Learning Targets Define parametric equations Graph curves parametrically within a given parametric...

Learning Targets• Define parametric equations• Graph curves parametrically within a

given parametric interval• Eliminate the parameter to obtain a

rectangular equation

Using Your Graphing Calculator

Function Mode vs. Parametric ModeVocabularyParametric Equations

Parameter

Parameter Interval

Rectangular Equation (Cartesian Equation) An equation with only x’s

and y’s.

The ordered pair (x, y) on a parametric curve is given by the parametric equations

…where t is called the parameter …

…and t is in the parameter interval, such as 0 ≤ t ≤ 2 .

Example:

t x y-3 (-3)2 - 2 =

73(-3) = -9

-2-10123

Example:

t x y-3 (-3)2 - 2 =

73(-3) = -9

-2 (-2)2 - 2 = 2

3(-2) = -6

-1 (-1)2 - 2 = -1

3(-1) = -3

0 (0)2 - 2 = -2

3(0) = 0

1 (1)2 - 2 = -1

3(1) = 3

2 (2)2 - 2 = 2 3(2) = 63 (3)2 - 2 = 7 3(3) = 9

Using Your Graphing Calculator

Using Your Graphing Calculator

Using Your Graphing Calculator

What is a good window for this parametric curve?

Parametric Interval:

Domain:

Range:

Using Your Graphing Calculator

Let’s start with Tstep = 1.

Using Your Graphing Calculator

What is a good value for Tstep?

Experiment with different values. What happens when you make the value bigger? Smaller?

Example:

Graph the parametric equations in our example for the following parametric intervals:

-3 ≤ t ≤ 1

-2 ≤ t ≤ 3

How are these different from the parametric curve we graphed earlier?

Learning Targets• Define parametric equations• Graph curves parametrically within a

given parametric interval• Eliminate the parameter to obtain a

rectangular equation

Eliminating the Parameter

In this example we will first solve one of the equations for t.

Then we will substitute this value for t in the other equation.

Your Turn!

Eliminate the parameter and identify the graph of the parametric curve:

Learning Targets• Define parametric equations• Graph curves parametrically within a

given parametric interval• Eliminate the parameter to obtain a

rectangular equation

Homework

Page 530#’s 1 – 25 odd, 65

For the remaining time in class, we will work on #65 from the homework assignment in small groups.

See page 18 in your textbook to review the equation of a circle.

#65. Parametrizing Circles

a) Graph the parametric equations for in the same square viewing window (ZOOM 5: ZSquare).

b) Eliminate the parameter t in the parametric equations to verify that they are all circles. What is the radius?

#65. Parametrizing Circles

c) Graph the parametric equations for using the following pairs of values for h and k:

d) Eliminate the parameter t in the parametric equations and identify the graph.

h 2 -2 4 3

k 3 3 -2 -3

#65. Parametrizing Circles

e) Write a parametrization for the circle with center (-1, 4) and radius 3.

From page 18:The standard form equation of a circle with center (h, k) and radius r is