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

Section 8.1: Midpoint & Distance Formulas

Midpoint

1 1( , )x y

2 2( , )x y1 2 1 2,

2 2x x y y

Section 8.1: Midpoint & Distance Formulas

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?

Section 8.1: Midpoint & Distance Formulas

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

Section 8.1: Midpoint & Distance Formulas

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

Section 8.1: Midpoint & Distance Formulas

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

Section 8.2: Parabolas

Does the parabola always open up or down?

-- No, it can also open left or right

Section 8.2: Parabolas

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

Section 8.2: Parabolas

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

Section 8.3: Circles

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.

Section 8.3: Circles

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)

Section 8.3: Circles

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

Section 8.4: Ellipses

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

Section 8.4: Ellipses

What Changes if Ellipse is not centered on the origin?

Standard Form of Equation

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

Section 8.4: Ellipses

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

Section 8.5: Hyperbolas

Table of Information about HyperbolasCentered at Origin

Standard Form of Equation

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

Section 8.5: Hyperbolas

a

b

by xa

Section 8.5: Hyperbolas

What Changes when Hyperbola is NOT Centered at the Origin

Standard Form of Equation

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

Section 8.5: Hyperbolas

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