Warm-up3/17/09

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Warm-up 3/17/09

1. Find the values of x for which sinx 0 is true.
2. Give the amplitude and period of the function
   y 3sin4x.
3. Find sin(sin-1 0.6)

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Did You Know?

• Q. What do bulletproof vests, fire escapes,
  windshield wipers, and laser printers all have in
  common?A. All invented by women.
• F One roach can live on a piece of gum for 5
  years.
• Q. If you were to spell out numbers, how far
  would you have to go until you would find the
  letter "A"?A. One thousand
• When a coffee seed is planted, it takes five
  years to yield it's first consumable fruit.
• The common goldfish is the only animal that can
  see both infra-red and ultra-violet light.

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• Tennessee is bordered by more states than any
  other. The eight states are Kentucky, Missouri,
  Arkansas, Mississippi, Alabama, Georgia, North
  Carolina and Virginia.
• Des Moines has the highest per capita Jello
  consumption in the U.S
• The Western-most point in the contiguous United
  States is Cape Alava, Washington.
• There are only three animals with blue tongues,
  the Black Bear, the Chow Chow dog and the
  blue-tongued lizard.

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• Grades, Papers, Folders

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• Conics Pre-Test

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Warm-up 3/18/09
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Did you know?

• In 1886, Coca-cola was first served at a pharmacy
  in Atlanta, Georgia for only five cents a glass.
  A pharmacist named John Pemberton created the
  formula for Coca-cola
• Flamingos are able to fly at a speed of
  approximately 55 kilometers an hour. In one night
  they can travel about 600 km
• Men are able to read fine print better than women
  can
• On average, 150 couples get married in Las Vegas
  each day
• Spiders usually have eight eyes, but still they
  cannot see that well
• In humans, the epidermal layer of skin, which
  consists of many layers of skin regenerates every
  27 days

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• Camel's milk does not curdle.
• The ball on top of a flagpole is called the
  truck.
• People generally read 25 slower from a computer
  screen compared to paper
• Certain female species of spiders such as the
  Australian crab spider, sacrifice their bodies as
  a food source for their offspring
• One grape vine produce can produce about 20 to 30
  glasses of wine.
• The Hubble telescope is so powerful that it is
  like pointing a beam of light at a dime that is
  two hundred miles away.
• On average people fear spiders more than they do
  death
• Every day, over five billion gallons of water are
  flushed down toilets in the United States
• In one trip, a honey bee visits about 75 flowers
• Jupiter is the fastest rotating planet, which can
  complete one revolution in less than ten hours
• A chicken loses its feathers when it becomes
  stressed

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• Conics Pre-Test

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Todays lesson

• Geometry Review
• ?

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By the end of the lesson today, you should be
able to

• Find the distance and midpoint between two points
  on a coordinate plane.
• Prove geometric relationships among points and
  lines using analytical methods.

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10.1 Analytic Geometry

• Distance Formula vs. Deriving the formula
• Can use graphing techniques and pythagorean
  theorem.
• Mid-point formula
• Use the words to make sense of the formula

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(No Transcript)
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Ex1

• Find the distance between points (-2,8) and
  (6,2).
• Answer
• 10 units

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Ex2

• Two children are playing hide and seek on the
  school playground. One of the children is hiding
  at the location (-4,3) on a map grid of the
  playground. The child who is doing the seeking
  is currently at the location (5, -2). Each side
  of a square on the playground grid represents 5
  feet. How far apart are the two children?
• Answer
• Map distance is 10.3 units unit 5 feet, so
  51.5 feet.

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Ex3

• Determine whether quadrilateral ABCD with
  vertices A(5,3), B(4,-2),C(-1,-2), and D(0,3) is
  a parallelogram.
• Parallelogram
• One pair of opposite sides are congruent and
  parallel.
• The measures of the two sides are equal, and the
  slopes are parallel, so ABCD is a parallelogram.

