Pre-Calculus: Notes 9.5 Parametric Equations
The equation: y = -x2 is an equation in rectangular form meaning its an equation of two variables.
A parametric equation tells you when an object is at a certain point. You can write both x and y as
functions of time (or any other parameter such as an angle).
Parametric Equations:
x=t
2
y=t
Plane Curve - If f and g are continuous functions of t on an interval, the set of ordered paris (f(t), g(t))
is a plane curve C. The equations given by:
x=f(t) and y = g(t)
are the parametric equations for C, and t is the parameter.
Sketching a Plane Curve
Plotting points in the order of increasing values of t allow you to trace the curve in a
specific direction. This is the ORIENTATION of the curve.
EX 1: Sketch the curve given by the parametric
equations:
x = t2
-3 < t < 3
y=t-1
t
x
y
-3
-2
-1
0
1
2
3
EX 2: Sketch the curve represented by: x = cosθ , y = 3sinθ. Then eliminate the parameter and
write the corresponding rectangular equation.
θ
x
y
0
π/4 π/2 3π/4 π
5π/4 3π/2 7π/4 2π
EX 3: Sketch the curve represented by: x = ln2t , y = 2t2. Then eliminate the parameter and write
the corresponding rectangular equation.
t
x
y
-1
0
1/2 1
3/2 2
5/2
3
EX 4: Graph the parametric equation with your graphing calculator.
x = sect y = tant
1. Set your calculator to parametric mode.
2. Type equations into y = . Be sure your PLOTS are off.
3. Set window to graph parametric equations.
4. GRAPH. OOOO, AHHHH!
Notice calculator graphs the asymptotes as well.
Rewrite the parametric equations in rectangular form. What type of equation do you think you will get?
Day 2: Notes Parametric Equations
Orientation
Sets of parametric equations can have the same rectangular equation, but have a different
orientation. Some may have different domains and some may be oriented from left to right or right
to left.
EX: Determine how the plane curves differ from each other.
a.) x = 2√t
y = 4 - √t
b.) x = -2t2
y = 4 + t2
c.) x = 2(t + 1)
y=3-t
Ex 2: Find two different parametric equations to represent the equation: y = 3x2 - 4
Ex 3: Graph the equations with your graphing calculator. Sketch graph on paper.