Algebra Quarter 4 Review

Transcription

1 Algebra Quarter 4 Review Chapters Covered (In Order) Chapter 11 Part 1 (probability and stats) Chapter 11 Part 2 (probability and stats) Chapter 10 (conics) Chapter 14-1, 13-2, 14-2, 14-4 (Trig functions) Best Ways to Study: As mentioned in class: complete this review and complete old tests. If you have time, flip through the chapters and take a look at the example problems at the beginning of each section. The website is likely easier to use than itunes U to track which sections we completed. Formulas You Will Need to Remember: The following is split into two categories. The first category will show up in multiple choice questions at the beginning of the exam. This means you will only have to be able to recognize them. Each multiple choice question will only have 3 options. For example: 1. What is the standard formula for a circle? a. x 2 + y 2 = 1 b. x + y = r c. x 2 + y 2 = r Recognizable Formulas:

2 Circle Equation: (x h) 2 + (y k) 2 = r 2 ½ bcsina = ½ acsinb = ½ absinc sina/a = sinb/b = sinc/c Memorize Formulas: npr used for permutations (order matters) (11.1) ncr used for combinations (order does not matter) (11.1) Experimental Probability = number of times event occurs / number of trials (11.2) Theoretical Probability: P(A) = m/n = number of outcomes/sample space (11.2) P(A and B) = P(A) * P(B) (11.3) P(A or B) = P(A) + P(B) (11.3) P(B A) = P(A and B) / P(A) (11.4) Mean, Median, Mode, and range formulas (which you all should know by now). Variance: σ! = (!!!)!! Standard Deviation: σ = (11.6) (!!!)!! (11.6) Binomial Probability: P(x) = ncx p x q n- x (11.8) Normal Distributions: 99.7% - 95% - 68% (of data within 3, 2, and 1 standard deviations) (11.9) Distance equation: d = (x! x! )! + (y! y! )! All of the trig/pythagorean identities (cos, sin, tan, cotan, cosecant, secant, etc.) Unit Circle If you feel that there is a mistake in any of the keys please let me know and I can take a look at it. As well, if you Google most of these questions they will come up online as there are many different schools that use this textbook.

3 Chapter 11 Part 1 Review 1. The following blanks may be filled with any letter or any single digit number. Letters and numbers may be repeated. How many possible combinations to fill in the blanks exist? 2. If the blanks in question 1 are to be filled again, but this time no letter or number may repeat, how many possible combinations are there to fill the blanks? 3. Mr. Hammel had 27 groups present science fair projects this week. If he was to going to send the top 10 groups to the big science fair in the gymnasium, how many different combinations could he send? 4. If Mr. Hammel was going to hand out metals (gold, silver, bronze, and copper) to first, second, third, and fourth place from his 27 science fair groups. How many different ways could the metals be handed out? 5. A consumer magazine rates televisions by identifying two levels of price, five levels of repair frequency, three levels of features, and two levels of picture quality. How many different ratings are possible? 6. In the Toronto Blue Jays first two games, Edwin Encarnacion has made 3 safe hits during 7 at bats. What is the experimental probability that Encarnacion does not get on base his next at bat? 7. What is the theoretical probability of being dealt two 7 s and one King in a five card hand from a standard deck of 52 playing cards? 8. A jar contains 30 red marbles, 50 blue marbles, and 20 white marbles. You pick one marble from the jar at random. Find the theoretical probability of picking a marble that is not white. 9. In a class of 147 students, 95 are taking math, 73 are taking science, and 52 are taking both math and science. One student is picked at random. Find each probability: a. P(taking math or science or both) b. P(taking math but not science) 10. A group of 30 students from your school is part of the audience for a TV game show. The total number of people in the audience is 150. What is the theoretical probability that 3 students from your school are selected as contestants out of 9 possible spots? 11. What is the probability of rolling an even number and a multiple of 3? 12. You have five books in your bag. Three are novels, one is a biography, and the other is a poetry book. During your first free period you pick one book from your bag and return it to the bag at the end of the period. If during your second free period of the day you choose a book again at random, what is the probability that you grab a novel both times. 13. S and T are mutually exclusive events. Find P(S or T) if P(S) = 3/5 and P(T) = 1/ A jar contains 4 blue marbles and two red marbles. Suppose you choose a marble at random and do not replace it. Then you choose a second marble. Find the probability of the following events: a. You select a red marble and then a blue marble b. Both red 15. Two cards are chosen at random from a deck of 52 cards. What is the probability that both of the cards you have chosen are queens? 16. Five coins are tossed together. What is the probability of getting exactly 2 heads?

