Parametric Representation of Functions of x

The graph of any function of x  can be given parametrically. For example the function  can be described parametrically by setting  and . The following command plots the graph given parametrically by   and  within the window in which x  varies from -5 to 5 and y  varies from -20..20.

plot([t,t^3,t=-5..5],x=-5..5,y=-20..20);

More generally the graph of  can be described parametrically by setting  and . It is often convenient to think of the parameter t  representing time. Then  represents the x -coordinate of a moving point at time t  and  represents the y -coordinate of the point at time t .

Recall that if a function  is one-one, then it has an inverse. The graph of the inverse function is obtained by reflecting the graph of  across the line . When a curve is described parametrically by    and , it is easy to generate its reflection across the line  by simply interchanging the functions  and .

>	plot([t^3,t,t=-5..5],x=-5..5,y=-5..5);

What function does the above graph represent?

To plot more than one curve parametrically on the same set of axes, simply list the parametric descriptions of the curves within curly brackets "{}" and separated by commas.

>	plot({[t,t^3,t=-5..5],[t^3,t,t=-5..5]},x=-10..10,y=-10..10,scaling=constrained);

Exercise

Plot the graph of  using parametric equations. Discuss whether this function is one-one. Then plot the reflection of this graph across the line  together with the graph of  on the same coordinate system. Is the reflected curve the graph of a function of x ? Explain. Determine the coordinates of the point of intersection of the two graphs by an appropriate use of the fsolve  command.

Parametric Representation of Circles and Ellipses.

Many curves that are graphs of a function of x can be described by parametric equations. One of the most basic examples is the unit circle, the circle centered at (0,0) with radius 1. We let  and , with the parameter t  varying from 0 to .

>	plot([cos(t),sin(t),t=0..2*Pi],x=-2..2,y=-2..2,scaling=constrained);

>

Run the following animation and determine whether the circle is traversed clockwise or counterclockwise as t  varies from 0 to .

Modify the above plot so that the circle has radius 0.6.

>

>

To shift a curve h  units horizontally, simply add h  to . If h  is positive, the shift is to the right  while if h  is negative, the shift is to the left . To shift a curve k  units vertically, simply add k   to . If k  is positive the shift is upwards  while if k  is negative the shift is downwards .

Exercise

Let h  denote the number of letters in your first name and k  denote the number of letters in your last name. Plot the circle of radius 1.5 centered at ( ) using a parametric plot.

>

>

The parametric representation of the circle can be easily modified to generate an ellipse.

>	plot([3*cos(t),1.5*sin(t),t=0..2*Pi],x=-4..4,y=-2..2,scaling=constrained);

>

Ellipses have two axes - a major axis  and a minor axis. The major axis is always the larger of the two axes. In the above example the major axis is the segment from the point (-3,0) to the point (3,0). In this case the length of the major axis is 6. The minor axis is the segment from the point (0,-1.5) to the point (0,1.5). In this case the length of the minor axis is 3. If the major axis is parallel to the x -axis, the ellipse is longer horizontally. If the major axis is parallel to the y -axis, the ellipse is longer vertically.

Exercise

Plot the ellipse centered at ( h,k ) having a vertical major axis of length 10 and a horizontal minor axis of length 7.

>

>

Parametric Representation of Lines

Since any non-vertical line is the graph of a function of x , any line can be represented parametrically. As an example, consider the line through the point (2,3) with slope -2.5. By the point-slope formula, an equation for this line is .

>	plot([t,-2.5*(t-2)+3,t=0..1],x=-5..5,y=-5..5,scaling=constrained);

Nothing appears in the window. Experiment with different ranges for t  until the line appears in the window. In order to avoid this type of problem, it would be convenient if the value t  = 0 corresponded to the point (2,3). We can accomplish this by setting . With this substitution, the equation of the line gives  or .

>	plot([2+t,3-2.5*t,t=-1..1],x=-5..5,y=-5..5);

In general, a parametric representation of the line through (a,b) with slope m  is given by setting  and . This representation of the line has the additional feature that  corresponds to the point ( ). This will be useful below when plotting the graph of a tangent line to a curve given parametrically.

