Accuplacer College Level Math Study Guide

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1 Testing Center Student Success Center Accuplacer 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 epanding polynomials Factor the following polynomials: a b 5a b 60a b Epand the following: y y y y 0. y y 6 y 8y 8 7y 6y y y y 6 8r 6s 8. y y II. Simplification of Rational Algebraic Epressions Simplify the following. Assume all variables are larger than zero III. Solving Equations A. Solving Linear Equations 0. y y y B. Solving Quadratic & Polynomial Equations 8 y y t t y y 5 August 0

2 C. Solving Rational Equations 0 y y a a D. Solving Absolute Value Equations 5 z y 7 y. 5 E. Solving Eponential Equations F. Solving Logarithmic Equations log 5 log log log 5.. log log 6. ln ln 0 ln ln ln G. Solving Radical Equations y w 7 IV. Solving Inequalities Solve the following inequalities and epress the answer graphically and using interval notation. A. Solving Linear Inequalities October 0

3 B. Solving Absolute value Inequalities: Solve and Graph C. Solving Quadratic or Rational Inequalities V. Lines & Regions Find the and y-intercepts, the slope, and graph 6 + 5y = 0.. Find the and y-intercepts, the slope, and graph =.. Find the and y-intercepts, the slope, and graph y = -.. Write in slope-intercept form the line that passes through the points (, 6) and (-, ). 5. Write in slope-intercept form the line perpendicular to the graph of - y = - and containing the point (, ). 6. Graph the solution set of - y. 7. Graph the solution set of - + y < -6. VI. Graphing Relations, Domain & Range For each relation, state if it is a function, state the domain & range, and graph it. y 6. y. y 7.. y 8 6 y 8. y f 9. y. f h 6 VII. Eponents and Radicals Simplify. Assume all variables are >0. Rationalize the denominators when needed y y 9. 5a b 9a b 6 8 7a ab 5 October 0

4 VIII. Comple Numbers Perform the indicated operation and simplify i i. 7. i 5 i i 5i IX. Eponential Functions and Logarithms f 6. Solve: log 9 Graph:. Graph: g 7. Graph: h log. Epress 8 in logarithmic form 8. Use the properties of logarithms to epand as 6. Epress log5 5 in eponential form much as possible: log y 5. Solve: log 9. How long will it take $850 to be worth $000 if it is invested at % interest compounded quarterly? X. Systems of Equations & Matrices Solve the system:. Solve the system: y 7 6 y y z y 6z y z 0. 0 Multiply: Find the determinant:. Perform the indicated operation: 6. Find the Inverse: 6 XI. Story Problems Sam made $0 more than twice what Pete earned in one month. If together they earned $760, how much did each earn that month?. A woman burns up three times as many calories running as she does when walking the same distance. If she runs miles and walks 5 miles to burn up a total of 770 calories, how many calories does she burn up while running mile?. A pole is standing in a small lake. If one-sith of the length of the pole is in the sand at the bottom of the Water Line lake, 5 ft. are in the water, and two-thirds of the total length is in the air above the water, what is the length Sand of the pole? October 0

5 XII. Conic Sections Graph the following, and find the center,. Identify the conic section and put it into standard foci, and asymptotes if possible. form. a) b) c) d) ( ) y 6 a) ( ) ( y) 6 9 b) ( ) ( y) 6 9 c) ( ) y d) XIII. Sequence and Series y y 6y y 6y 99 y 0 Write out the first four terms of the sequence whose general term is a n. Write out the first four terms of the sequence whose general term is n an n. n Write out the first four terms of the sequence whose general term is an. Find the general term for the following sequence:,5,8,,, Find the general term for the following sequence:,,,,, Find the sum: 6 k 0 k 7. Epand the following: XIV. Functions k0 k k y k Let f ( ) 9 and g( ) 6. Find the following. f( ) g() 5. ( g f)( ). f(5) g() 6. f ( g( )). f( ) g( ) 7.. f (5) g(5) f () 8. f f () XV. Fundamental Counting Rule, Factorials, Permutations, & Combinations Evaluate: 8!! 8!. 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?. In a horse race, how many different finishes among the first three places are possible for a ten-horse race?. 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. October 0 5

