Amy's Reflection #28

here are some stuff from chapter 6 + examples...

Ellipses

Steps:
1. find the center
2. determine the major axis
3. find the vertex (± √big denom)
4. find the other intercept ( ± √small denom)
5. find the focus (c^2 = a^2 + b^2)
6. determine the length of the major axis (2√big denom)
7. find the length of the minor axis (2√small denom)
8. finally graph

Example 1: Graph the following ellipse. Find its major intercepts, length of the major axis, minor intercepts, length of the minor axis, and foci.

x^2/4 + y^2/9 = 1

This ellipse is centered at (0, 0). Since the larger denominator is with the y variable, the major axis lies along the y-axis.

Since a^2 = 9 then a = 3 & Since b^2 = 4

then b = 2Major intercepts: (0, 3), (0, –3)

Length of major axis: 2 √9 = 6

Minor intercepts: (2, 0), (–2, 0)

Length of minor axis: 2√4 = 4
c^2 = a^2 + b^2
= 9 - 4
= 5
= √5

Foci: (0, √5) , (0, -√5)

then you graph your points..

Parabolas:

how to find the axis of symmetry, vertex, focus, & directrx??

1.) to find the axis of symmetry: x = -b/2a

2.) for the vertex: (-b/2a, f(-b/2a)) or use complete the square:
y = (x+a)^2 + b.....a & b are numbers and (-a,b) = vertex

3.) to find the focus: 1/4p= the coefficient of x^2 and then add p

Note:
*If opens up, add to y value from vertex, if opens down, subtract
*If opens right, add to x value to vertex, if opens left, subtract)

4.) directrix: is p units behind the vertex

Note:
*If opens up, subtract; if opens down, add from y-value of vertex.*If opens right, subtract x-value*If opens left, add x-value

Example: x^2 + 1

~vertex:
x = -b/2a
x = 0/2(1) = 0
0^2 + 1 = 1
(0,1)

~Focus:
1/4p = 1
4p = 1
p = 1/4
(0, 1 + 1/4)
(0, 5/4)

~directrix:
y = 1 - 1/4
y = 3/4

can someone please explain recursive definition? now someone has a question to answer..