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Ex4

• Find the coordinates of the midpoint of the
  segment that has endpoints (4,8) and (5,3).
• Answer
• (1/2, 11/2)

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Ex5

• Prove that the diagonals of a rectangle are
  congruent.
• Use corners
• (0,0)
• (a,0)
• (0,b)
• (a,b)

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Lesson Overview 10-1A
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Lesson Overview 10-1B
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5-Minute Check Lesson 10-2B
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Assignment

• Summary Quiz
• Hw
• p.620-621
• 12-36 Even

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Warm-up 3/19/09

• A circle has a radius of 12 inches. Find the
  degree measure of the central angle subtended by
  an arc 11.5 inches long.
• 54.9
• Find sin390.
• ½ or 0.5
• 3) Solve z2 8z -14 by completing the square.
• 4 v2
• 4) If x2 16 and y2 4, what is the greatest
  possible value of (x - y)2?
• 36

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Did you know?

• Sharks are immune to cancer
• Manicuring the nails has been done by people for
  more than 4,000 years
• The study of the iris of the eye is called
  iridology
• Back in 1919, the Russian transplant pioneer
  Serge Voronoff made headlines by grafting monkey
  testicles onto human males.
• In 1946, the New York Yankees became the first
  baseball team to travel by plane
• By recycling just one glass bottle, the amount of
  energy that is being saved is enough to light a
  100 watt bulb for four hours

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• The Mall of America, located in Bloomington,
  Minnesota is so big that it can hold 24,336
  school buses
• If you have three quarters, four dimes, and four
  pennies, you have1.19. You also have the largest
  amount of money in coins without being able to
  make change for a dollar.
• In the United States, poisoning is the fourth
  leading cause of death among children
• In 1916, Charlie Chaplin was making 10,000 a
  week, making him the highest paid actor of his
  time
• It's possible to lead a cow upstairs...but not
  downstairs.
• People that smoke have 10 times as many wrinkles
  as a person that does not smoke
• Thomas Edison was afraid of the dark. (Hence, the
  light bulb?)

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Check, go over hw

• 10.1
• p.620-621
• 12-36 Even
• Questions

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By the end of todays lesson, you should be able
to

• Use and determine the standard and general forms
  of the equation of a circle
• Graph circles

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• From
• Algebra
• II

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Definitions (you already know?)

• Circle
• Set of all pints in a plane an equal distance
  from a center point.
• Radius
• Distance from the center to any point on the
  circle.

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You read

• Learning to read a textbook
• Read over p. 623 in your book
• notice pictures, vocabulary, etc.

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Equation of a circle

• If r represents the radius of a circle with its
  center at the origin, then the equation of the
  circle can be written in the form
• x2 y2 r2
• The equation of the circle with center that is
  Not at (0,0) but at another point (h,k) is
• (x h)2 (y h)2 r2
• This is the standard form of the equation for a
  circle.

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Application

• A tree in your garden has a ring of flowers.
  Each flower is four feet from the center of the
  tree trunk. You draw a coordinate grid to model
  your yard. The tree trunk is located at (-4,3).
  Write an equation that models the ring of
  flowers.
• (x - -4) 2 (y 3)2 42
• (x 4)2 (y 3)2 16

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Try This

1. Write the equation of a circle whose center is at
   (5,-3) and whose radius is 5.
2. Describe the translation that gives you the
   equation (x 2)2 (y 5)2 1.

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Using the center and radius to graph a circle

• Find the center and radius of the circle
• (x 6) 2 (y 7) 2 25
• Find the center and radius of the circle whose
  equation is
• (x 16) 2 (y 9) 2 144

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• Describe how you could use a stake and a long
  piece of rope to mark the perimeter of a circular
  home that will be 20 feet across.

