Name: Period: Final Exam Review

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1 Name: Period: Final Exam Review Note: These problems are NECESSARY but not SUFFICIENT to prepare for the final exam. Be sure to look over your notes, quizzes, and textbook problems as well. Try to summarize concepts on your own. Conic Sections Topics / Things to Know Identify the equation of a circle, ellipse, hyperbola, and parabola in standard form. Find the equation of a circle, ellipse, or hyperbola (including the center) using coordinate geometry, the Pythagorean Theorem/Distance Formula, or by analyzing given graphs. Identify vertices, co-vertices, and foci of ellipses and hyperbolas. Also, identify asymptote lines of a hyperbola. Find the constant sum or difference that defines an ellipse or hyperbola. Identify the vertex, focus, and directrix of a parabola. Use the discriminant (B 4AC) and completing the square to identify a conic. Circles 1. Find the center and equation of the circle that has a diameter with endpoints ( 3, ) and (5, 8).. Write the equation of the circle with center ( 5, ) that contains the point (7, 7). 3. Write the equation of a circle with center ( 9, 1) and radius Write the equation of the line tangent to the circle (x 1) (y ) = 169 at the point (13, 3). 5. A radio tower is 65 miles west and 45 miles north of Jason and emits a signal that reaches all receivers in a 75 mile radius. Does the signal reach Jason? Write an equation that represents the locations that are furthest from the radio tower. Ellipses 6. Find the constant sum for the ellipse with: a) Foci F 1 ( 3, 0), F (3, 0) and point P (0, 4) b) Foci F 1 (0, 6), F (0, 6) and point P (0, 10) 7. Write the equation of the ellipse and sketch a graph. a) focus (0, 8) and vertex (0, 10) b) focus (3, 0) and co-vertex (0, 4) 8. Graph the equation (x 3) 5 (y ) 16 = Given the following equation co-vertices of the ellipse. (x ) (y 3) 36 = 1, write the coordinates of the vertices and 1

2 Hyperbolas 10. Find the constant difference for the hyperbola with a) foci ( 5, 0), (5, 0), and point (5, 4) on the hyperbola. b) foci ( 7, 0), (7, 0), and point (4, 0) on the hyperbola. 11. Write the equation in standard form of the hyperbola with: a) center (0, 0), focus (10, 0), and vertex (8, 0). b) center (0, 0), focus (0, 6), and co-vertex (4, 0). c) Find the vertices, co-vertices, and asymptotes of the hyperbola then graph it. (x 4) 5 (y 3) 36 = 1, and Parabolas 1. a) Identify the vertex, focus, and directrix line of the parabola defined by the equation 1y10 = x 4x 14 Identifying Conics 13. Identify the conic sections below as either circle, ellipse, or hyperbola. a) (x 1) 54 (y ) 33 = 1 b) (x 1) 54 (y ) 33 = 1 c) (x 1) (y ) = Find the standard form of the equation x y 8x 10y 8 = 0 by completing the square. 15. Find the standard form of the equation 5x 0y 30x40y 15 = 0 by completing the square. 16. Find the standard form of the equation 1x 6y 4x 36y 90 = 0 by completing the square. Probability and Stats Topics / Things to Know Know the difference between combinations ( n C r ) and permutations ( n P r ). Use combinations and permutations to count the number of arrangements for a given situation. Find the probability of an event using the complement set. Find the conditional probability of an event. Determine whether two events are independent or dependent and calculate probabilities accordingly. Use Venn Diagrams or charts to find probabilities. Use the Empirical Rule ( ) and use a Z-table to find z-scores for a normally distributed data set. Also, identify any outliers.

3 17. Esteban is a stylish guy. Each day he wears a hat, a top, and a bottom. Hugh owns 10 hats, shorts, 4 t-shirts, a red polo shirt, a collared shirt, 4 pants, 3 dress shirts, and 3 khaki pants. How many different outfits can Hugh wear? 18. a) 8 swimmers are competing in a swim meet, how many ways can a gold, silver, and bronze medal be given? b) From a group of 1 Olympic swimmers, 4 are selected for a swim meet. How many ways can they be selected. 19. Alana owns pairs of shoes. Next week, she wants to wear a different pair of shoes to school each day. How many ways can she choose 5 pairs of shoes? 0. If two number cubes (one red, one blue) are rolled at the same time: a) What is the probability that the sum of the two number cubes is greater than or equal to 5? b) What is the probability that the sum of the two dice is prime? 1. If a baby born in 013 is selected at random, what is the probability that they were not born on the first, middle, or last day of any month that year?. A survey of 50 students find that 35 students favor a new school schedule while 15 favor keeping the school schedule the same. What is the probability that in a random sample of 3 students, exactly two will favor the new schedule and one will favor the current schedule? 3. A bag contains 6 red marbles and 9 blue marbles. a) An experiment consists of taking one marble out of the bag, replacing it, and then selecting another marble. What is the probability of selecting a blue marble and then a red marble? Are these events independent or dependent, why or why not? Express your answer as a decimal. b) A second experiment consists of taking one marble out of the bag, without replacing it, and then selecting another marble. What is the probability of selecting a blue marble and then a red marble? Are these events independent or dependent, why or why not? Express your answer as a decimal. 4. Ryan, Kevin, and Michael are playing a board game. The game requires them to roll two dice. a) During Kevin s turn, he rolls a 1, but falls asleep so Ryan re-rolls for him. What is the probability that he either rolls a sum that is even or a multiple of 3? b) During Michael s turn, what is the probability that he either rolls a sum of 7 or an even number? 5. A survey is taken of 100 people. Of the 100 people surveyed, 6 people are employed and 50 are female. Of the 6 employed, 30 are male. What is the probability that a randomly chosen person surveyed is male or unemployed? 6. Ben, Jayce, and Becky are all in a music store looking for violins. There are 10 different brands of violins. What is the probability that at least of them buy the same brand? 3

