Circles and ParabolasReview

Equations of the Curves centered at the origin (0,0)

• Circle

• Parabola orpyx 42 pxy 42

222 ryx

Equations for Curves not centered at the origin

• Circle

Centered at (h, k)

• Parabola

or

222 )()( rkyhx

)(4)( 2 kyphx )(4)( 2 hxpky

Circles

Circles are a special type of ellipses. There is a center that is the same distance from every point on the diameter. In the equation the center is at (h, k). The distance from the center to any point on the line is called the radius of the circle. From the equation to find the radius you take the square root of r2.

222 )()( rkyhx 222 ryx

Parabolas

A parabola is a curve that is oriented either up, down, left, or right. The vertex of the parabola is at (h, k). In the equation the h value added or subtracted to x moves the parabola left and right. If you subtract the value of h the parabola moves to the right. If you add the value of h the parabola moves to the left. Parabolas are symmetrical across the line through the vertex of the parabola.

)(4)( 2 kyphx )(4)( 2 hxpky

Problem 1Circle

Graph the following equation of a circle

(x- 3)2 + (y- 3)2 = 16*Find first before graphing

• the center for the circle • the radius for the circle.

Solution to Problem 1

8

6

4

2

-2

-5 5

r

C: (3.00, 3.00)

E: (-1.00, 3.00)

D: (3.00, -1.00)

B: (7.00, 3.00)

A: (3.00, 7.00)

BE

A

D

C

Center: (3,3)Radius: 4

Problem 2Parabola

Graph the following equation of the parabola.

(x + 2)2 = ½ (y – 1)

Determine the following before graphing the equation:

•Which way does the parabola open?

•The vertex of the parabola.

•The focus and the directrix

Solution to Problem 2Opens upVertex: (-2, 1)To get the Focus: ½ ÷ 4 = ½ ∙ ¼ = 1/8, sofrom the vertex (-2, 1)we stay at -2 and add 1/8 to the y coordinate (1).

Focus: (-2, 9/8)To get the Directrix:From the vertex we subtract 1/8 from the y coordinate (1).

Directrix: y = 7/8

10

8

6

4

2

-2

-5

D: (-4.00, 9.00)

C: (-3.00, 3.00)

B: (0.00, 9.00)

A: (-1.00, 3.00)

V: (-2.00, 1.00)

f x = 2x+2 2+1D B

C A

V

(-2, 9/8)y = 7/8

Problem 3 Click on the correct answer to move to the next problem.

What type of object/ curve is given by the equation below? What is the center of the equation?

)10(4)2( 2 yx

A. Circle Center (-2,-10)

B. Circle Center (2,10)

C. Parabola Center (2,10)

D. Parabola Center (-2,-10)

Problem 4 Click the correct answer to continue.

What is the center and the radius of the following circle equation?

100)1( 22 yx

A. Center (-1,0) Radius = 10

B. Center (0,1) Radius = 100

C. Center (0,1) Radius = 10

D. Center (-1,0) Radius = 100

Begin Homework

Pages 89-90

(Math Mate 7 is due next class)