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Lesson Overview 10-2A
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General Form of Circle Equation

• x2 y2 Dx Ey F 0
• When the equation of a circle is given in general
  form, it can be rewritten in standard form by
  completing the square for the terms in x and the
  terms in y.
• Example

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Lesson Overview 10-2B
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Example

• The equation of a circle is
• x2 y2 4x 6y 4 0.
•  
• a. Write the standard form of the equation.
•  
• b. Find the radius and the coordinates of the
  center.

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• x2 y2 4x 6y 4 0
• Group to form perfect square trinomials.
• (x2 4x ?) (y2 6y ?) -4
• (x2 4x 4) (y2 6y 9) -4 4
  9 Complete the square.
• (x 2)2 (y 3)2 9 Factor the trinomials.
• (x 2)2 (y 3)2 32
•  
• b. The center of the circle is located at (-2, 3)
  and the radius is 3.

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Now, you try

• The equation of a circle is
• 2x2 2y2 4x 12y 18 0
• Write the standard form of the equation
• Find the radius and the coordinates of the
  center.

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Finding the equation of a circle that passes
through three points.

• Substitute in the (x,y) points for x and y in the
  general form of the equation
• Use matrices to determine what D,E,F are.
• Write the equation
• Use completing the square to simplify.

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Ex)

• Write the standard form of the equation of the
  circle that passes through the points at (5,3),
  (-2,2), and (-1, -5)
• Plug in all three sets of points for x y
• x2 y2 Dx Ey F 0
• (5)2 (3) 2 D(5) E(3) F 0
• (-2) 2 (2) 2 D(-2) E(2) F 0
• (-1) 2 (-5) 2 D(-1) E(-5) F 0

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SOLVE THE 3X3 MATRIX

• 5D 3E F -34
• -2D 2E F -8
• -D 5E F -26
• D-4, E 2, F -20

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• NOW, JUST PUT THOSE NUMBERS INTO THE GENERAL FORM
  AND SOLVE FOR THE CIRCLE
• x2 y2 Dx Ey F 0
• x2 y2 -4x 2y -20 0
• AFTER COMPLETING THE SQUARE, YOU GET
• (X 2)2 (Y 1)2 25

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Now, you try

• Write the standard form of the equation of the
  circle that passes through the points (-2, 3),
  (6, -5), and (0, 7). Then identify the center and
  the radius of the circle.

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• Substitute each ordered pair for (x, y) in
• x2 y2 Dx Ey F 0, to create a system of
  equations.
•   (-2)2 (3)2 D(-2) E(3) F 0 (x, y)
  (-2, 3)
• (6)2 (-5)2 D(6) E(-5) F 0 (x, y) (6,
  -5)
• (0)2 (7)2 D(0) E(7) F 0 (x, y) (0,
  7)
•  Simplify the system of equations.
• -2D 3E F 13 0
• 6D 5E F 61 0
• 7E F 49 0
•  The solution to the system is D -10, E -4,
  and F -21.
•  

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• The equation of the circle is
• x2 y2 10x 4y 21 0.
• After completing the square, the standard form is
  (x 5)2 (y 2)2 50.
• The center of the circle is (5, 2)
• and the radius is v50 or 5v2.

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Assignment

• Section 10.2
• p. 628-630
• 25,28,35-39all, 41a,
• 43, 48, 55, 58
• 15 problems total

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Warm-up 3/20/09
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Did you know?

• In Colorado, there are about 83,000 dairy cows
• Just by recycling one aluminum can, enough energy
  would be saved to have a TV run for three hours.
• The first telephone call from the White House was
  from Rutherford Hayes to Alexander Graham Bell
• Turtles can breathe through their butts
• A glockenspiel is a musical instrument that is
  like a xylophone. It has a series of metal bars
  and is played with two hammers
• Teenage suicide is the second cause of death in
  the state of Wisconsin
• Diamonds were first discovered in the riverbeds
  of the Golconda region of India over 4,000 years
  ago.

