Parametric Equations

I.  Quiz

II.  Homework

III.  Lines

Recall that a line has equation y = mx + b.  Suppose that one airplane moves along the line y = 2x + 3, while the other airplane moves along the line y = 3x - 2.  Can you tell whether the airplanes collide?  Even though the lines intersect, the equations themselves do not tell us whether there will be a mid air collision.  To be able to mathematically model this scenario, we use parametric equation.  We introduce the variable t for time and write x and y as a function of t.  

Consider the two equations:

A)  x(t) = t, y(t) = 2t + 1 and

B)  x(t) = 2t, y(t) = 4t + 1.

These describe the same line, but the second one travels twice as fast.

Definition:  A curve given by x = x(t), y = y(t) is called a parametrically defined curve and x = x(t) and y = y(t) are called the parametric equations for the curve.

Finding the parametric equations for a line given two points

Example:  Find the parametric equations for the line through the points (3,2) and (4,6) so that when t = 0 we are at the point (3,2) and when t = 1 we are at the point (4,6).

Solution:  We write symbolically:

(x,y) = (1 - t)(3,2)+ (t)(4,6) = (3 - 3t + 4t,2 - 2t + 6t) = (3 + t,2 + 4t)

so that x(t) = 3 + t and y(t) = 2 + 4t

IV.  Functions

if y = f(x) is a function of x we can write parametric equations by writing

x = t and y = f(t).

Example:  The parabola  y = x²  can be represented by the parametric equations:

x = t and y = t²

V.  Circles

Consider the circle centered at (0,0) with radius 2.  We can write it parametrically as

x(t) = 2cos(t) and y = 2sin(t)

We see that the circle is drawn in a counterclockwise direction.

We can draw the same circle as

x(t) = 2cos(-t) and y(t) = 2sin(-t)

now the circle is drawn clockwise.

We can also write

x(t) = 2cos(t²) and y = 2sin(t²)

now the circle begins slowly and speeds up.

VI.  A Cool Example

We will draw x(t) = 11cost - 6cos(11/6 t) and y(t) = 11sin(t) - 6sin(11/6 t)

VII.  Eliminating the Parameter:

Example:  

Consider the parametric equations

x(t) = t² and y(t) = sint(t)  for  t > 0

To find the conventional form of the equation we solve for t:

t = sqrt(x) hence

y = sin(sqrt(x))

is the equation.

Example:  Eliminate the parameter for

x(t) = e^t and y(t) = e^2t + 1

We write:

y(t) = (e^t)² + 1

Hence y = x² + 1

Intersections

Let x₁(t) = 2t + 1 and y₁(t) = 4t²  

and x₂(t) = 3t and y₂(t) = 3t

Do they intersect?

If so then there is a c with

2c + 1 = 3c and 4c²  = 3c

the first equation gives us that c = 1, Putting this into the second equation we have

4 = 3 which tells us that they do not intersect.

Do their graphs intersect?

If so then there exists a c and a k such that

2c + 1 = 3k and 4c²  = 3k

Hence we see that

2c + 1 = 4c²  

or that

4c²  - 2c + 1 = 0.  

We solve to get tow intersection points

c = .5 + sqrt(13)/6 and .5 - sqrt(13)/6 hence they intercept.