4 17. Five coins are tossed together. When the coins land, four heads are showing and one tail is showing. What is the experimental probability of receiving four heads and one tail on the next toss? What is the theoretical probability of receiving four heads and one tail on the next toss? 18. Three balls are drawn randomly from a bag containing 3 black, 5 red, and 4 blue balls. What is the probability that the balls drawn are different color? 19. Three balls are drawn randomly from a bag containing 3 black, 5 red, and 4 blue balls. What is the probability of drawing exactly 2 black balls? 20. A bag contains 5 white balls and 3 black balls. What is the probability of selecting two balls that are the same color if you select both balls at the same time? Answers for Chapter 11 Part ,176,782, ,402,410, ,436, , /7 = 57.1% /2,598,960 = 0.87% 8. 4/5 = 80% 9. a. 116/147 = 78.9% b. 43/147 = 29.3% 10. (30C3 * 120C6)/150C9 = 17.9% 11. 1/6 = 16.7% 12. 9/25 = 36% % 14. a. 26.7% b. 1/15 = 6.7% 15. 4C2 / 52C2 = 0.45% 16. 5C2 / 2 5 = 31.3% 17. Experimental: 100% Theoretical: 5C4/2 5 = 15.6% 18. 3C1 * 5C1 * 4C1 / 12C3 = 3/11 = 27.2% 19. 3C2 * 9C1 / 12C3 = 12.2% 20. (5C2 + 3C2)/8C2 = 46.4%

5 Chapter 11 Part 2 1. Use the following numbers to answer questions a, b, and c: 5, 7, 9, 10, 12, 11, 14, 17 a. What is the median of the data? b. What is the 75 th percentile of the data? c. Your friend tells you that the standard deviation of the data is 1.0. Without calculating the standard deviation, does your friend s answer seem correct? 2. During a flu epidemic, 35% of the school s students have the flu. Of those with the flu, 90% have high temperatures. However, it is estimated that 12% of the students without the flu, also have high temperatures. If a student has a high temperature, what is the probability that he or she has the flu? 3. The mean length of Beethoven s nine symphonies is 37 minutes and the standard deviation is 12 minutes. Give the whole number of standard deviations from the mean that includes all of the following data values. 27 min, 30 min, 47 min, 35 min, 30 min, 40 min, 35 min, 22 min, 65 min 4. The following table is composed of data from a survey taken by a restaurant. Customers who disliked their visit rated the restaurant a 1, while those who really enjoyed it rated it a 5. They found the mean from the survey to be How many people in total took the survey? Score Frequency x 6 5. You are given 4 multiple choice questions, each with 5 possible answers (A E). You run out of time on the test and your teacher takes your booklet with the test questions in it. You are allowed to randomly select an answer for each question though before handing it in. What is the probability that you get at least 3 questions correct? 6. A local fruit company guarantees that 90% of the pineapple it ships will ripen within four days of delivery. Find the probability that at least 10 pineapples are ripe within four days if 12 pineapples are to be delivered. 7. To win a prize, the diameter of a tomato must be greater than 4 inches. The diameters of a crop of tomatoes grown in a special soil or normally distributed, with a mean of 3.2 inches and a standard deviation of 0.4 inches. What is the probability that a tomato grown in the special soil will be the winners? 8. A normal distribution has a mean of 100 and a standard deviation of 10. What is the probability that a value selected is at most 110?

