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Study guide

Accuplacer Math Study Guide 2020

Accuplacer Math Study Guide The following sample questions are similar to the format and content of questions on the Accuplacer College Level Math test. Reviewing these samples will give you a good idea of how the test works and just what mathematical topics you may wish to review before taking the test itself. Our purposes in providing you with this information are to aid your memory and to help you do your best. I. Factoring and expanding polynomials Factor the following polynomials: Expand the following: 1. 15a3b2  45a2b3  60a2b 2. 7x3 y3  21x2 y2 10x3 y2  30x2 y 9.  x 1 x 1 x  3 10. 2x  3y2 3. 6x4 y4  6x3 y2  8xy2  8 11. x 3  3x 6  6  4. 2x2  7xy  6y2 5. y4  y2  6 6. 7x3  56 y3 7. 81r4 16s4 12. x2  2x  32 13.  x 15 14.  x 16 8.  x  y2  2 x  y 1 II. Simplification of Rational Algebraic Expressions Simplify the following. Assume all variables are larger than zero. 1. 32  5  4  40 4. 2 18  5 32  7 2. 9 35 8  2  27 81 3. 5. 6x 18 12x 16 x4 3x2  2x  8 4x 12 III. Solving Equations A. Solving Linear Equations 1. 3  2 x 1  x 10 2. x  x  1 2 7 3. y  y  2  y2  6 4. 2 x 1 3x  3 x 1 B. Solving Quadratic & Polynomial Equations 1.  y  8  y  2   0 5. t2  t 1  0  3  3     2. 2x3  4x2  30x  0 6. 3x3  24 3. 27x3  1 7.  x 12  x2  25 4.  x  3 x  6  9x  22 8. 5y2  y  1 C. Solving Rational Equations 1.  y 1 2  0 y 1 11 4. x2  25 2  x  5 1 x  5 2. 2  3  12 5. 1  6 x  3 x  3 x2  9 a a2  5 3. 1  2  5x 6. 1  1  x 6  x x  3 x2  3x 18 x2  3x x x  3 D. Solving Absolute Value Equations 1. 5  2z 1  8 4. 1 x  3  1 2 4 4 2. x  5  7  2 3. 5x 1  2 5. y 1  7  y E. Solving Exponential Equations 1. 10x  1000 4. 3x2 9x   1 2. 103x5  100 3. 2x1  1 8 5. 2x2 42 x   1 F. Solving Logarithmic Equations 1. log2  x  5  log2 1 5x 2. 2log3  x 1  log3 4x 3. log2  x 1  log2  x 1  3 4. ln x  ln2x 1  0 5. ln x  ln x  2  ln 3 6. 32 x  4x1 G. Solving Radical Equations 1. 4 2 y 1  2  0 4. x2  9  x 1  0 2. 2x 1  5  8 5. 3 3x  2  4  6 3. 5x 1  2 x 1  0 6.  2 IV. Solving Inequalities Solve the following inequalities and express the answer graphically and using interval notation. A. Solving Linear Inequalities 1. 3 x  4  2 5 3. 3 x  2  6   x  3 14 2. 3 x  3  5 x 1 4. 2  3x 10  5 B. Solving Absolute value Inequalities: Solve and Graph. 1. 4x 1  6 2. 4x  3  2  9 3.  5 4. 5  2x  15 C. Solving Quadratic or Rational Inequalities 1. 3x2 11x  4  0 3. x  2  0 3  x 2. 6x2  5x  4  x 1 x  3 4. 0 2x  7 V. Lines & Regions 1. Find the x and y-intercepts, the slope, and graph 6x 5y = 30. 2. Find the x and y-intercepts, the slope, and graph x = 3. 3. Find the x and y-intercepts, the slope, and graph y = -4. 4. Write in slope-intercept form the line that passes through the points (4, 6) and (-4, 2). 5. Write in slope-intercept form the line perpendicular to the graph of 4x - y = -1 and containing the point (2, 3). 6. Graph the solution set of x - y ≥ 2. 7. Graph the solution set of -x 3y < -6. VI. Graphing Relations, Domain & Range For each relation, state if it is a function, state the domain & range, and graph it. 1. y  2. y  x  2 x  2 6. x  y2  2 7. y  x2  8x  6 3. y  x 1 x  2 8. y  4. f  x   x 1  3 9. y  3x 5. f  x  2x  5 x2  9 10. 2 h  x  3x2  2x 1 VII. Exponents and Radicals Simplify. Assume all variables are >0. Rationalize the denominators when needed.  54a6b2 2 1. 6.   2. 5 147  4  9a3b8  3. 5  15  3 8. 2  2  4 3 5  3  x 3 y 3 x 4.  5 9.  3  5. VIII. Complex Numbers Perform the indicated operation and simplify. 1.  4  9 5. 4  3i2 3. 4. 4  3i4  3i 6. i25 3  2i 7. 