Learning Targets Define parametric equations Graph curves parametrically within a given parametric...
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Transcript of Learning Targets Define parametric equations Graph curves parametrically within a given parametric...
![Page 1: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/1.jpg)
Learning Targets• Define parametric equations• Graph curves parametrically within a
given parametric interval• Eliminate the parameter to obtain a
rectangular equation
![Page 2: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/2.jpg)
Using Your Graphing Calculator
![Page 3: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/3.jpg)
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 .
![Page 4: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/4.jpg)
Example:
t x y-3 (-3)2 - 2 =
73(-3) = -9
-2-10123
![Page 5: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/5.jpg)
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
![Page 6: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/6.jpg)
Using Your Graphing Calculator
![Page 7: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/7.jpg)
Using Your Graphing Calculator
![Page 8: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/8.jpg)
Using Your Graphing Calculator
What is a good window for this parametric curve?
Parametric Interval:
Domain:
Range:
![Page 9: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/9.jpg)
Using Your Graphing Calculator
Let’s start with Tstep = 1.
![Page 10: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/10.jpg)
Using Your Graphing Calculator
What is a good value for Tstep?
Experiment with different values. What happens when you make the value bigger? Smaller?
![Page 11: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/11.jpg)
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?
![Page 12: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/12.jpg)
Learning Targets• Define parametric equations• Graph curves parametrically within a
given parametric interval• Eliminate the parameter to obtain a
rectangular equation
![Page 13: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/13.jpg)
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.
![Page 14: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/14.jpg)
Your Turn!
Eliminate the parameter and identify the graph of the parametric curve:
![Page 15: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/15.jpg)
Learning Targets• Define parametric equations• Graph curves parametrically within a
given parametric interval• Eliminate the parameter to obtain a
rectangular equation
![Page 16: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/16.jpg)
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.
![Page 17: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/17.jpg)
#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?
![Page 18: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/18.jpg)
#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
![Page 19: Learning Targets Define parametric equations Graph curves parametrically within a given parametric interval Eliminate the parameter to obtain a rectangular.](https://reader037.fdocuments.us/reader037/viewer/2022110400/56649dcf5503460f94ac44e9/html5/thumbnails/19.jpg)
#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