Post on 31-Dec-2015
description
POD
Describe this curve.
A circle with center at (-2, 3), and radius 11.
Is it a function? Nope.
121)3()2( 22 yx
New vocabulary
y = f(x) is a function, if f is a function. We know how this works.
This is an example of a plane curve, but plane curves also include non-functions, like the conics we’ve been studying.
New vocabulary
Various examples P(a) and P(b) are the endpoints of curve C.If P(a) = P(b), the curve is a closed curve.If there is not overlap, it is a simple closed curve.
Is there a way to describe these non-functions in terms of functions? (Of course there is, or we wouldn’t be doing this.)
P(a)
P(b)
P(a) = P(b)
New vocabulary
Plane curve:a set C of ordered pairs (f(t),g(t)), where f and g are functions defined on interval I.
In other words, a graph may not be a function, but we substitute the coordinates x and y with separate components that are functions of t– a mathematical sleight of hand.
Then we specify an interval I to run t in.
Parametric equations
We use parametric equations to describe plane curves.
The format is a bit different from the y = form we’re used to– we add a third variable and base the x and y functions on it.
The curve C with parameter t:x = f(t)y = g(t) for t in interval I.
The final result is a curve– which could be the same as curves we’ve seen– which runs in a particular direction (orientation).
Parametric equations– use it
The curve C with parameter t:x = f(t)y = g(t) for t in I.
1. x = 2t y = t2 -1 -1 ≤ t ≤ 2
a. On calculators, graph the curve, and determine its orientation. Change the T window and T step to see how the graph changes.
b. Do this with a triple-column chart on the next slide.
c. Combine them to find an equation in x and y (a more familiar form).
Parametric equations– use it1. x = 2t y = t2 -1 -1 ≤ t ≤ 2
a. Graph the curve, and show its orientation.
We can do this easily on thegraphing calculators.
Change the MODE to “PARA” for parametric mode. The Y= window is different, and we include both functions.
Parametric equations– use it1. x = 2t y = t2 -1 -1 ≤ t ≤ 2
a. Plot the curve, and show its orientation.
The window screen is also different– we include the final x and y dimensions, of course, but add the interval and increments for t.
Parametric equations– use it
1. x = 2t y = t2 -1 -1 ≤ t ≤ 2
a. Plot the curve, and show its orientation.
The final graph looks like something we’ve seen before.
Why does it stop?
Parametric equations– use it
1. x = 2t y = t2 -1 -1 ≤ t ≤ 2
b. Plot the curve, and show its orientation.
t x y
-1
-½
0
½
1
3/2
2
It’s like there are two dependent variables (x and y) based on one independent variable (t).
How does the curve “run”?
Parametric equations– use it
1. x = 2t y = t2 -1 -1 ≤ t ≤ 2
c. Combine them to find an equation in x and y (a more familiar form).
x = 2t t = x/2y = t2 -1y = (x/2)2 -1 y= ¼ x2 -1
Parametric equations– use it
1. x = 2t y = t2 -1 -1 ≤ t ≤ 2
One curve can be expressed by an infinite number of parametric equations.
x = t y = ¼ t2 – 1 -2 ≤ t ≤ 4
x = t3 y = (1/8) t6 – 1 -21/3 ≤ t ≤ 41/3
Try graphing them on calculators.
Parametric equations– use it
2. P(x, y) is x = a cos t and y = a sin t, for all real number values of t, and a >0. Describe the motion of P.
Don’t graph, just think.
Parametric equations– use it
2. P(x, y) is x = a cos t and y = a sin t, for all real number values of t, and a >0. Describe the motion of P.
Don’t graph, just think.
Graph on calculators. Set a = 5.
Parametric equations– use it
2. P(x, y) is x = a cos t and y = a sin t, for all real number values of t, and a >0. Describe the motion of P.
222
2
2
2
2
22
1
1sincos
ayx
a
y
a
x
tt
ta
y
ta
x
tay
tax
22
2
22
2
222
222
sin
cos
sin
cos
Parametric equations– use it
2. P(x, y) is x = a cos t and y = a sin t, for all real number values of t, and a >0. Describe the motion of P.
Why look, a circle with radius a, centered on the origin.
The curve follows a counter-clockwise rotation.
Graph it on calculators to check. Radian or degree mode?
222 ayx
Parametric equations– use it
3. From p. 824, #4. Graph the parametric equation and give its orientation. What is the equation in x-y notation?
x = t3 + 1 y = t3 – 1 -2 ≤ t ≤ 2
Parametric equations– use it
3. From p. 824, #4. Graph the parametric equation and give its orientation. What is the equation in x-y notation?
x = t3 + 1 y = t3 – 1 -2 ≤ t ≤ 2
What does this curve look like?
Parametric equations– use it
3. From p. 824, #4. Graph the parametric equation and give its orientation. What is the equation in x-y notation?
x = t3 + 1 y = t3 – 1 -2 ≤ t ≤ 2
t x y
-2
-1
0
1
2