4  5i IX. Exponential Functions and Logarithms 1. Graph: f  x  3x 1 6. Solve: logx 9  2 2. Graph: g  x  2x1 7. Graph: h  x  log3 x 3. Express 82  1 64 in logarithmic form 8. Use the properties of logarithms to expand as 3 4. Express log5 25  2 in exponential form much as possible: log4 y 5. Solve: log2 x  4 9. How long will it take $850 to be worth $1000 if it is invested at 12% interest compounded quarterly? X. Systems of Equations & Matrices 2x  3y  7  1 1 1  0 2 1 1. Solve the system: 6x  y  1 4. Multiply:  0 2 0  1 2 0 2. Solve the system: x  2 y  2z  3 2x  3y  6z  2 x  y  z  0 2 1 3 0 0 1 1 2 5. Find the determinant: 3 1 3. Perform the indicated operation: 6. Find the Inverse:  1 2 1 2   3 1   1 2 2   3  3  1 2 1 6  XI. Story Problems 1. Sam made $10 more than twice what Pete earned in one month. If together they earned $760, how much did each earn that month? 2. A woman burns up three times as many calories running as she does when walking the same distance. If she runs 2 miles and walks 5 miles to burn up a total of 770 calories, how many calories does she burn up while running 1 mile? 3. A pole is standing in a small lake. If one-sixth of the length of the pole is in the sand at the bottom of the lake, 25 ft. are in the water, and two-thirds of the total length is in the air above the water, what is the length of the pole? XII. Conic Sections 1. Graph the following, and find the center, 2. Identify the conic section and put it into standard foci, and asymptotes if possible. form. a) (x  2)2  y2  16 a) x2  4x 12  y2  0 (x 1)2 b)  ( y  2)2  b) 9x 2 18x 16y2  64y  71 16 9 (x 1)2 c)  ( y  2)2  c) 9x 2 18x 16y2  64y  199 16 9 d) (x  2)2  y  4 d) x2  y  4x  0 XIII. Sequence and Series 1. Write out the first four terms of the sequence whose general term is an  3n  2 2. Write out the first four terms of the sequence whose general term is a  n2 1 3. Write out the first four terms of the sequence whose general term is a  2n 1 4. Find the general term for the following sequence: 2,5,8,11,14,17.... 5. Find the general term for the following sequence: 6 4, 2,1, 1 , 1 ,.... 6. Find the sum: 2k 1 k 0 7. Expand the following:   4  xk y4k k k 0   XIV. Functions Let 1. f (x)  2x  9 f (3)  g(2) and g(x)  16  x2 . Find the following. 5. (g f )(2) 2. f (5)  g(4) 3. f (1) g(2) f (5) 4. g(5) 6. f (g(x)) 7. f 1 (2) 8. f  f 1 (3) XV. Fundamental Counting Rule, Factorials, Permutations, & Combinations 8! 1. Evaluate: 3!8  3! 2. A particular new car model is available with five choices of color, three choices of transmission, four types of interior, and two types of engines. How many different variations of this model car are possible? 3. In a horse race, how many different finishes among the first three places are possible for a ten-horse race? 4. How many ways can a three-Person subcommittee be selected from a committee of seven people? How many ways can a president, vice president, and secretary be chosen from a committee of seven people. XVI. Trigonometry 1. Graph the following through on period: 2. Graph the following through on period: f (x)  sin x g(x)  cos(2x) 3. A man whose eye level is 6 feet above the ground stands 40 feet from a building. The angle of elevation from eye level to the top of the building is 72 . How tall is the building? 4. A man standing at the top of a 65m lighthouse observes two boats. Using the data given in the picture, determine the distance between the two boats. Answers I. Factoring and Expanding Polynomials When factoring, there are three steps to keep in mind. 1. Always factor out the Greatest Common Factor 2. Factor what is left 3. If there are four terms, consider factoring by grouping. Answers: 1. 