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Parameter
A third variable “t” that is related to both x & y
Ex)The ant is LOCATED at a point (x,
y)
Its location changes based on TIME (t)
x(t) = the ant’s horizontal location at time “t”
y(t) = the ant’s vertical location at time “t”
x(t) = t2 – 2 , y(t) = 3tInterval: -3 ≤ t ≤ 1t x y
Rectangular Form vs. Parametric
Rectangular Form: an equation written in terms of only two variables (what you have used in math up to this point).
Parametric Form: an equation defined by a third variable “t”
Parameterization: changing from rectangular to parametric form
Eliminating the parameter: changing from parametric to rectangular form.
Parametric Rectangular
Step 1: solve one of the equations for t
Step 2: Substitute into the other
equation
Step 3: Simplify
*If the graph is a circle a different process is used
Ex 1) Change to the parametric equation below to rectangular form & identify the type of curve:
x = 1 – 2t , y = 2 – t
Graph: x =2cosѲ, y = 4sinѲ
What type of graph is it?
What is the general equation for this type of graph?
Eliminate the paramter
Rectangular Parametric
“Parametization” Let x = t (or whatever you want!)
Sub “t” (or whatever) in for “x” into y =
*Ellipse & circles – sub in “cos” & “sin”
A car is about to drive off a cliff. What are all the different aspects of the situation? What different measurements exist?
•Driving forward (horizontally)
•Falling downwards (vertically)
•Driving at a certain speed
(velocity)
•Time is passing
50 ft
10 ft
Velocity = 25 ft/s
x(t) = horizontal position @ time t
x(t) = 10 + 25t
Initial location
Rate
50 ft
10 ft
Velocity = 25 ft/s
y(t) = height @ time t
y(t) = 50 - 16t2
Initial location
“Free Fall”ft/s
2. As a cargo plane ascends after takeoff, its altitude increases at a rate of 40 ft/s. while its horizontal distance from the airport increases at a rate of 240 ft/s.
Use the distance formula d = rt.
x = 240t
y = 40t
Describe the location of the cargo plane 20 seconds after take off.
x = 240t = 240(20) = 4800
y = 40t = 40(20) = 800Substitute t = 20.
At t = 20, the airplane has a ground distance of 4800 feet from the airport and an altitude of 800 feet.
3. A helicopter takes off with a horizontal speed of 5 ft/s and a vertical speed of 20 ft/s.
x = 5t
y = 20t