How to use this Algebra II - Semester 2 Study Packet

Transcription

1 Excellence is not an act, but a habit. Aristotle Dear Algebra II Student, First of all, Congrats! for making it this far in your math career. Passing Algebra II is a huge mile-stone Give yourself a pat on the back, for giving it your best thus far. If you re hesitating on giving yourself that pat on the back, well, buckle up! We ve got a lot of math practice to catch up on. How to use this Algebra II - Semester 2 Study Packet 1. Complete all review homework assigned by teacher 2. Re-learn and/or study topic, as needed 3. After review is complete, take the corresponding practice test 4. For multiple choice exams in math, for most question types, cover answer choices and complete the problem. Double check. Then look for the correct answer choice. 5. Come to tutoring for answer key and for multiple choice test taking strategies Algebra II Topics Included Multiple Choice Practice Tests These are basic concept questions, level easy to medium. These practice tests are not designed to teach concepts. They are designed to review concepts after studying and practice multiple choice style exams. Note: Graphing is minimal on this review, but may be tested on your exam! (Discuss with your teacher!) Powers, roots, complex numbers (important semester 1 review) Quadratics (important semester 1 review) Functions & Transformations Conics Polynomials Exponential and logarithmic expressions Sequences, series, probability Statistics Trigonometry Happy Studying, Kristy Kristy Arthur Tutoring Services k.arthur.tutoring@gmail.com

2 Excellence is not an act, but a habit. Aristotle Page 2 of 16 Powers, Roots, and Complex Numbers Subtract. (11 y 2 ) (11 y)(11+y) (11 y) 11 a Multiply. Evaluate the third root of Rationalize the denominator of. 10x 2 11x 6 10x x 6 10x Simplify by multiplying and factoring Solve the equation 22 x =. 13 no real solution 13 and 36 4 Rewrite (4x 5 ) 3/7 without rational exponents and simplify if necessary. 2x Divide. 16x 11 x 2 x 2 Find. Simplify (f 4/3 g 5/6 ) 12. f 16 g 10

3 Excellence is not an act, but a habit. Aristotle Page 3 of 16 Quadratic Equations Find the quadratic equation whose solutions are + 5i and 5i. x 2 x + x x 2 25 x 2 x Substitute values from the following equation into the quadratic formula: x 2 2x 6 = 0. Solve x 2 + 8x + 15 = 0. 1 and 15 3 and 5 3 and 5 1 and 15 What should you do as a first step in solving this equation x 2 4x = 7 by completing the square? add 4 to both sides square 7 add 4x to both sides add 2 to both sides Suppose a coin that is tossed upward can be modeled by the quadratic function h(t) = 16t t, where h(t) is the height in feet and t is the time in seconds. At what time will the coin be at a height of 5 ft? If necessary, round your answer(s) to the nearest hundredth of a second. Solve 3x 2 + 3x = 8x and 0, 1 and and 3 1 and Solve z 4 6z = 0. 1 and 5 1 and 5 1 and 0.25 s and 1.25 s 280 s 0 s and 1.5 s 0.25 s and 1.25 s Determine the nature of the solution(s) of 5x 2 + 2x 1 = 0. two real two complex two fake one real 1, 1, and

4 Excellence is not an act, but a habit. Aristotle Page 4 of 16 Solve x = 10x. 5 + i and 5 i + and Choose the graph that represents y x i and 5 i and 13 What must be true of a and c so that is a real number? either a = 0 or c = 0 both a < 0 and c < 0 either a > 0 or c > 0 ac < 0 Quadratic Functions and Transformations Which of the following is symmetric with respect to the origin? y 2 = 9 (x + 2) 2 x 2 = y + 3 x = x y y = x 3 3x

5 Excellence is not an act, but a habit. Aristotle Page 5 of 16 Graph the set of functions on the same set of axes. f(x) = x 2 and h(x) = (x 2) 2 Here is a graph of y = f(x). How would the graph of y = 4f(x) be different? The peaks would be at ( 6, 16), (0,16), and (6, 16); the valleys would be at ( 4, 16), (2, 16), and (8, 16). The peaks would be at ( 16, 4), (8,4), and (32, 4); the valleys would be at ( 24, 4), (0, 4), and (24, 4). The peaks would be at ( 24, 4), (0,4), and (24, 4); the valleys would be at ( 16, 4), (8, 4), and (32, 4). The peaks would be at ( 4, 16), (2,16), and (8, 16); the valleys would be at ( 6, 16), (0, 16), and (6, 16).