2. 3. 15a2b(ab  3b2  4) x2 y(7 y 10)(xy  3) 7x3 y3  21x2 y2 10x3 y2  30x2 y x2 y(7xy2  21y 10xy  30) x2 y (7xy2  21y)  (10xy  30) x2 y 7 y(xy  3) 10(xy  3) 2(3x3 y2  4)(xy2 1) Since there are 4 terms, we consider factoring by grouping. First, take out the Greatest Common Factor. When you factor by grouping, be careful of the minus sign between the two middle terms. 4. (2x  3y)(x  2y) 5. ( y2  2)( y2  3) y4  y2  6 u2  u  6 (u  2)(u  3) When a problem looks slightly odd, we can make it appear more natural to us by using substitution (a procedure needed for calculus). Let u  y2 Factor the expression with u’s. Then, substitute the y2 back in place of the u’s. If you can factor more, proceed. Otherwise, you are done. 6. 7(x  2y)(x2  2xy  4y2 ) 7. (3r  2s)(3r  2s)(9r2  4s2 ) Formula for factoring the sum of two cubes: a3  b3  (a  b)(a2  ab  b2 ) The difference of two cubes is: a3  b3  (a  b)(a2  ab  b2 ) 8. 9. 10. (x  y 1)2 x3  3x2  x  3 4x2 12xy  9 y2 Hint: Let u=x y 11. 3x2 2  3 2 12. 13. 14. x4  4x3 10x2 12x  9 x5  5x4 10x3 10x2  5x 1 x6  6x5 15x4  20x3 15x2  6x 1 When doing problems 13 and 14, you may want to use Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 II. Simplification of Rational Algebraic Expressions 1. 13 2. 38 9 If you have , you can write 4 as a product of primes (2 2) . In square 3. roots, it takes two of the same thing on the inside to get one thing on the x2 4. 49 2 6 outside:  5. x  2 III. Solving Equations A. Solving Linear Equations 14 4 1. x  5 2. x  or 2 5 5 3. y  3 4. x  1 B. Solving Quadratic & Polynomial Equations 8 2 1. y  ,  3 3 2. x  0, 3,5 Solving Quadratics or Polynomials: 1. Try to factor 2. If factoring is not possible, use the quadratic formula 3. x  1 , 1  x  where ax2  bx  c  0 3 6 6 4. x  10, 4 5. t  1  2 2 6. x  2, 1 i 7. x  3, 4 Note: 2a i  Example:  i  i 8. y  1 21 10 C. Solving Rational Equations 1 1. y  3 1  2  0 y 1 y 1 ( y 1)( y 1)  1  y 1 2   0( y 1)( y 1) y 1 Solving Rational Equations: 1. Find the lowest common denominator for all fractions in the equation 2. Multiply both sides of the equation by the lowest common denominator 3. Simplify and solve for the given variable 4. Check answers to make sure that they   do not cause zero to occur in the ( y 1)( y 1) 1 y 1  ( y 1)( y 1) 2  0 y 1 denominators of the original equation. ( y 1)  2( y 1)  0 3y 1  0 2. Working the problem, we get x  3 . However, 3 causes the denominators to be zero in the original equation. Hence, this problem has no solution. 3. x   15 4 4. x  2 5. a  1, 5 6. x  2,1 D. Solving Absolute Value Equations 1. z  2 or z  7 Solving Absolute Value Equations: 5  2z 1  8 5  2z  9 1. Isolate the Absolute value on one side of the equation and everything else on the other side 2. Remember that x  2 means that the object inside 5  2z  9 or 5  2z  9 2z  4 or 2z  14 2. x  0 or x  10 the absolute value has a distance of 2 away from zero. The only numbers with a distance of 2 away from zero are 2 and -2. Hence, x  2 or x  2 . Use the 3. No solution. An absolute value cannot equal a negative number same thought process for solving other absolute value equations 4. x  2 or x  1 5. y  3 Note: An absolute value cannot equal a negative value. y 1  7  y y 1  7  y or y 1  (7  y) x  2 does not make any sense. 