6 XVI. Trigonometry Graph the following through on period: f ( ) sin. Graph the following through on period: g( ) cos( ). A man whose eye level is 6 feet above the ground stands 0 feet from a building. The angle of elevation from eye level to the top of the building is 7. How tall is the building?. 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. October 0 6

7 Answers I. Factoring and Epanding Polynomials When factoring, there are three steps to keep in mind. Always factor out the Greatest Common Factor. Factor what is left. If there are four terms, consider factoring by grouping. Answers:. 5 a b( ab b ) y y y (7 0)( ) y y y y y y y y (7 0 0) (7 ) ( 0 0) y y y y y y y y 7 ( ) 0( ) Since there are 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.. ( y )( y ). ( y)( y) 5. ( y )( y ) y u y 6 u6 ( u)( u) 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 y Factor the epression with u s. Then, substitute the y back in place of the u s. If you can factor more, proceed. Otherwise, you are done. 6. 7( y)( y y ) Formula for factoring the sum of two cubes: a b ( a b)( a ab b ) The difference of two cubes is: a b ( a b)( a ab b ) (r s)(r s)(9r s ) ( ) y Hint: Let u=+y y 9y When doing problems and, you may want to use Pascal s Triangle 6 October 0 7

8 II. III. Simplification of Rational Algebraic Epressions. 8 If you have Solving Equations A. Solving Linear Equations 5. or 5 B. Solving Quadratic & Polynomial Equations, you can write as a product of primes ( ). In square roots, it takes two of the same thing on the inside to get one thing on the outside:. y. 5 8 y,. 0,,5. i, 6. 0, 5. i t 6., i 7., 8. y 0 Solving Quadratics or Polynomials: Try to factor. If factoring is not possible, use the quadratic formula Note: i b b ac where a a b c 0 Eample: i i i C. Solving Rational Equations y 0 y y ( y )( y ) 0( y )( y ) y y ( y )( y ) ( y )( y ) 0 y y ( y) ( y) 0 y 0 Solving Rational Equations: Find the lowest common denominator for all fractions in the equation. Multiply both sides of the equation by the lowest common denominator. Simplify and solve for the given variable. Check answers to make sure that they do not cause zero to occur in the denominators of the original equation.. Working the problem, we get. However, causes the denominators to be zero in the original equation. Hence, this problem has no solution. October 0 8

9 a, 5 6., D. Solving Absolute Value Equations z or z 7 5 z 8 5 z 9 5z 9 or 5 z 9 z or z or No solution. An absolute value cannot equal a negative number. or 5. y y 7 y y 7 y or y (7 y) 0 8 or y 6 No solution or y Hence, y is the only solution Solving Absolute Value Equations: Isolate the Absolute value on one side of the equation and everything else on the other side. Remember that means that the object inside the absolute value has a distance of away from zero. The only numbers with a distance of away from zero are and -. Hence, or. Use the same thought process for solving other absolute value equations Note: An absolute value cannot equal a negative value. does not make any sense. Note: Always check your answers!! E. Solving Eponential Equations October , 5., F. Solving Logarithmic Equations log ( 5) log ( 5 ) , log ( ) log ( ) log ( ) log ( ) ( ) 0 Some properties you will need to be familiar with. If If a a r s a, then r s r r b, then a b Properties of logarithms to be familiar with: If log M log N, then M N 9 b b If log b y, then this equation can be rewritten in y eponential form as b log ( M N) log M log N b b b log M log M log N b b b r log M r log M b b Always check your answer!! Bases and arguments of logarithms cannot be negative N

10 is the only solution since - causes the argument of a logarithm to be negative is the only solution since - causes the argument of a logarithm to be negative is the only solution since - causes the argument of a logarithm to be negative ln ln ln ln ( )ln ln ln ln (ln ln ) ln G. Solving Radical Equations 5 y 8 y 0 y y 5 y No solution. does not work in the original equation w, Solving Equations with radicals: Isolate the radical on one side of the equation and everything else on the other side. If it is a square root, then square both sides. If it is a cube root, then cube both sides, etc. Solve for the given variable and check your answer Note: A radical with an even inde such as 6,,, cannot have a negative argument (The square root can but you must use comple numbers). IV. Solving Inequalities A. Linear 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 For eample: 5 5 Interval Notation:, 0 October 0 0