4 7. Of 600 juniors and seniors surveyed at SPHS, 40 are athletes and 335 are female. It is also known that there are 105 female athletes. What is the probability that a randomly selected student from the survey is male or an athlete? Female or not an athlete? A male, given that they are an athlete? 8. The average travel time for Southwest Airlines from Los Angeles to Houston is normally distributed with a mean of 3.5 hours and standard deviation of 0.5 hours. What is the probability that a randomly selected flight from L.A. to Houston takes more than 4 hours? Between 3. and 3.8 hours? 9. Using the empirical rule ( ), if a normally distributed data set has a mean of 10 and standard deviation of.3, 67% of the data will be between what values? 95% of the data will be between what values? Would 18 be an outlier? 30. Battery lifetime is normally distributed for large samples. The mean lifetime is 500 days and the standard deviation is 61 days. To the nearest percent, what percent of batteries have lifetimes longer than 561 days? Between 378 days and 439 days? 31. A group of 65 students has a mean age of 15.8 years with a standard deviation of 0.6 years. The ages are normally distributed. How many students are between 15.8 and 17 years? Express answer to the nearest student. 3. Five hundred values are normally distributed with a mean of 15 and a standard deviation of 10. a. What percent of the values lies in the interval , to the nearest percent? b. What percent of the values are less than 145? c. What interval about the mean includes 95% of the data? d. What interval about the mean includes 99.7% of the data? 33. If 10 is added to each data point in a set of data, are the mean, median, mode, IQR, and standard deviation affected. Describe the effect (or lack thereof) of doing this for each measure of center and spread. 34. Suppose you have a deck of 5 cards: a) What is the probability of selecting a Jack, Ace, and then another Jack, if you replace the cards each time? b) What is the probability of selecting a Jack, Ace, and then another Jack, without replacing the cards? Trigonometric Functions and the Unit Circle Topics / Things to Know Use the six trigonometric functions and right triangle trigonometry to solve applications and use degrees and radian measures interchangeably. Find all coordinates on the unit circle. Use inverse trigonometric functions to identify angles on the unit circle within the restricted domains. Use the Law of Sines and the Law of Cosines to solve problems. Solve problems involving arc length. 4

5 Graph trigonometric functions with transformations from the parent functions. 35. For the following angles, find the six trigonmetric ratios, without a calculator. Based on this information, write the values that correspond to the coordinates (x, y) on the unit circle a) 135 b) 300 c) 7π 6 d) π Find the following inverse trig. function values. ( ) a) arcsin b) cos 1 ( ) 1 c) tan 1 ( 1) d) sin 1 () 37. Find ALL possible values of sin ( 1 ) A surveyor whose eye level is 5 feet above the ground determines the angle of elevation to the top of an office building to be If the surveyor is standing 40 feet from the base of the building, what is the height of the building to the nearest foot? 39. The minute hand of Big Ben s Clock Tower in London is 8 feet long. How far does it travel in 0 minutes? 40. Two airplanes leave the airport at the same time. One airplane flies due east at a speed of 300 miles per hour. The other airplane flies east-northeast at a speed of 350 miles per hour (the angle between the two directions is.5 ). If the planes are at the same altitude, how far apart are they after hours? Round your answer to the nearest mile. 41. Use the Law of Cosines or Law of Sines to find the missing lengths and angles. If you are using the Law of Sines, determine how many triangles are possible and include information for all possible triangles. Sketching your own triangle will help. a) m A = 40, m B = 85, a = 8 cm b) b = 3, c = 18, m A = 173 c) m A = 106, m B = 56, c = 1.5 m d) a = 10 ft, b = 6 ft, m A = 105 e) a = 4. mi, b = 5.7 mi, m A = Graph the following functions for two full periods. Identify the transformations from the parent function. ( (a) f(x) = sin x π ) ( x ) ( (b) f(x) = cos (c) f(x) = tan x π ) 4 3 5