Showing posts with label intermediate algebra. Show all posts
Showing posts with label intermediate algebra. Show all posts

Saturday, April 27, 2013

Interactive: Ellipses


Interactive Instructions: Move any of the green points.



[ NOTESEllipses ]


Friday, April 26, 2013

Ellipses

An ellipse is the set of all points whose distances to two fixed points add to the same constant. In this section, we will focus on the equation of ellipses.
 
[InteractiveEllipses ]

The long axis of an ellipse is called the major axis and the short axis is called the minor axis.  These axes terminate at points that we will call vertices.  The vertices along the horizontal axis will be (± a,k) and the vertices along the vertical axes will be (h± b). These points, along with the center, will provide us with a method to sketch an ellipse given standard form.

Given the equation of an ellipse in standard form we can graph it using the following steps:
  1. First plot the center (h, k).
  2. Plot a point a units up and down from the center.
  3. Plot a point b units left and right of the center.
  4. Sketch the ellipse and label the four points.
In this example the major axis is the vertical axis and the minor axis is the horizontal axis.  The major axis measures 2b = 10 units in length and the minor axis measures 2a = 6 units in length.  There are no x- and y- intercepts in this example.

Graph the ellipse. Label the center and 4 other points.
 
 
 
It will often be the case that the ellipse is not given in standard form.  In this case, we will begin by rewriting it in standard form.

Graph the ellipse. Label the center and four other points.
 
  
 
The next set of examples give the equations in general form.  Complete the squares to obtain standard form. Remember to factor the leading coefficient out of each variable grouping before using (B/2)^2 to complete the square.

Graph the ellipse. Label the center and four other points.
 
 
Answer the questions.
 
 
When the calculation for intercepts yields complex results then there are no real intercepts.  This just means that the graph does not cross that particular axis.


Example: Find the equation of the ellipse centered at (3, -1) with a horizontal major axis of length 10 units and vertical minor axis of length 4 units.

Example: Find the equation of the ellipse given the vertices ( ±3,0) and (0, ±8).

Example: Find the equation of the ellipse whose major axis has vertices (-1, -2) and (-1, 10) and minor axis has co-vertices (-3, 4) and (1, 4).

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Circles

A circle is the set of points equidistant from a fixed point called the center. The distance from the center to any point is called the radius.  Recall the following formulas from Geometry:
 

The circumference can be thought of as the perimeter of the circle.  In this section, we will be interested in the equations that describe circles.
 
From the equation of a circle in standard form we can see that a circle is completely determined by its center and radius.  In addition, notice that it is not a function because it fails the vertical line test.
[ InteractiveCircles ]

Given the equation of a circle in standard form, you can graph it using the following steps:
  1. Plot the center (hk).
  2. Use the radius r to plot four points up, down, left and right of the center.
  3. Connect the points and label them.
 
Even though a circle is not a function, it is a relation and we could still find the domain and range.  In addition, we will be asked to find the x- and y-intercepts.  This particular example does not have any but the technique to find them is the same method used throughout this study guide.


To find y-intercepts set x = 0 and solve for y.
To find x-intercepts set y = 0 and solve for x.


Graph the circles and label the x- and y-intercepts.
 

 
 
 
  
  
Graphing circles in standard form is just a matter of identifying the center and the radius.  The difficulty comes when the circle is not given in standard form.  In this case, when given general form, we will complete the square twice as illustrated below.
  1. Group all terms with variable x and group all terms with variable y.
  2. Complete the square for each grouping.
  3. Be sure to add (b/2)^2 to both sides of the equation.
Rewrite the circles in standard form and identify the center.
 
Find the area of the following circles.
 
Graph and find the x- and y-intercepts, area, and circumference.
 
Given the center and radius, find the equation of the circle.
 
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