Plane Curves and Parametric Equations

New Ways to Describe Curves (10.4)

POD

Describe this curve.

Is it a function?

121)3()2( 22 yx

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

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

t3 = x – 1 y = (x – 1) – 1

y = x – 2