorean Theorem gives , so:
The plot shows an ellipse. The major axis runs from to , and the minor axis runs from to .
Figure10.2.19.Graphing the parametric equations , in Example 10.2.18
This final equation should look familiar — it is the equation of an ellipse! Figure 10.2.19 plots the parametric equations, demonstrating that the graph is indeed of an ellipse with a horizontal major axis and center at .
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.
Figure 10.2.21 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.
A plot of the parametric curve ,, known as an astroid. The curve has four cusps, at the points , and . The appearance is not unlike what one would get if one held these four points fixed on the unit circle, and then bent the rest of the circle so that it curves inward (toward the origin) rather than outward.
(a)Astroid where and
The plot of the curve resembles a flower. It consists of eight loops, equal in size and shape, each passing through the origin. The shape of each loop resembles a narrow tear-drop.
(b)Rose Curve where and
The curve , known as a hypotrochoid, is shown. The curve is symmetric about the axis, and resembles a five-pointed star, with one point on the axis at .
The points of the star are rounded, rather than sharp. The curve passes through itself several times, in a manner similar to how a pentagram is drawn.
(c)Hypotrochoid where and
The epicycloid is shown. The curve travels once around the origin, and consists of three arcs, with each pair of arcs meeting at a cusp.
The effect is not unlike the appearance of a three-leaf clover, if the leaves were much more puffy, and further from the center.
(d)Epicycloid where and
Figure10.2.21.A gallery of interesting planar curves
A curve defined by , is smooth on an interval if and are continuous on and not simultaneously 0 (except possibly at the endpoints of ). A curve is piecewise smooth on if can be partitioned into subintervals where is smooth on each subinterval.
We see at both and are 0; thus is not smooth at , corresponding to the point . The curve is graphed in Figure 10.2.24, illustrating the cusp at .
A plot of the curve is shown. Despite the fact that and are both differentiable functions of , there is a sharp cusp at the point corresponding to , which happens to be the point where the derivatives of and with respect to are simultaneously zero.
The appearance of the curve is not unlike that of a set of tweezers.
Figure10.2.24.Graphing the curve in Example 10.2.23; note it is not smooth at
If a curve is not smooth at , it means that as defined. This, in turn, means that rate of change of (and ) is 0; that is, at that instant, neither nor is changing. If the parametric equations describe the path of some object, this means the object is at rest at . 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 , produce the familiar 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 , 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.
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. 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 . Verify that at , the point on the graph has a tangent line with slope of 1.