The graph of the ordered pairs (x, y) where x=f(t), y=g(t) are functions defined on an interval I of t-values is a parametric curve. The equations are parametric equations for the curve, the variable t is a parameter, and I is the parametric interval.
Parametric function have 2 variables that are depending on a common variable. Imagine that a rock is dropped from a 420-foot tower. The rock's height y in feet above the ground t seconds later (ignoring air resistance) is modeled by y=-16t^2+420. The line of the rock's fall is on the vertical line x=2.5. The rock's original position and its position after each of the first five seconds are the points (2.5, 420), (2.5, 404), (2.5, 356), (2.5, 276), (2.5, 164), (2.5, 20) which are described by the pair of equations x=2.5, y=-16t^2+420 when t= 0, 1, 2, 3, 4, 5. These two equations are an example of parametric equations with parameter t. The parameter t represents time.
Eliminating the Parameter: To eliminate the parameter of two equations, first solve for the parameter of the first equation and plug the resulting expression in for the second.
x=1-2t, y=2-t
Solve for t:
x=1-2t
2t=1-x
t=1/2(1-x)
Substitute for t in second equation:
y=2-t
y=2- 1/2(1-x)
y=0.5x+1.5
x=1-2t, y=2-t
Solve for t:
x=1-2t
2t=1-x
t=1/2(1-x)
Substitute for t in second equation:
y=2-t
y=2- 1/2(1-x)
y=0.5x+1.5
More Elimination of Parameters: Here are two functions used in physics that are parametric-
h=1/2 at^2+Vot+ho (up and down)
h=height now
ho= height to start
a=acceleration
t=time passed
Vo=velocity in the beginning
d=Vot (side to side)
d=distance traveled
Vo=velocity to start
t=time passed
Here are the steps to eliminate the parameter:
First solve for t: t=d/Vo. This is then substituted in for the h equation: h=1/2 a(d/Vo)^2+Vot+ho
h=1/2 at^2+Vot+ho (up and down)
h=height now
ho= height to start
a=acceleration
t=time passed
Vo=velocity in the beginning
d=Vot (side to side)
d=distance traveled
Vo=velocity to start
t=time passed
Here are the steps to eliminate the parameter:
First solve for t: t=d/Vo. This is then substituted in for the h equation: h=1/2 a(d/Vo)^2+Vot+ho
Here is another example of eliminating the parameter for a different type of equation. You need the same substitution skills, and a bit of further knowledge of functions. It looks at sine and cosine functions, and involves the equation cos^2t+sin^2t=1.