2 Parametric representation of a function

Suppose we write x and y in terms of t in the form

x = 4 t y = 2 t 2 ,  for  − 1 ≤ t ≤ 1 (1)

For different values of t between − 1 and 1, we can calculate pairs of values of x and y . For example when t = 1 we see that x = 4 ( 1 ) = 4 and y = 2 × 1 2 = 2 . That is, t = 1 corresponds to the point with ( x , y ) coordinates ( 4 , 2 ) .

A table of values is given in Table 3.

Table 3

t − 1 − 0.5 0 0.5 1
x − 4 − 2 0 2 4
y 2 0.5 0 0.5 2

If the resulting points are plotted on a graph then different values of t correspond to different points on the graph. The graph of (1) is plotted in Figure 9.

Figure 9 :

{ Graph of the function defined parametrically by $x=4t$, $y=2t^2$, $-1\le t\le 1$}

It is often possible to convert a parametric representation of a function into the more usual form by combining the two expressions to eliminate the parameter. Thus if x = 4 t we can write t = x 4 and so

y = 2 t 2 = 2 x 4 2 = 2 x 2 16 = x 2 8

Using y = x 2 8 we can, by giving x values, find corresponding values of y . Plotting these ( x , y ) values gives, of course, exactly the same curve as in Figure 9.

Task!

Consider the function x = 1 2 t + 1 t , y = 1 2 t − 1 t , 1 ≤ t ≤ 8 .

  1. Draw up a table of values of this function.
  2. Plot a graph of the function
t 1 2 3 4 5 6 7 8
x 1 1.25 1.67 2.13 2.60 3.08 3.57 4.06
y 0 0.75 1.33 1.88 2.40 2.92 3.43 3.94

It is possible to eliminate t between the two equations so that the original parametric form can be expressed as x 2 − y 2 = 1 .

Task!

A particle with mass m falls under gravity so that at time t its distance from the y -axis is 2 t and its distance from the x -axis is − m g t 2 2 + 3 where g is a constant (the acceleration due to gravity). Find the value of t when the particle crosses the x -axis and, at this time, find the distance from the y -axis.

Begin by obtaining the parametric equations of the path of the particle:

x = 2 t y = − m g t 2 2 + 3

Now find the value of t when y = 0 :

t = 6 ∕ ( m g ) Finally, obtain the value of x at this value of t :

x = 2 6 ∕ ( m g )

Exercises
  1. Explain what is meant by the term ‘parameter’.
  2. Consider the parametric equations x = t , y = t , for t ≥ 0 .
    1. Draw up a table of values of t , x and y for values of t between 0 and 10.
    2. Plot a graph of this function.
    3. Obtain an explicit equation for y in terms of x .

2. (c) y = x 2 , 0 ≤ x ≤ 10