Assignment 10

Jonathan Lawson

This exploration will look at parametric equation.  "A parametric equation is an equation that introduces a third variable, t, which is called a parameter.  By writing both x and y as functions of t, you obtain a parametric equation." -- This paragraph was taken from Calculus by Larson, et.al., 5th edition.

Now let us consider the parametric equation

x = a + t

y = b + kt

We will start off by letting a = b = 0 and look at the two different values for k.

Ex.

x = t and y = 3t

graph1

Ex.

x = t and y = -2t

graph2

Graphing these equations for t > 0 we see that the equations start at the origin and are linear equations with k equal to the slopes of the lines.

If we solve for t in terms of x in the original equation and plug that value into the y function we get

t = x - a, so

y = b + kt

y = b + k(x - a)

y = kx + (b - ka)

We can see that this parametric equation is actually a linear function with slope k and y-intercept (b - ka).  Here is an example of another parametric equation:

Ex.

x = 3 + t

y = -1 + 2t

graph3