6 Chapter 11 Part 2 Answers: 1. a) 10.5 b) 13 c) No, not reasonable. The data is too spread out and there are only 8 terms or 80.2% 3. 3 Standard deviations people took the survey % or or 88.91% % or % or 0.84 Chapter 10 Review Questions 1. What are the focus and directrix of the parabola with equation y =!!" x!? 2. Write an equation of a parabola with vertex at the origin and a directrix of x = What are the vertex, focus, and directrix of the parabola with equation y = x 2 4x + 8? 4. Write an equation of a parabola with the a vertex of (- 5, 4) and a focus of (- 5, 0) 5. Write an equation of a parabola with vertex at (1, 1) and a directrix of y=- 1/2 6. What are the center and radius of a circle with the equation: x 2 + y 2 + 8x 10y = 8 7. What is the center and radius of the following circle: (x + 2) 2 + (y + 4) 2 = Write an equation for a circle with center (- 1.5, - 3) and a radius of 2 9. Write an equation for a circle that passes through (12, - 5) and has a center at the origin. (Hint use the distance formula) 10. What are the foci of the ellipse with equation 25x 2 + 9y 2 = 225? 11. Write the equation for the ellipse with a focus of (3, 0) and an x- intercept of An opening to a tunnel makes a perfect ellipse and is 82 feet wide and 58 feet high. What is the equation of the ellipse? 13. Find the distance between the foci of the ellipse if the length between the major axis is 18 and the length between the minor axis is What are the vertices, foci, and asymptotes of the hyperbola with equation 9y 2 7x 2 = 63? 15. The path of a comet aound the sun followed one branch of a hyperbola. Find the equation that models its path around the sun, given that a = 40 million miles and c = 250 million miles. Use the horizontal model 16. Write an equation with foci (0, +/- 2) and vertices (0, +/- 1)

7 Answers for Chapter 10: 1. F: (0, - 3) d: y = 3 2. x = 1/15 y 2 3. V: (2, 4) F: (2, 4.25) directrix: y = y = - 1/16 (x + 5) y = 1/6(x 1) Center: (- 4, 5) and radius is 7 7. Center: (- 2, - 4) and radius (x + 1.5) 2 + (y + 3) 2 = 4 9. x 2 + y 2 = (0, 4) and (0, - 4) 11. x 2 /36 + y 2 /27 = x 2 / y 2 /841 = units 14. V: (0, +/- 7) F: (0, +/- 4) Asymptotes: y = +/- ( 7/3)x 15. (x 2 / 1.6x10 15) (y 2 /6.09x10 16) = y 2 (x 2 /3) = 1 Review Questions from 14-1 (answers can be found below) 1. For the following four equations, provide the domain of validity: a. cotθ = cscθ cosθ b. cscθ sinθ = cotθ cosθ c. cosθ cotθ = (1/sinθ) sinθ d. sinθ secθ = tanθ 2. Simplify the following trigonometric expressions: a. cscθ cosθcotθ b. cscθ / (sinθ + cosθcotθ) c. (sin 2 θcscθsecθ)/tanθ 3. Verify the following identities a. secθ sinθtanθ = cosθ b. sin 2 θtan 2 θ = tan 2 θ sin 2 θ c. sinθcosθ(tanθ + cotθ) = 1 d. (sin 4 θ cos 4 θ) = 2sin 2 θ 1 (hint: use difference of squares on the left hand side) Answers for 14-1: 1a) all real numbers except even multiples of pi/2. 1b) All real numbers except even multiples of pi/2 1c) All real numbers except even multiples of pi/2 1d) All real numbers except odd multiples of pi/2 2a) sinθ 2b) 1 2c) 1 3) No answers. You should work with either the left or the right side of the equation until it looks the same as the other.

8 Review for 13-2: 1. Give an angle between 0 and 360 degrees that is coterminal to the following angles: a. 385 b c Find the measure of each angle given a coordinate point that falls on the terminal side of the angle. a. (2, 2) b. (- 3/2, ½) 3. Find the cosine and sine of the following angles and write as a set of coordinate points (cosx, sinx): a b. 390 c. 315 Solutions for a) 25 1b) 150 1c) 55 2a) 45 or b) 150 or a) (- 1/2, 3/2) 3b) ( 3/2, ½) 3c) ( 2/2, - 2/2) *You may have had 1/ 2. To get to 2/2, simply rationalize the denominator (multiply the top and bottom by 2) 14-2 Problems 1. For the following, give the values for θ that satisfy the equation for 0 θ < 2π. a. 2cosθ - 3 = 0 b. 2sinθ - 2 = 0 c. (sinθ 1)(sinθ + 1) = 0 Solutions from a) π/6, 11π/6 1b) π/4, 3π/4 1c) π/2, 3π/2 For 14-4 problems, refer to the textbook example problems (this is the section on Area of a triangle and Law of Sines)