Algebra 2 Warm up3.10.14

Find the center and radius of the circle and then graph it:

116822 yxyx

The Lame Joke of the day..

What did Sushi A say to Sushi B?

And now it’s time for..

Wasabi!(What’s up B, get it?)

The Lame Joke of the day..

What did one ocean say to the other ocean?

And now it’s time for..

Nothing , it just waved

THE

An ellipse is the set of all points such that the sum of the distances from two fixed points, called the foci, is a constant.

You can draw an ellipse by taking two push pins in cardboard with a piece of string attached as shown:

The place where each pin is is a focus (the plural of which is foci). The sum of the distances from the ellipse to these points stays the same because it is the length of the string.

Major AxisMin

or A

xis

Focus 1 Focus 2

PointPF1 + PF2 = constant

3.4.2

The Ellipse

PARTS OF AN ELLIPSE

center focifoci

major axis minor axis

vert

ices

vertices

The major axis is in the direction of the longest part of the ellipse

The vertices are at the ends of the major axis

The foci are always on the major axis

The standard equation for an ellipse centered at the origin:

2222

2

2

2

and 0 where,1 cbabab

y

a

x

a a

c c

b

b

The values of a, b, and c tell us about the size of our ellipse.

Find the vertices and foci and graph the ellipse:

149

22

yx

From the center the ends of major axis

are "a" each direction. "a" is the

square root of this value

The ends of this axis are the vertices

(-3, 0) (3, 0)a a

From the center the

ends of minor axis are "b"

each direction. "b" is the

square root of this value

b

b

We can now draw the ellipse

To find the foci, they are "c" away from the center in each direction. Find "c" by the equation:

222 bac 492 c 5 2.25 c

0,5 0,5

An ellipse can have a vertical major axis. In that case the a2 is under the y2

12

2

2

2

a

y

b

x

Find the equation of the ellipse

shown

You can tell which value is a because a2 is always greater than b2

(0, -4)

(0, 4)

From the center the

ends of minor axis are 1 each

direction so "b" is 1

To find the foci, they are "c" away from the center in each direction along the major axis. Find "c" by the equation:

222 bac 1162 c 15 9.315 c

From the center, the vertices are 4 each way so "a" is 4.

1

4 2

2

2

2

y

b

x

1

41 2

2

2

2

yx2 2

11 16

x y

Graph the ellipse:

16416

22

yx

Is the ellipse horizontal or vertical?Vertical because the larger number is under “y”a is always the larger number:

864 a 416 b

Graph the ellipse:

2525 22 yx

This equation is not in standard form (equal to 1) , so we must divide both sides by 25.

Is the ellipse horizontal or vertical?Horizontal because the larger number is under “x”a is always the larger number.

525 a 11 b

125

22

yx

Graph each ellipse:

369 22 yx 2525 22 yx

There are many applications of ellipses.

A particularly interesting one is the whispering gallery. The ceiling is elliptical and a person stands at one

focus of the ellipse and can whisper and be heard by another person standing at the other focus because all

of the sound waves that reach the ceiling from one focus are reflected to the other focus.