Accelerated Math 3 – 10.6 Parametric Equation Notes
Name: __________________
Big Idea: Points on a graph can be represented by two separate equations for the x and y coordinates in
terms of a third variable (usually called t), instead of a single equation. The variable t represents time in
many applications, which makes parametric equations useful for describing the location of objects as a
function of time.
Plane curves are “paths of motion”
Up until now you have studied equations with two variables, x & y, whose graphs can be used to
describe the path of an object.
We can use those equations to describe ___________ an object has been or is going, yet we
cannot tell you ___________ it will be at a particular location.
In order to be able to tell when an object is at a certain location, we need a third variable called a
_________________.
Where the variables (x & y) describing the location of an object become functions of the
parameter themselves.
Example: (Projectile Motion) Thrown objects follow parabolic paths.
Curvilinear Motion: Two
Variables for Position, One
Variable for Time.
For this particular motion
problem, x and y are
continuous functions of t, and
the resulting path is a plane
curve. (Recall that a
continuous function is one
whose graph can be traced
without lifting the pencil
from the paper.
Definition – A plane curve C is the collection of points generated by two functions that depend on a
parameter. The equations given by: x = f(t) and y = g(t) are parametric equations for C and the parameter
is t. The _____________ of a plane curve is the direction that the curve is being drawn.
Sketching a Plane Curve
A plane curve results when the ordered pairs ( x(t), y(t) ) are graphed for all values of t on some interval.
One way to sketch the plane curve is to make a table of values. The parameter t has several values listed
for it and the corresponding values for x(t) and y(t) are computed. Then the ordered pairs are plotted and
the curve is drawn in between the plotted pairs.
Orientation or Direction
When sketching a plane curve, the "direction of increasing t" or "orientation" of the curve is indicated by
little arrows indicating which direction the curve is moving in when the value of the parameter t increases.
Graphing:
Ex 1:
Sketch the graph of the plane curve described by the parametric equations:
8
x  t 2  4 and y  2t
t
-2
-1
0
1
2
3
x
6
4
y
2
-10
-5
5
10
15
10
15
-2
-4
-6
-8
Ex. 2: Sketch the graph of the plane curve described by the parametric equations:
8
x  6  t 2 and y 
t
2
6
4
t
-2
-1
0
1
2
3
4
x
y
2
-10
-5
5
-2
-4
-6
-8
*Make sure to include arrows on your curve to indicate the path/direction of the plane curve.
The same plane curve can be drawn with different parametric equations,
But how the plane curve is drawn will be different.
Ex 3:
Sketch the following plane curve given by the parametric equations using 0  t  2 .
x  t and y  4  t
t
x
x  2  t and y  2  t
x
y
x  2t and y  4  2t
t
y
0
0
0
1
1
1
2
2
2
4
4
4
2
2
2
Ex 4:
x
Graph the path of a projectile given by the parametric equations: x  20t and y  64  16t 2 .
When does the projectile hit the ground? What is the shape of the trajectory?
t
0
0.5
1
1.5
2
2.5
3
y
x
y
Converting between Rectangular Equations and Parametric Equations
Converting parametric equations into rectangular equations uses the process called:
“Eliminating the Parameter”
1. Solve for t in one of the equations (usually x, but you should solve for t in the simpler equation)
2. Substitute the result into the other equation
3. Simplify the rectangular equation that appears and solve for y
Note: the rectangular equation will draw the entire path for all values of x. In order to match the graph
with the plane curve the domain might need to be restricted.
Ex. 5: Convert x  2t and y  4  2t into a rectangular equation.
Ex. 6: Convert x  t 2  4 and y 
t
into a rectangular equation.
2
Eliminating the Angle Parameter
Another way to eliminate the parameter is using trigonometry. If the parametric equations are defined in
terms of sine and cosine we can use the fact that sin2   cos2   1 to eliminate the parameter  .
Ex. 7:
Eliminate the angle parameter and find the corresponding rectangular equation. Don’t solve for y, you
may leave the equations as is when the parameter is an angle.
x  3cos  and y  4sin  , 0    2 .
Converting a Rectangular Equation into a Parametric Equation
Let x = some expression of t (it will be defined in your instructions, you will need to solve for x), then make
the substitution into the rectangular equation for y.
Ex. 8: Convert y  x 2  2 x into a parametric equation using the following choices for t.
x
let t  x
let t 
let t  x  1
2