Transcript Circles, Parabolas, Ellipses, and Hyperbolas

Circles and Parabolas
Review
Equations of the Curves
centered at the origin (0,0)
• Circle
x y r
2
2
2
• Parabola
x2  4 py or y 2  4 px
Equations for Curves
not centered at the origin
• Circle
( x  h)  ( y  k )  r
2
2
2
Centered at (h, k)
• Parabola
( x  h)  4 p( y  k )
2
or
( y  k )  4 p( x  h)
2
Circles
( x  h)  ( y  k )  r
2
2
2
x y r
2
2
2
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.
Parabolas
( x  h)  4 p( y  k )
2
( y  k )  4 p( x  h)
2
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.
Problem 1
Circle
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
10
Center: (3,3)
Radius: 4
8
A
6
C: (3.00, 3.00)
A: (3.00, 7.00)
B: (7.00, 3.00)
D: (3.00, -1.00)
E: (-1.00, 3.00)
4
r
E
B
C
2
-10
-5
5
D
-2
Problem 2
Parabola
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 2
Opens up
Vertex: (-2, 1)
To get the Focus:
10
fx  = 2x+22 +1
½ ÷ 4 = ½ ∙ ¼ = 1/8, so
from the vertex (-2, 1)
we stay at -2 and add 1/8
to the y coordinate (1).
V: (-2.00, 1.00)
8
C: (-3.00, 3.00)
D: (-4.00, 9.00)
6
4
C
A
2
From the vertex we subtract 1/8
from the y coordinate (1).
Directrix: y = 7/8
B
A: (-1.00, 3.00)
B: (0.00, 9.00)
Focus: (-2, 9/8)
To get the Directrix:
-15
D
(-2, 9/8)
y = 7/8
-10
V
-5
-2
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?
( x  2)2  4( y  10)
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?
( x 1)  y  100
2
A. Center (-1,0) Radius = 10
B. Center (0,1) Radius = 100
C. Center (0,1) Radius = 10
D. Center (-1,0) Radius = 100
2
Begin Homework
Pages 89-90
(Math Mate 7 is due next class)