Project

A parametric representation of a curve can be interpreted as a particular motion along the curve. More specifically the parametric representation contains information as to where a point is on the curve at time t . The same curve can have many different representations. For example, the unit circle is represented parametrically by  and  as t  varies from 0 to . The unit circle is also represented parametrically by  and  as t  varies from 0 to .

>	plot([cos(2*t),sin(2*t),t=0..Pi],x=-2..2,y=-2..2,scaling=constrained);

In the second case the circle is traversed once as t  varies from 0 to . If we interpret these representations as motion along the circle, then in a certain sense the second case represents motion that is twice as fast as the first case since a point covers the same path in half the time. When a curve is represented parametrically, it makes sense to talk about how fast a point is moving at a particular value of t . In other words we can discuss the speed  of a particle moving along the curve at time t . When motion along a curve C  is represented parametrically by x( t ) and y( t ), the speed of a particle at ( ) is defined by

>

restart:

For the standard representation of the unit circle we have

>	x1:=t->cos(t);y1:=t->sin(t);

>	speed1:=t->sqrt((D(x1)(t))^2+D(y1)(t)^2);

>	speed1(t);

>	simplify(speed1(t));

Notice that in this case the speed of the particle is constant or independent of t . In other words the standard representation of the unit circle represents motion around the unit circle at a constant speed of 1.

For the second representation of the unit circle we have

>	x2:=t->cos(2*t);y2:=t->sin(2*t);

>	speed2:=t->sqrt((D(x2)(t))^2+D(y2)(t)^2);

>	speed2(t);

>	simplify(speed2(t));

Again the speed is independent of t , but now the speed is 2 instead of 1. Thus the second parametric representation of the unit circle represents motion around the unit circle at a constant speed of 2.

Motion along a Circle

Show that  and  with t  varying between 0 and 1 is a parametric representation motion along the unit circle, but that the speed is not constant. Specifically determine the speed of a particle at times t  = 0.25, t  = 0.5 and t  = 0.75. Verbally describe the motion along the circle.

>

restart:

>

>

>

>

The Bug Revisited

Consider the path of the bug at the beginning of this lab. The path of the bug was given parametrically by  and  as t  varies from 0 to 10. Determine the speed of the bug at times t  = 2, t  = 4, t  = 6, t  = 8 and t  = 10. Is the speed of the bug constant? Find a formula for the speed of the bug at time t . Plot the speed function as a function of t  for t from 0 to 10. Use this plot to describe the speed of the bug. Estimate the maximum and minimum

>

restart:

>

>

>

>

Path of Points on a Wheel

Suppose a tire on a car has a radius of 15 inches or 1.25 feet and the tire rotates as the car travels on a horizontal road. Suppose a point on the edge of the tire is marked with a small dot of paint. What path does this point on the edge of the tire trace out as the car moves. The path is known as a cycloid  and is most easily described parametrically. It can be shown that this path is given parametrically by  and . Note that when t  = 0, the marked point is at the origin and when , the wheel has made one complete revolution and the marked point is at the bottom of the wheel again. Since the wheel has made one revolution, the marked point will be at a distance equal to the circumference of the wheel down the road from where it started.

Execute the following set of commands to generate an animation of the rolling wheel.

>

restart:with(plots): s:=plot([1.25*(t-sin(t)),1.25*(1-cos(t)),t=0..4*Pi],x=0..6*Pi,scaling=constrained): r:=t->plot({[1.25*(2*Pi/6)*t+1.25*cos(w),1.25+1.25*sin(w),w=0..2*Pi]},x=-2..5*Pi,color=blue,scaling=constrained): q:=t->plot([[1.25*((2*Pi/6)*t-sin((2*Pi/6)*t)),1.25*(1-cos((2*Pi/6)*t))]],style=point,symbol=circle,color=black): z:=t->display({s,r(t),q(t)}): display(seq(z(t),t=0..12),insequence=true,scaling=constrained);

>

In the above animation the wheel is turning at the rate of 1 radian per unit of time. Determine the speed of the marked point on the rim of the wheel when the when the wheel has made a quarter of a revolution. Determine the speed of the marked point on the rim of the wheel when the wheel has made half of a complete revolution. From the graph of the path of the marked point on the wheel, it appears that the path does not have a unique tangent line at the points where the marked point is at the bottom of the wheel. Explain why this is so.