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• French artist, Michel Vienkot, uses cow dung as
  paint when he creates his pictures
• There are 122 pebbles per square inch on a
  Spalding basketball
• The seventeenth president of the United States,
  Andrew Johnson did not know how to read until he
  was 17 years old
• The fastest growing tissue in the human body is
  hair
• A cubic yard of air weighs about 2 pounds at sea
  level.
• Pancakes are served for breakfast, lunch and
  dinner in Australia
• A lion feeds once every three to four days
• A honey bee has four wings
• Cheddar cheese is the best selling cheese in the
  USA
• In the movie "The Matrix Reloaded" a 17 minute
  battle scene cost over 40 million to produce

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Assignment

• Section 10.2
• p. 628-630
• 25,28,35-39all, 41a,
• 43, 48, 55, 58
• 15 problems total

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10.3 Ellipses

• LEQ How do you identify the characteristics of
  an ellipse?
• What do you already know about the ellipse?

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Review Words

• Major Axis
• Vertices
• Focus Center
• co-vertices minor axis

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Standard Form of an Ellipse

• x2 y2 1
• a2 b2
• If a gt b
• Major axis horizontal(x)
• Vertices (a,0)
• Co-Vertices(0, b)

• x2 y2 1
• b2 a2
• If b gt a
• Major axis vertical(y)
• Vertices (0,a)
• Co-Vertices(b,0)

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The Foci

• The foci are important points in an ellipse.
  The foci of an ellipse are always on the major
  axis and are c units from the center.
• C2 a2 b2

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• EX2 Find the foci of the ellipse with the
  equation 25x2 9y2 225
• Divide the coefficients by 255 (must 1)
• X2 y2 1
• 9 25
• The major axis is y. Vertices are (0,5)(0,-5)
• The minor axis is x. Co-vertices are (3,0)
  (-3,0)
• Foci are on the major axis using the formula
• C2 a2 b2 a2 is always the larger
• C 4 so the foci are at (0,4) and (0,-4)

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Eccentricity

• The eccentricity of an ellipse, e, is a measure
  that describes the shape of an ellipse.
• e is defined as ca
• Since c is smaller than a, c/a is always a
  fraction between 0 and 1.
• The closer e is to 0, the more it looks like a
  circle
• The closer e is to 1, the more if looks like an
  oval.

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Ex1

• Find the equation of the ellipse with the given
  information.

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Ex2

• Find the equation of the ellipse with the given
  information.

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Ex3

• SPACE The graph models the elliptical path of a
  space probe around two moons of a planet.The
  foci of the path are the centers of themoons.
  Find the coordinates of the foci.

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• The center of the ellipse is the origin.
• a (314) or 157
• b (110) or 55
• The distance from the origin to the foci is c km.
• c2 b2 a2
• c2 552 1572
• c ? 147
•  
• The foci are at (147, 0) and (-147, 0).

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Ex4

• Consider the ellipse graphed below.
•  
• a. Write the equation of the ellipse in standard
  form.
•  
• b. Find the coordinates of the foci.

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• The center of the graph is at (-2, 1). Therefore,
  h -2 and k 1.
• Since the ellipses vertical axis is longer than
  its horizontal axis, a is the distance between
  points at (-2, 1) and (-2, -3) or 4. The value of
  b is the distance between points at (-2, 1) and
  (1, 1) or 3.
•  
•  
• b. Using the equation c , we find that c .
  The foci are located on the vertical axis, units
  from the center of the ellipse. Therefore, the
  foci have coordinates (-2, 1 - v7) and (-2, 1
  v7).

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Ex5

• Find the coordinates of the center, the foci,
  and the vertices of the ellipse with the equation
  9x2 16y2 54x 32y 47 0. Then graph the
  equation.
• First, write the equation in standard form.
• 9x2 16y2 54x 32y 47 0
• 9(x2 6x ?) 16(y2 2y ?) 47 ? ?
• 9(x2 6x 9) 16(y2 2y 1) 47 9(9)
  16(1) Complete the square.
• 9(x 3)2 16(y 1)2 144
• Write in standard form and graph.

