Parametric Equations

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 sets of equations:

1. x(t) = t,       y(t) = 2t + 1 
   

2. 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 the functions            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

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

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)  =  2 cos(t²)     and     y  =  2 sin(t²)

now the circle begins slowly and speeds up.

A Cool Example

The graph of 

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

is pictured below:

       

Eliminating the Parameter

If a curve is given by parametric equations, we often are interested in finding an equation for the curve in standard form:  

        y = f(x)

Example

Consider the parametric equations

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

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


        t =  

hence

        y = sin()

is the equation.

Example

Eliminate the parameter for

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


Solution

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 two intersection points

       

 

hence their graphs intercept.  Their graphs are shown on the right.

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