Equations of Circles

(x – a)2 + (y – b)2 = r2

(a, b) = center of the circle r = radius of the circle

(x – 4)2 + (y + 3)2 = 36

(4, -3) = center of the circle

6 = radius of the circle

What is the center and the radius of this circle?

Let’s Look at the Graph of

(x – 4)2 + (y + 3)2 = 36

center of the circleis (4, -3)

radiusof the circleis 6

Equations of an Ellipse

The x-intercepts a

2 2a cbx y Standard Form

The y-intercepts bThe center is the origin

2 23616 576x y

The x-intercepts 6

The y-intercepts 4

The center is the origin

Intercept Form22

22 1yxba

The x-intercepts aThe y-intercepts b

The center is the origin22

22 461

yx

22

1361

6yx

The x-intercepts 6

The y-intercepts 4

The center is the origin

Let’s Look at the Graph of

16x2 + 36y2 = 576

center ofthe ellipse

is the origin

x-interceptsare 6

y-interceptsare 4

Rewriting the Equation of an Ellipse

Rewrite the following equation of an ellipse in intercept form.

2 21625 400x y Since the equation must be equal to 1, divide by 400.

222 400400 4

16500 400yx

Simplify the fractions.

22

2161

5yx I think I can do this

without having to do too much work.

Rewrite the following equation of an ellipse in standard form. 22

811

4yx

Multiply by the common denominator.

22

4324 324 2

811 3 4

yx

Simplify the equation.

2 28 3 414 2x y

Just switch the coefficients of x and

y.

Graphing Circles and Ellipses 2 2( ) ( 15)3 6yx

Graph the following circle.Write the equation of the circle whose graph looks like this.

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

2 234) 25( ( )yx

Graph the following ellipse.

2 22 2 559 2x y

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

22

141

8yx

Write the equations of the following two ellipses in both standard form and intercept form.

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

2 24 12 5925 2x y 22

2491

5yx

2 21610 16000x y 22

0161

10yx

Equations of a Hyperbola

xy = kAs x increases, y decreases so that the product of x and y is

always k

xy = 8As x increases, y decreases so that the product of x and y is

always 8

•A hyperbola is a function•The coordinate axes are asymptotes of the graph•When k > 0, the graph is in quadrant I and quadrant III•When k < 0, the graph is in quadrant II and quadrant IV•Each branch of the hyperbola is the reflection of the other in the origin

Let’s Look at the Graph of

xy = 8Since k > 0,

the graph is in quadrant I

and quadrant III

The x and y axes are

asymptotes of the graph

Let’s Look at the Graph of

xy = -8Since k < 0,

the graph is in quadrant II

and quadrant IV

The x and y axes are

asymptotes of the graph

Graphing Hyperbolas

6xy Solve the equation for y. 6y

x

Make a table of values.

1

2

3

6

12

24

x y6

3

2

1

.5

.25

Since k is positive,

the graph is

in quadrants I and III

4xy

4y

x

x y1

2

4

8

16

32

4

2

1

.5

.25

.125

Since k is negative,

the graph is

in quadrants II and

IV

More Graphing Hyperbolas

12xy Solve the equation for y. 12y

x

Make a table of values.

1

2

3

6

12

24

x y12

6

4

2

1

.5

Since k is positive,

the graph is

in quadrants I and III

24xy

24y

x

x y1

2

3

4

6

8

12

24

24

12

8

6

4

3

2

1

Since k is positive,

the graph is

in quadrants I and III

6xy 6xy

12xy

6xy

12xy

24xy

6xy

12xy

24xy

72xy