1540 hyperbolas

Elipse

Circle

Hyperbola

Parabola

Section 8.1: Midpoint & Distance Formulas

What does the midpoint formula do & why is it useful?

•Midpoint formula allows you to find the middle of something as an EXACT POINT

A

B

Midpoint – Half Way


Midpoint

1 1( , )x y

2 2( , )x y1 2 1 2,

2 2x x y y


What does the distance formula do & why is it useful?

•Distance formula allows you to find the length of something as an EXACT VALUE

A

B

How long is the linefrom point A to point B?


1 1,x y

2 2,x y

2 1x x

2 1y y

How does this help with the distance of the line?

* Ask Pythagoras: 2 2 2a b c

A

B

C


1 1,x y

2 2,x y

2 1x x

2 1y yA

B

This gives the Distance Formula:

2 22 1 2 1( )d x x y y


End Day #1

Homework:

Pg. 414 ( 13 – 19 odd, 25 – 31 odd, 34, 35, 43, 44 )

Section 8.2: Parabolas

What should we remember from chapter 6?

•Standard form of the equation of a Parabola

•How a Vertex is written

•How to tell if the parabola opens up or down

2( )y a x h k

( , )h k

If a > 0, parabola opens upIf a < 0, parabola opens down


Does the parabola always open up or down?

-- No, it can also open left or right


Table of Concept Summary for Parabolas

Form of Equation

Vertex (h, k) (h, k)

Axis of Symmetry x = h y = k

Focus

Directrix

Direction of Opening Up, if a > 0Down, if a < 0

Right, if a > 0Left, if a < 0

2( )y a x h k 2( )x a y k h

1,4

h ka

1 ,4

h ka

14

y ka

14

x ha


End Day #2

Homework:

Pg. 424 ( 12 – 14, 16 – 18, 21 – 23, 25, 30 – 34, 48, 49 )

Directions for (16 – 18, 21 – 23, 25):

Write each equation in standard form.

Find vertex, axis of symmetry,y-intercept if y= and

x-intercept if x=, tell the direction of opening, and graph.

Section 8.3: Circles

How do write out the equation of a circle with center at (0,0)? 2 2 2x y r

What is r? r is the radius, which is the distance from the center of the circle to the edge

What if center is not (0,0)? new center is written as (h,k)and we use the formula

2 2 2( ) ( )x h y k r


What if we are given two points and need to find the equation of the circle? 1 1,x y

2 2,x y

1. Use Midpoint Formula- this gives the center (h,k)

2. Use Distance Formula- this gives the radius length (r)

3. Plug values into general equation.


What if we are given the center and a tangent?

1. Substiute in the center (h,k) and point that is tangent (x,y) into general equation

2. Solve for radius (r)

3. Plug center (h,k) and radius (r) into general equation.

(x, y)

(h,k)


End Day #3

Homework:

Pg. 429 ( 17 – 25 odd, 28, 29 – 45 odd )

Section 8.4: Ellipses

X-axis

Y-axis

(-a,0) (a,0)

F (-c,0) F (c,0)

baa

Major Axis

MinorAxis


Table of Information for Ellipses with center at Origin (0,0):

Standard Form of Equation

Direction of Major Axis

Horizontal Vertical

Foci (c, 0) & (-c, 0) (0, c) & (0, -c)

Length ofMajor Axis

2a 2a

Length ofMinor Axis

2b 2b

2 2

2 2 1x ya b

2 2

2 2 1y xa b


What Changes if Ellipse is not centered on the origin?


Foci

2 2

2 2

( ) ( ) 1x h y ka b

2 2

2 2

( ) ( ) 1y k x ha b

( , )h c k ( , )h k c


End Day #4

Homework:

Pg. 438 (13 – 19 odd, 22, 24 – 35 Left Hand Column, do not worry about Foci)

Section 8.5: Hyperbolas

What are Hyperbolas? * Hyperbolas can be thought of as two parabolas going in opposite directions


Table of Information about HyperbolasCentered at Origin


Direction of Transverse Axis

Horizontal Vertical

Vertices ( a, 0 ) & ( -a, 0 ) ( 0, a ) & ( 0, -a )

Equations of Asymptotes

2 2

2 2 1x ya b

2 2

2 2 1y xa b

by xa

ay xb


a

b

by xa


What Changes when Hyperbola is NOT Centered at the Origin


Equations of Asymptotes

2 2

2 2

( ) ( ) 1x h y ka b

2 2

2 2

( ) ( ) 1y k x ha b

( )by k x ha

( )ay k x hb


Homework:Pg. 445 – 6 – 8: graph, give coordinates of vertices, & equations of asymptotes

21 – 31 odd: do NOT find the foci

41, 42

End Day #5

Education

1540 hyperbolas