0  8 or 2 y  6 Note: Always check your answers!! No solution or y  3 Hence, y  3 is the only solution E. Solving Exponential Equations 1. x  3 10x  1000 10x  103  x  3 2. x  1 3. x  4 4. x  1, 1 5. x  1, 3 Some properties you will need to be familiar with. If ar  as , then r  s If ar  br , then a  b F. Solving Logarithmic Equations 1. x  2 3 log2 (x  5)  log2 (1 5x) Properties of logarithms to be familiar with:  If logb M  logb N , then M  N  If logb x  y , then this equation can be rewritten in x  5  1 5x exponential form as by  x 6x  4 2. x  1,1 2 log3 (x 1)  log3 (4x) log (x 1)2  log (4x)  logb (M N)  logb M  logb N  M       r 3 3  logb M  r (x 1)2  4x x2  2x 1  0  Always check your answer!! Bases and arguments of logarithms cannot be negative 3. x  3 is the only solution since -3 causes the argument of a logarithm to be negative 1 4. x  is the only solution since -1 causes the argument of a logarithm to be negative 2 5. x  1 6. x  is the only solution since -3 causes the argument of a logarithm to be negative ln 4 2 ln 3  ln 4 32 x  4x1 2x ln 3  (x 1) ln 4 2x ln 3  x ln 4  ln 4 x(2 ln 3  ln 4)  ln 4 G. Solving Radical Equations 5 1. y  8 4 2 y 1  2  0  1 2 2 y 1  1 4 Solving Equations with radicals: 1. Isolate the radical on one side of the equation and everything else on the other side 2. If it is a square root, then square both sides. If it is a cube root, then cube both sides, etc… 3. Solve for the given variable and check your answer Note: A radical with an even index such as , 4 , 6 ,… 2 y  5 4 cannot have a negative argument (The square root can but you must use complex numbers). 2. x  4 3. x  5 5x 1  2 5x 1  2  0 x 1  5x 12  2 x 12 5x 1  4  x 1 4. No solution. x  4 does not work in the original equation. 5. x  2 6. w  3, 3 IV. Solving Inequalities A. Linear 1. 3 x  4  2 5 3 x  6 When solving linear inequalities, you use the same steps as solving an equation. The difference is when you multiply or divide both sides by a negative number, you must change the direction of the inequality. 5  3 5 x  10 Interval Notation: , 10 For example: 15  13 5  3 2. x  7 3. x  4 Interval Notation: ,7 Internal Notation: 4,  4. 4  x  5 Internal Notation: 4, 5 B. Absolute Value 4x 1  6 Think of the inequality sign as an alligator. If the alligator is 6  4x 1  6 1. 7  4x  5 facing away from the absolute value sign such as, x  5 , then one can remove the absolute value and write  7  x  5 4 4 5  x  5. This expression indicates that x cannot be farther than 5 units away from zero. If the alligator faces the absolute value such as, x  5 , then one can remove the absolute value and write x  5 or x  5 . These expressions express that x cannot be Interval:  7 , 5  less than 5 units away from zero.  4 4  2. x  1 or x  5 2 Interval:  ,  5  1,   2    3. x  20 or x  10 Interval: , 2010,  4. 5  x  10 Interval: 5,10 C. Quadratic or Rational 3x2 11x  4  0 3x 1 x  4  0 1 Steps to solving quadratic or rational inequalities. 1. Zero should be on one side of the inequalities while everything else is on the other side. 2. Factor x  and 3 x  4 make the above factors zero. 3. Set the factors equal to zero and solve. 4. Draw a chart. You should have a number line and lines dividing regions on the numbers that make the factors zero. Write the factors in on the side. 5. In each region, pick a number and substitute it in for x in each factor. Record the sign in that region. 6. In our example, 3x 1 is negative in the first region when we substitute a number such as -2 in for x. Moreover, 3x 1 will be negative everywhere in the first region. Likewise, x-4 will be Answer:   1 ,4  x  3y  6   2.  ,  4    1 ,      3. 2, 3 negative throughout the whole first region. If x is a number in the first region, then both factors will be negative. Since a negative times a negative number is positive, x in the first region is not a solution. Continue with step 5 until you find a region that satisfies the inequality. 