Suppose that instead of marking a point on the rim of the wheel, we mark a point that is at a distance p  from the center of the wheel. Then the path of such a point can also be described parametrically. The equations are  and . Again as t  varies from 0 to , the wheel makes one revolution. Plot the paths of such marked points for p  = 0.2, 0.5, 0.75 and 1.0

>	plot({seq([1.25*t-p*sin(t),1.25-p*cos(t),t=0..4*Pi],p=[0.2,0.5,0.75,1.0])},x=0..6*Pi,y=0..2.5,scaling=constrained);

>

How do the paths vary as the marked point moves from being close to the the center of the the wheel to being close to the rim of the wheel? How do the speeds of the four different marked points compare when the wheel has made a quarter of a turn. Do the paths of a point strictly inside the wheel have tangent lines when the marked point is at its lowest position? Explain.

You can create an animation of the wheel turning with the marked point being at a distance p  from the center. Here p  should be a positive number that is less than 1.25. To create different animations simply change the value of p  in the second line and then execute the list of commands.

>

restart:with(plots): p:=1.0: s:=plot([1.25*t-p*sin(t),1.25-p*cos(t),t=0..4*Pi],x=0..6*Pi,y=0..2.5,scaling=constrained): r:=t->plot([[1.25*(2*Pi/6)*t+p*cos(w),1.25+p*sin(w),w=0..2*Pi],[1.25*(2*Pi/6)*t+1.25*cos(w),1.25+1.25*sin(w),w=0..2*Pi]],x=-2..5*Pi,color=[green,blue],scaling=constrained): q:=t->plot([[1.25*2*Pi/6*t-p*sin((2*Pi/6)*t),1.25-p*cos((2*Pi/6)*t)]],style=point,symbol=circle,color=black): z:=t->display({s,r(t),q(t)}): display(seq(z(t),t=0..12),insequence=true,scaling=constrained);

>

Extra Credit - Tangent Lines and Parametric Plots

A basic question in calculus is to determine the equation of the tangent line to a curve at a point on the curve when the curve is given parametrically. Suppose a curve C  is given parametrically by  and  where both  and  are differentiable with respect to t . To determine the tangent line we need to determine  at a specific point (a,b) on the curve C . Suppose (a,b) correspones to a specific value of t, say . Specifically, this means  and . In order to determine  we use the Chain Rule.  We have

 =  /

To determine  at the point (a,b), we evaluate  and  at . This formula for  is only valid when  is non-zero.

As an example we consider the circle of radius 1 centered at the origin with (a,b) = ( , ). If we represent the circle parametrically with  and  with t  ranging between 0 and . Then the point (a,b) corresponds to .

>	x1:=t->cos(t);y1:=t->sin(t);

>	slope:=t->D(y1)(t)/D(x1)(t);

>	slope(Pi/3);

The tangent line at (a,b) is the line through ( , ) with slope . We represent this line parametrically by setting  and   .

>	plot({[cos(t),sin(t),t=0..2*Pi],[1/2+t,sqrt(3)/2-(sqrt(3)/3)*t,t=-1..1]},x=-3..3,y=-3..3,scaling=constrained);

Exercise

Consider the ellipse given parametrically by the equations  and  as t  varies from 0 to ..

1. Determine the equation of the line tangent to the graph of this curve when t  = .

2. Determine parametric equations for the tangent line when t  = .

3. Plot both the ellipse and the tangent line on the same set of coordinate axes.

4. As t  varies from 0 to  is the motion around the ellipse clockwise or counterclockwise. Explain your answer.