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• 10.3 Summary Quiz
• Assignment (10 problems)
• p. 638-640
• 18,24,26,34,36
• 38,49,52,53a,63

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5-Minute Check Lesson 10-4A
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5-Minute Check Lesson 10-4B
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Warm-up 3/23/09

• Find the coordinates of the vertex, foci,
  vertices, and sketch the graph.
• 25x2 9y2 100x 18y 116
• 9x2 4y2 18 16y 11
• Write the equation of the ellipse that satisfies
  the following information
• The center is at (-2,-3), the length of the
  vertical major axis is 8 units , and the length
  of the minor axis is 2 units.
• The foci are located at (-1,0) and (1,0) and a
  4.
• The center is at (1,2), the major axis is
  parallel to the x-axis, and the ellipse passes
  through points at (1,4) and (5,2).

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Did you know?

• According to research, Los Angeles highways are
  so congested that the average commuter sits in
  traffic for 82 hours a year
• Over one million Pet Rocks were sold in 1975,
  making Gary Dahl, of Los Gatos, California, a
  millionaire. He got the idea while joking with
  friends about his pet that was easy to take care
  of, which was a rock
• Because metal was scarce the Oscars given out
  during World War II were made of plaster
• The game Monopoly has been played by
  approximately 500 million people in the world,
  and the game is available in 26 languages

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• In 1983, a Japanese artist, Tadahiko Ogawa, made
  a copy of the Mona Lisa completely out of
  ordinary toast
• Average number of days a West German goes without
  washing his underwear 7
• Cotton crops can be sprayed up to 40 times a year
  making it the most chemical-intensive crop in the
  world
• Every year in the U.S., there are 178,000 new
  cases of lung cancer
• On average, the American household consumes six
  pounds of peanut butter annually
• A housefly can only ingest liquid material. They
  regurgitate their food to liquefy the food that
  they are going to eat
• You can send a postcard from Hell. There is a
  small town located in the Cayman Islands called
  "Hell." They even have a post office

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• 10.3 Summary Quiz
• Assignment (10 problems)
• p. 638-640
• 18,24,26,34,36
• 38,49,52,53a,63

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10.4 By the end of the lesson today, you should
be able to

• Use and determine the standard and general forms
  of the equation of a hyperbola
• Graph a hyperbola.

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Lesson Overview 10-4A
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Lesson Overview 10-4B
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Drawing a Hyperbola

• Use the standard form
• Find the values of a and b
• Draw a central rectangle
• Draw the asymptotes (diagonals of square)
• Draw branches through vertices

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• Ex1) Graph 9x2 25y2 225
• Re-write in standard form
• 9x2 25y2 1
• 225 225
• x2 y2 1
• 25 9 a 5, b 3
• Transverse axis (major one) is under x
• Vertices are (5,0) and (-5,0)
• Use a and b to draw the rectangle
• Draw the asymptotes and diagonals
• Equation of diagonals (ys/xs) y 3/5x

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Finding the Foci

• The foci of a hyperbola follow a similar rule to
  those of an ellipse.
• However, it is exactly the same as the
  Pythagorean theorem
• c2 a2 b2

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• Ex2) Find the foci of the hyperbola x2 y2 1
• 36 4
• Transverse axis is x-axis.
• a 6, b 2 Vertices are at (6,0) (-6,0)
• C v(62 22)
• C v40 6.32 foci are at (6.32, 0)
  (-6.32,0)
• Draw square, Draw diagonals,
• Equation of asymptotes (ys/xs) y 2/6x
• Draw curves through vertices and curving at
  asymptotes.

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Ex3)

• Find the coordinates of the center, foci, and
  vertices, and the equations of the asymptotes of
  the graph, then graph the equation.
• (y 4)2 (x 2)2 1
• 25 .

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• 10.4 Summary Quiz
• Assignment/HW
• p. 650-651
• 21-25 odd, 31-41 odd, 45