11 Interval Notation:,7. 7. Internal Notation:,. 5 Internal Notation:,5 B. Absolute Value Think of the inequality sign as an alligator. If the alligator is facing away from the absolute value sign such as, 5, then one can remove the absolute value and write 5 5. This epression indicates that cannot be farther than 5 units away from zero. If the alligator faces the absolute value such as, 5, then one can remove the absolute value and write 5 or 5. These epressions epress that cannot be less than 5 units away from zero. Interval: 7 5,. or 5 5 Interval:,,. 0 or 0 Interval:, 0 0,. 5 0 Interval: 5,0 October 0

12 C. Quadratic or Rational. 0 0 and make the above factors zero. + - Answer:, y 6,,., ,, Steps to solving quadratic or rational inequalities. Zero should be on one side of the inequalities while everything else is on the other side.. Factor. Set the factors equal to zero and solve.. 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 in each factor. Record the sign in that region. 6. In our eample, + is negative in the first region when we substitute a number such as - in for. Moreover, + will be negative everywhere in the first region. Likewise, - will be negative throughout the whole first region. If is a number in the first region, then both factors will be negative. Since a negative times a negative number is positive, 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 epression, check that your endpoints do not make the original inequality undefined. V. Lines and Regions intercept: (5,0) y intercept: (0,6) slope: 6 5. intercept: (,0) y intercept: None slope: None October 0

13 . intercept: None y intercept: (0,-) slope: y y 6. y 7. y 6 October 0

14 VI. Graphing Relations y Domain:, Range: 0,. y Domain: Range:, 0,. y Domain: All Real Numbers ecept - Range:,,. f Domain:, Range:, October 0

15 f Domain: All Real Number ecept Range: All Real Numbers 6. y Domain: Range:,, 7. y 8 6 Domain: Range:,, 8. y Domain: Range: 0,,0 October 0 5

16 9. y Domain: Range:, 0, 0. h 6 Domain: All Real Numbers ecept: Range:,,0, VII. Eponents and Radicals y 7 5 y 6 6 5a b 6 a b 8 6 9a b a b 6 7a a a ab a ab ab a b a b a b ab ab b October 0 6

17 VIII. Comple Numbers 6 9 i i 8i 6 9 i i i.. 6 i i i i i i i 9i 9 9 i i 6 9i i i i 6 i 9i 6 i 9 7 i i i i i i i i i 5i i 0i i 0 i 5i 5i 6 5i 6 5 IX. Eponential Functions and Logarithms f g. log log 6 October 0 7

18 6. =; - is not a solution because bases are not allowed to be negative h log log log log y y 9. A P r n nt where A = Money ended with P = Principle started with r = Yearly interest rate n = Number of compounds per year t = Number of years October t t t 0 0. log log log t log 7 0 log 7 t 0. log 8

19 X. Systems of Equations,,. k k 5 5 k for k Natural numbers XI. Story Problems Let = the money Pete earns Pete earns $50 0 the money Sam earns Sam earns $50. = burned calories walking = burned calories running Answer: 0 calories. = length of pole 5 Answer: 50 feet 6 XII. Conic Sections y 6 a) Center: (,0) Radius: October 0 9

20 b) y 6 9 Center: (-,) Foci: 7, c) y 6 9 Center: (-,) Foci: (-6,), (,) Asymptotes: y 5 y y d) Verte: (,) Foci: Directri: 5, 7 y y 6. a) Circle b) Ellipse c) Hyperbola y 6 9 y 6 9 y d) Parabola October 0 0

21 XIII. Sequence and Series,, 7, 0. 0,, 8, 5., 5, 9, 7. a n n n 5. n a n 6 6. k k k0 k y y y 6 y y k k XIV. Functions f g 5. f g f g 7 8. f g g f g f g 5 9 f g f f ; f October 0

22 XV. Fundamental Counting Rule, Permutations, & Combinations Committee 5 Elected 0 XVI. Trigonometry sin f. g cos. 6 0tan7 9.. Distance between the boats 65tan 65tan meters October 0