#21: PARAMETRIC EQUATIONS

The graph of an equation of the form y = f (x), is the set of all points (x, f (x)) for every value of x in the domain of the function. The graph of an equation of the form y = y(t), x = x (t) is the set of all points (x(t) , y(t)) for every value of t in the domain of x (t) and y(t). The second form is call a parametric form of a curve C in the xy- plane. The parametric equation of a particular graph is not unique. However, in many physical applications, the parameter t has a physical meaning and is important to the application. Therefore, the graph of the parametric equations is considered to be just the set of points (x(t) , y(t)) while the curve C is considered to be the graph and the information about the parameter. This information can be included with the graph by labeling points on the graph with their t value and by putting an arrow on the graph denoting increasing values of t.

1. Let x (t) = t 1 / 3 and y (t) = t - 3.

  1. Complete the following table.
    t : -3 -2 -1 0 1 2 3
    x(t) :              
    y(t) :              

  2. Draw the graph of the parametric equations. Include the t values for the points in the table and an arrow indicating increasing values of t.


2. Let x(t) = 4 sin ( t / 3 ) and y(t) = 2 cos ( t / 3 ) .

  1. Complete the following table.
    t : -3 -2 -1 0 1 2 3
    x(t) :              
    y(t) :              

  2. Draw the graph of the parametric equations. Include the t values for the points in the table and an arrow indicating increasing values of t.


Notice that the graph you get for a parametric equation is not necessary one that is a function of x. This is one of the strengths of the parametric form, you can work with relations as well as functions and uniquely determine the point of interest by giving its parameter value.

3. Let x(t) = t + 1 and y(t) = 3t + 1.

  1. Complete the following table.
    t : -3 -2 -1 0 1 2 3
    x(t) :              
    y(t) :              

  2. Draw the graph of the parametric equations. Include the t values for the points in the table and an arrow indicating increasing values of t.


4. Let x(t) = t 2 + 1 and y(t) = 3t 2 + 1.

  1. Complete the following table.
    t : -3 -2 -1 0 1 2 3
    x(t) :              
    y(t) :              

  2. Draw the graph of the parametric equations. Include the t values for the points in the table and an arrow indicating increasing values of t.


5. Let x(t) = sin ( t / 2 ) +1 and y(t) = 3 sin ( t / 2 ) +1.

  1. Complete the following table.
    t : -3 -2 -1 0 1 2 3
    x(t) :              
    y(t) :              

  2. Draw the graph of the parametric equations. Include the t values for the points in the table and an arrow indicating increasing values of t.


6. HOMEWORK #1: Describe the graphs of above three parametric equations in such a way that a 10 year old could draw the graphs from your description. Why are the graphs different?



ELIMINATING THE PARAMETER

It is sometimes possible to eliminate t from the two equations x = x(t) and y = y(t) to obtain the equation of the graph in terms of x and y alone. It is important to include any restrictions on x and y the parametric form puts on these variables.

7. Let x (t) = t 1 / 3 and y (t) = t - 3

  1. Determine what restrictions, if any, there are for x and y as t varies through the real numbers.

  2. Solve for t in one of the equations and substitute it into the other.

  3. Graph the function.


8. Let x(t) = 4 sin ( t / 3 ) and y(t) = 2 cos ( t / 3 ) .

  1. Determine what restrictions, if any, there are for x and y as t varies through the real numbers.

  2. Use a trigonometric identity to elliminate t from the two equations

  3. Graph the function.


9. Let x(t) = t + 1 and y(t) = 3t + 1.

  1. Determine what restrictions, if any, there are for x and y as t varies through the real numbers.

  2. Solve for t in one of the equations and substitute it into the other.

  3. Graph the function.


10. Let x(t) = t 2 + 1 and y(t) = 3t 2 + 1.

  1. Determine what restrictions, if any, there are for x and y as t varies through the real numbers.

  2. Solve for t 2 in one of the equations and substitute it into the other.

  3. Graph the function.


11. HOMEWORK #2:Let x(t) = sin ( t / 2 ) +1 and y(t) = 3 sin ( t / 2 ) +1.

  1. Determine what restrictions, if any, there are for x and y as t varies through the real numbers.

  2. Solve for t in one of the equations and substitute it into the other.

  3. Graph the function.






A smooth curve described by y = y(t), x = x (t) is one where

A piece-wise smooth curve is one that satisfies the definition of a smooth curve except at a finite number of points.





12. HOMEWORK #3: Which of the curves described in the above problems are smooth curves? If a curve is not smooth, state the values of t that violate the definition.



WRITING PARAMETRIC EQUATIONS

Since the parametric form of a graph is not unique, there is no straight forward method to go from a graph to a parametric representation. Some suggestions are given below.

13. If y = f (x), let x = t and y = f (t). Find a parametric representation for y = x / ( x + 1 )

14. If x = g(y), let y = t and x = g(t). Find a parametric representation for x = y 2.

15. The straight line through the points (x 1, y 1) and (x 2, y 2) can be written as x(t) = x 1 + t ( x 2 -x 1) and y(t) = y 1 + t ( y 2 - y 1 ). Find a parametric representation for a straight line through the points (2,-1) and (3,4)

16. Use sin 2 t + cos 2 t = 1 for a circle or an ellipse. Find a parametric representation for ( x - 2 ) 2 / 4 + ( y + 1 ) 2 / 9 = 1.

17. HOMEWORK #4: Use tan 2 t + 1 = sec 2 t for a hyperbola. Find a parametric representation for ( x - 2 ) 2 / 4 - ( y + 1 ) 2 / 9 = 1.

18. Use geometry. Many interesting parametric curves can be developed by looking at the coordinates of a point on a circle which rolls along inside or outside another fixed line or circle. Use geometry to find the equation of the epicycloid generated by rolling, without slipping, a circle of radius B around the outside of a fixed circle of radius A.

  1. Draw a picture

  2. You will use the central angle, t, of the fixed circle measured from the x-axis to the radius drawn to the point of tangency of the two circles. Also, you will assume the outer circle starts out tangent to the fixed circle at the point (A,0). Find the arc length on the fixed circle from (A,0) to the point of tangency as a function of t.

  3. Since the outer circle does not slip, the point P has moves this same distance from the point of tangency along the outer circle. Find the central angle in the outer circle from the point of tangency to point P.

  4. Use the two right triangles in the figure to calculate the coordinates of the point P as a function of t.

  5. HOMEWORK #5: Draw two separate graphs, one for A = 3, B = 1 and a second for A = , B = 1.




HOMEWORK ASSIGNMENT:

REQUIRED PROBLEMS #1, 2, 3, 4, 5; Section 9.2: 10, 18, 26
SUGGESTED PROBLEMS Section 9.2: 7, 15, 23, 29, 35, 37, 45, 53, 65-69
Some useful web sites:



READING ASSIGNMENT BEFORE NEXT WORKSHEET: Section 9.3

rbyrne@math.cudenver.edu
 ROXANNE BYRNE :UNIVERSITY OF COLORADO AT DENVER: ©:1999, Roxanne Byrne