7. Especially with rational expression, check that your endpoints do not make the original inequality undefined. 4.  ,  7  1, 3  2    V. Lines and Regions 1. x – intercept: (5,0) y – intercept: (0,6) 6 slope:  5 2. x – intercept: (3,0) y – intercept: None slope: None 3. x – intercept: None y – intercept: (0,-4) slope: 0 4. y  1 x  4 2 1 1 5. y   x  3 4 2 6. x  y  2 7. x  3y  6 13 VI. Graphing Relations 1. y  Domain: 2,  Range: 0,  2. y  Domain: 0,  Range: 2,  3. y  x 1 x  2 Domain: All Real Numbers except -2 Range: ,1 1,  4. f  x   x 1  3 Domain: ,  Range: , 3 5. f  x  2x  5 x2  9 Domain: All Real Number except 3 Range: All Real Numbers 6. x  y2  2 Domain: 2,  Range: ,  7. y  x2  8x  6 Domain: ,  Range: 22,  8. y  Domain: , 0 Range: 0,  9. y  3x Domain: ,  Range: 0,  10. 2 h  x  3x2  2x 1 Domain: All Real Numbers except: Range: 2,  , 0  1 ,1 3 VII. Exponents and Radicals 1. 2x 2. 5 147  4  35 3 16 19 3. 5 3  x7 4. y4 5. y3  54a6b2 2  6 2 a6b12 6.  9a3b8    a3b6   36     3 27a3  3a  3a • 3 4ab   3 2a2b2 3 2a2b2 8. 5  3 2a2b2 3 4ab 2ab 2b  x  x  3  9.  x  3   x  3   x  9    VIII. Complex Numbers 1. 16  4  4i 12i  8i 2. 16 • 9  4i3i  12i2  12 16 4i 4i 3i 12i2 12 4 3.   •        9 3i 3i 3i 9i2 9 3 4. 4  3i4  3i  16  9i2  16  9  25 5. 4  3i2  4  3i4  3i  16  24i  9i2  16  24i  9  7  24i 6. i25  i • i24  i i2 12  i 112  i 3  2i 7. 4  5i • 4  5i  4  5i 12  23i 10i2 16  25i2  12  23i 10  2  23i 16  25 41 41 IX. Exponential Functions and Logarithms 1. f  x  3x 1 2. g  x  2x1 3. log 1 8 64  2 4. 52  25 log2 x  4 5. 24  x 16  x 6. x=3; -3 is not a solution because bases are not allowed to be negative 7. h  x  log3 x 8. log 3  log 3  log y 4 y 4 4  r nt 9. A  P 1    where A = Money ended with P = Principle started with r = Yearly interest rate n = Number of compounds per year t = Number of years  0.12 4t 1000  850 1 4    20  0.12 4t 17  1 4     20   0.12 4t log  17   log 1 4      log  20   4t log 1 0.12   17   4      log  20     t 4 log 1 0.12   4    X. Systems of Equations  3 k  2    1 3   5 5   8 2  1.  , 2,    2.  k   for k  Natural numbers  2 4   5 5   k      5 8 1 0 2  3.  5 14  4.  2 4 0   1 2 5  1  1   2 2  5. 5 6.  1 1    4 4  XI. Story Problems 1. Let x = the money Pete earns 2x 10  x  760 Pete earns $250 2x 10  the money Sam earns Sam earns $510 2. x = burned calories walking 23x  5x  770 x  70 3x = burned calories running Answer: 210 calories 3. x = length of pole 2 x  25  1 x  x 3 6 Answer: 150 feet XII. Conic Sections 1. a)  x  22  y2  16 Center: (2,0) Radius: 4  x 12 b) y  2 2   1 16 9 Center: (-1,2) Foci: 1 7, 2  x 12 c) y  2 2   1 16 9 Center: (-1,2) Foci: (-6,2), (4,2) y  3 x  3 Asymptotes: 4 4 y   3 x  5 4 4 d)  x  22  y  4 Vertex: (2,4) Foci:  2, 15   4    17 Directrix: y  4 2. a) Circle  x  22  y2  16 b) Ellipse  x 12 y  2 2   1 16 9 c) Hyperbola  x 12 y  2 2   1 d) Parabola 16 9 y   x  22  4 XIII. Sequence and Series 1. 1, 4, 7, 10 2. 0, 3, 8, 15 3. 3, 5, 9, 17 4. an  3n 1  1 n1 5. a  4 •     2 3n 6. 2k 1  11 3  5  7  9 11  35 k 0 7.   4  xk y4k  y4  4xy3  6x2 y2  4x3 y  x4 k k 0   XIV. Functions 1. f 3  g 2  3 12  15 2. f 5  g 4  19  0  19 3. f 1• g 2  7 •12  84 4. f 5  19   19 g 5 9 9 5.  g f 2  g  f 2  g 5  9 6. f g  x  f 16  x2   216  x2  9  2x2  41 7. f 1  x  x  9 ; 2 f 1 2  2  9   7 2 2 8. 3 XV. Fundamental Counting Rule, Permutations, & Combinations 1. 56 2. 120 3. 720 4. Committee 35 Elected 210 XVI. Trigonometry 1. f  x  sin x 2. g  x  cos2x 3. x  6  40 tan 72 129.1 4. Distance between the boats  65 tan 42  65 tan 35 50 11.59meters

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