6 Excellence is not an act, but a habit. Aristotle Page 6 of 16 Graph the set of functions on the same set of axes. f(x) = x 2 and h(x) = x 2 4 Consider the graph of y = x 2. Which graph represents y = 3(x 1) 2? Which of the following is not symmetric with respect to the y axis? 3x 2 6y 2 = 42 5x + 2y = 8 y = x 4 9 y 9 = x 4 10x 2

7 Excellence is not an act, but a habit. Aristotle Page 7 of 16 Find the vertex and the line of symmetry of f(x) = 2(x + 4) 2. Consider the graph of y = x vertex: ( 4, 0); line of symmetry: x = vertex: ( 2, 4); line of symmetry: x = vertex: (4, 0); line of symmetry: x = Which graph represents y =? vertex: ( 2, 4); line of symmetry: x = 2 Which of the following functions is neither even nor odd? f(x) = x 3 + x 2 3x f(x) = x 4 + 5x 2 f(x) = x f(x) = 3x 3 x

8 Excellence is not an act, but a habit. Aristotle Page 8 of 16 Conics What is the equation for this hyperbola in standard form? Find the distance between (3, 7) and ( 4, 5). Graph the ellipse coordinates of its foci. and give the Give the standard form for the equation of the graph of the ellipse shown below.

9 Excellence is not an act, but a habit. Aristotle Page 9 of 16 Find an equation of a parabola that has a focus at (3, 5) and a directrix at x = 11. (y 5) 2 = (x 3) (y 3) 2 = 16(x 5) (y 5) 2 = 16(x 7) (y 5) 2 = 4(x 11) Write an equation for the conic y 2 2y 9x 2 = 35 in standard form. Then give the center of the conic. ; (0,1) ; (1,0) ; (3,1) ; (1,3) Find the coordinates of the midpoint of the segment having the endpoints ( 3, 2) and ( 1, 5). ( 2.5, 3) ( 2, 3.5) ( 2, 1.5) (1, 3.5) Find the vertex, focus, and directrix of y = (x + 1) vertex: ( 1, 3); focus: ( 1, 3.25); directrix: y = 2.75 vertex: (3, 1); focus: (2.75, 1); directrix: x = 3.25 vertex: (1, 3); focus: ( 1, 2.75); directrix: x = 3.25 vertex: ( 1, 3); focus: ( 1, 2.75); directrix: y = 3.25 Complete the square to find the center and radius of the circle. x 2 + y 2 6x 2y 26 = 0 Center: (6, 2); radius: Center: (3, 1); radius: 6 Center: ( 3, 1); radius: 6 Center: ( 6, 2); radius: Write an equation of the circle with center at ( 8, 3) and radius 4. (x 8) 2 + (y 3) 2 = 576 (x 8) 2 + (y 3) 2 = 96 (x + 8) 2 + (y + 3) 2 = 96 (x + 8) 2 + (y + 3) 2 = 4

10 Excellence is not an act, but a habit. Aristotle Page 10 of 16 Exponential and Logarithmic Functions Suppose f(x) = 2x + 5 and g(x) = 6x 2. Find f(g( 7)) If log and log 3 3 = , then approximate x = log x x x x Which of the following are properties of logarithms? log (xy) = log x + log y = log (x y) log (x n ) = nlog x log x log y = log (x + y) IV only I, II, and III I and III II and III Convert log x 32,768 = 5 to an exponential equation. x 5 = 32,768 5 x = 32,768 x 32,768 = 5 log x = Convert 8 2/3 = 4 to a logarithmic equation. = log 84 log 8 = log 4 2/3 log 4 = log 8 log 8 = log 4 Find an equation for f 1 (x) if f(x) = f 1 (x) = x 2 18 f 1 (x) = x 2 18; x 0 f 1 (x) = x ; x 0 f 1 (x) = x ; x > 0 Express 6log b x 5log b y 7 log b z as a single logarithm. log b log b log b (6x 5y 7z) log b (x 6 y 5 z 7 ) Sequences, Series Which of the following sequences are arithmetic or geometric? 343, 49, 7, 1, 1, 4, 9, 16, 2, 9, 16, 23, 2, 3, 5, 8, 13, Arithmetic: c and d Geometric: a only Arithmetic: c only Geometric: a and b Arithmetic: c and d Geometric: a and b Arithmetic: c only Geometric: a only

11 Excellence is not an act, but a habit. Aristotle Page 11 of 16 A geometric series converges if its an absolute value that is less than 1. common ratio sigma notation common difference first term has Find the first five terms of the sequence a n = n n. 2, 4, 6, 8, 10 1,,,, 1,,,, 1, 4, 27, 256, 3125 Find S 7 for the sequence 12, 14, 16, 18, Give a formula for the nth term of the geometric sequence 16, 24, 36, 54,. Give the sum (1.6) n 26(1.5) n (1.4) n 1 Find the common ratio for the geometric sequence, 3, 5,. 5 This is not a geometric sequence. Give the sum of the infinite geometric series:

12 Excellence is not an act, but a habit. Aristotle Page 12 of 16 Counting and Probability A pizza parlor offers a choice of mozzarella or Colby cheese. Available toppings are mushrooms, olives, and sausage. How many different medium-size cheese pizzas with one topping can be ordered? Evaluate 25C , ,014, Find the number of distinguishable permutations of the letters in the word INFINITY Expand: (x + 2) 6. x 6 + 6x x x x 2 + 6x + 1 x 6 + 2x 5 + 4x 4 + 8x x x + 64 x 6 12x x 4 160x x 2 192x + 64 x x x x x x + 64 Ten names are put into a hat to be drawn for four different door prizes. In how many ways can four names be drawn from the hat without replacement? Find the 5th term of the expansion: (2x 3y) 7. 35x 3 y 4 210x 3 y 4 22,680x 3 y 4 22,680x 3 y 4 The symbol 6! is read as "permutation six." "six factorial." "six chosen randomly." none of the above Evaluate 4P

13 Excellence is not an act, but a habit. Aristotle Page 13 of 16 Statistics and Data Analysis Find the mean, median, and mode of the following set of data. 6, 9, 7, 4, 7, 6, 3, 5, 3, 1, 1, 6, 7, 8, 6, 7, 4, 2, 1, 7 mean : 5, median: 5, mode: 6 mean : 5, median: 1, mode: 7 mean : 5, median: 6, mode: 7 mean : 7, median: 5, mode: 1 What is the range of this data? 894, 718, 241, 823, 197, 379, 593, The mean of the following data is 12. Compute the standard deviation. 17, 11, 12, 9, 10, What is the mean deviation of this data? 894, 718, 241, 823, 197, 379, 593, Suppose that a data set is normally distributed with a mean of 40 and a standard deviation of 5. Approximately 68% of the data values lie between what two numbers? 25, 55 30, 50 35, 45 40, 70 Construct a frequency distribution for the data using the intervals i) 0 24, ii) 25 49, iii) 50 74, and iv) Which interval has a relative frequency of 20%? 4, 98, 81, 6, 18, 20, 26, 14, 89, 21, 17, 65, 2, 23, 5, 1, 89, 9, 15, 19, 0, 91, 84, 11, 7, 29, 16, 72, 24, 13 i ii iii iv

14 Excellence is not an act, but a habit. Aristotle Page 14 of 16 Construct a stem-and-leaf diagram for the data set. 45, 98, 35, 21, 79, 45, 23, 89, 65, 32, 78, 58, 54, 28, 89, 31, 67, 9, 20, 91, 84, 26, 37, 68, 24, 13 Construct a box-and-whisker plot of the data. 15, 49, 42, 19, 12, 6, 1, 42, 42, 4, 7, 2, 10, 36, 43

15 Excellence is not an act, but a habit. Aristotle Page 15 of 16 Trigonometric Functions Use ΔABC to find cos A and tan B. Give the exact value of the trigonometric ratio for tan Undefined Given sin θ = , find θ in degrees and minutes, then convert the measure to decimal form. cos A = ; tan B = cos A = ; tan B = cos A = ; tan B = cos A = ; tan B = Convert 245 to radian measure ', ', ', ', Find the sine, cosine, and tangent for q. Find cos θ, cot θ, and the length of side a for the following triangle in which θ = 60 and c = 24. cos θ =, cot θ =, a = 12 cos θ =, cot θ =, a = 12 sin q = ; cos q = ; tan q = sin q = ; cos q = ; tan q = sin q = ; cos q = ; tan q = sin q = ; cos q = ; tan q = cos θ =, cot θ =, a = 12 cos θ =, cot θ =, a = 12 For an angle measure of 809, give an equivalent angle measure between 0 and 360, and tell in which quadrant the terminal side lies. 89, Quadrant I 89, Quadrant II 209, Quadrant III 209, Quadrant IV

16 Excellence is not an act, but a habit. Aristotle Page 16 of 16 Find the reference angle for angle shown. The expression sin (A + B) is identical to which expression? sin A cos B + cos A sin B sin A sin B cos A cos B sin A cos B cos A sin B cos A sin B sin A cos B Use the identity cos (a b) cos a cos b Trigonometric Identities and Equations Find two angles, in radians, between 0 sin a sin b to evaluate cos form. in simplest and 2π which satisfy sin x = and and If sin θ = and θ is an angle in the third quadrant, find sin 2θ. and and To find the exact value of sin 165 from a sum or difference identity, choose a suitable replacement for Find a value of cos (A + B) if tan A = and csc B =, both angles being acute. If cot θ = and sin θ is negative, find the value of tan θ. 2 3