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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) 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. • 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. Grades, Papers, Folders Conics Pre-Test Warm-up 3/18/09 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 • 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 Conics Pre-Test Today’s lesson: Geometry Review 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. 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 Ex1 Find the distance between points (-2,8) and (6,2). Answer: 10 units 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. 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. Ex4 Find the coordinates of the midpoint of the segment that has endpoints (4,8) and (5,3). Answer: (1/2, 11/2) Ex5 Prove that the diagonals of a rectangle are congruent. Use corners: (0,0) (a,0) (0,b) (a,b) Lesson Overview 10-1A Lesson Overview 10-1B 5-Minute Check Lesson 10-2B Assignment Summary Quiz Hw: p.620-621 #12-36 Even Warm-up 3/19/09 1) 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 ° 2) Find sin390°. ½ or 0.5 3) Solve z2 – 8z = -14 by completing the square. 4 ± √2 4) If x2 = 16 and y2 = 4, what is the greatest possible value of (x - y)2? 36 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 • 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 have$1.19. 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?) Check, go over hw 10.1 p.620-621 #12-36 Even Questions By the end of today’s lesson, you should be able to: • Use and determine the standard and general forms of the equation of a circle • Graph circles From Algebra II… 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. You read… Learning to read a textbook… Read over p. 623 in your book; notice pictures, vocabulary, etc. 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. 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 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. 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 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. Lesson Overview 10-2A 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… Lesson Overview 10-2B 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. a. 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 + 9Complete the square. (x + 2)2 + (y – 3)2 = 9Factor 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. 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. 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. 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 SOLVE THE 3X3 MATRIX: 5D + 3E + F = -34 -2D + 2E + F = -8 -D – 5E + F = -26 D=-4, E = 2, F = -20 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 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. 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. 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 √50 or 5√2. Assignment Section 10.2 p. 628-630 #25,28,35-39all, 41a, 43, 48, 55, 58 15 problems total Warm-up 3/20/09 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. • 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 Assignment Section 10.2 p. 628-630 #25,28,35-39all, 41a, 43, 48, 55, 58 15 problems total §10.3: Ellipses LEQ: How do you identify the characteristics of an ellipse? What do you already know about the ellipse? Review Words Major Axis Vertices Focus Center co-vertices minor axis Standard Form of an Ellipse x 2 + y2 = 1 x2 + y2 = 1 a2 b 2 b 2 a2 If a > b If b > a Major axis: Major axis: horizontal(x) vertical(y) Vertices: (±a,0) Vertices: (0,±a) Co-Vertices:(0, ±b) Co-Vertices:(±b,0) 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 • EX2 Find the foci of the ellipse with the equation 25x2 + 9y2 = 225 • Divide the coefficients by 255 (must = 1) X 2 + 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) Eccentricity • The eccentricity of an ellipse, “e”, is a measure that describes the shape of an ellipse. • “e” is defined as c÷a • 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. Ex1 Find the equation of the ellipse with the given information. Ex2 Find the equation of the ellipse with the given information. 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 the moons. Find the coordinates of the foci. 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). Ex4 Consider the ellipse graphed below. a.Write the equation of the ellipse in standard form. b.Find the coordinates of the foci. a. The center of the graph is at (-2, 1). Therefore, h = - 2 and k = 1. Since the ellipse’s 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 - √7) and (-2, 1 + √7). 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. 10.3 Summary Quiz Assignment (10 problems) p. 638-640 #18,24,26,34,36 38,49,52,53a,63 5-Minute Check Lesson 10-4A 5-Minute Check Lesson 10-4B Warm-up 3/23/09 Find the coordinates of the vertex, foci, vertices, and sketch the graph. 1) 25x2 + 9y2 + 100x – 18y = 116 2) 9x2 + 4y2 – 18 + 16y = 11 Write the equation of the ellipse that satisfies the following information: 3) 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. 4) The foci are located at (-1,0) and (1,0) and a = 4. 5) 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). 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 • 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 10.3 Summary Quiz Assignment (10 problems) p. 638-640 #18,24,26,34,36 38,49,52,53a,63 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. Lesson Overview 10-4A Lesson Overview 10-4B 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 Ex1) Graph 9x2 – 25y2 = 225 Re-write in standard form: 9x2 – 25y2 = 1 225 225 x 2 – 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 (y’s/x’s): y = ±3/5x 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 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 = √(62 + 22) C = √40 ≈ 6.32 foci are at (6.32, 0) & (-6.32,0) Draw square, Draw diagonals, Equation of asymptotes (y’s/x’s) y = ±2/6x Draw curves through vertices and curving at asymptotes. 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 36 25 . 10.4 Summary Quiz Assignment/HW: p. 650-651 #21-25 odd, 31-41 odd, 45