Parametric curves
When given separate expressions for the variables x and y in terms of a common variable, that variable (t for example) is known as the parameter and the expressions are parametric expressions.
You can use the parametric expressions to eliminate the common parameter (get an equation in terms of x and y only), and then sketch the resulting curve in an x,y plane corresponding to intervals of t - this is a parametric curve.
This curve has direction and orientation dictated as t increases over the interval.
From equation to curve
Sketch the curve defined by x = t^2 and y = t - 2 where -2 <= t <= 1.
y = t-2 => t = y+2
x = t^2 => x = (y+2)^2 // notice this is an equation of a parabola
use t interval [-2, 1]:
t x=t^2 y=t-2
-2 4 -4
-1 1 -3
0 0 -2
1 1 -1
sketch these points and use arrows to show the direction of the curve as t increases
From curve to equation
Express either x or y and then set the other variable equal to t for your second equation.
Express the curve: x^3 - 6x^2 - y + 15 = 0 from (x,y) = (-2, -17) to (x,y) = (3, -12) in parametric form.
x^3 - 6x^2 - y + 15 = 0 => y = x^3 - 6x^2 + 15
use x = t as parametric equation for x
=>
y = t^3 - 6t^2 + 15
using x = t find the limits of t
-2 <= t <= 3