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