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1 Princeton High School Mathematics Department PreCalculus Summer Assignment Summer assignment vision and purpose: The Mathematics Department of Princeton Public Schools looks to build both confidence and competence in students as they strengthen their mathematical understanding. Success in mathematics is dependent upon mastering critical concepts. Such concepts will be extended and applied in more challenging contexts in successive years. It is for this reason the department is supporting and providing summer assignments for students. These assignments will serve as a reinforcement of previously learned skills. Students will be expected to complete the summer assignment in preparation for the coming school year. Teachers will be reviewing the work and assessing students knowledge of previously learned material to help students understand key areas of strength and weakness as they begin a new course. Directions: Please complete all of the following questions. Be sure to show all of your work in the space provided or attach all work that is completed on additional pages. Questions are divided into subgroups based on skill and concept. Sample procedures and examples have been provided to help reinforce or remind you of previously covered material. Also, please be sure to complete the following information: Name: School(s) Attended Last Year: Math Class(s) Taken Last Year: Math Teacher(s) Last Year:
2 Rationalizing the denominator and rational exponents 1. Convert to radical notation: x a) x b) x x c) 4x d) x 2. Rationalize the denominator: a) " b) " " c) " d) " " Factoring (GCF, factor by grouping, all types of trinomials, special patterns) and solving basic quadratic equations 3. What is a factor of 3x 15x + 4x 20? a) 3x 5 b) x + 5 c) 3x + 4 d) x 4 4. Solve: 2x x 15 = 0 a) 3, b), 3 c) 3, d), 3 5. Solve: x 7 = 0 a) -7,7 b) -49, 49 c) 7, 7 d) 0,7
3 Definition of a function, function notation, graphs of functions 6. Determine which relation is not a function. a) {(7,6),(8,5),(-4,2),(7,3)} b) {(6,5),(3,-2),(-4,5),(-2,1)} c) {(-1,-1),(2,3),(5,-8),(-4,2)} d) {(-8,-6),(-6,-8),(5,3),(3,5)} 7. Given that f x = 4 x, find f( 3). a) 13 b) 31 c) -23 d) Given that f x = 4 x, find f(x 2). a) 8 2x b) x c) 4x x d) x 4x Domain of different types of algebraic functions 9. Find the domain of the function {(-5,6),(2,-8),(7,12),(-2,-8)} a) {6,-8,12} b) {-5,6,2,-8,7,12,-2} c){-5,2,7,-2} d) { 6,2,7,12} 10. Find the domain of f x =. a) x 5 b) x 2 c) x 2 or x 5 d) x 2 Slope 11. What is the slope of the line with the equation: 3x 2y = 6? a) b) c) d)
4 12. Find the slope of the line containing the points (8,-5) and (3,4). a) b) c) d) Linear equation forms: slope intercept, point slope, and standard form 13. Find the y-intercept of the graph 6x 4y = 5. a) 0, b), 0 c) 0, d) 0, 14. Find the equation of the line that passes through (4,-3) and (8,-5). a) y = x 1 b) y = 2x + 11 c) y = 2x + 5 d) y = x + 1 Parallel and Perpendicular lines 15. Find the equation of the line containing the point (-1,-3) and perpendicular to the line 2x + 3y = 6. a) y = x b) y = x c) y = d) y = x
5 Solve quadratic equations by factoring, completing the square, and the quadratic formula, find the discriminant of quadratics. 16. Solve: 2x + x 28 = 0. a) 4, b) -4,7 c) 4, d) 4, 17. Solve: x 6x + 1 = 0. a) 3 ± 2 2 b) 1,5 c) 3 ± 10 d) 3 ± Solve: x + 81 = 0. a) -9,9 b) 9i, 9i c) -81 d) 81i 19. Solve x + 5x = 8 by completing the square. a) ± b) ± c) ± d) ± "
6 Solve rational equations, radical equations, and absolute value equations 20. Solve: 3x 7 x 20 = 0. a) 4 b), 4 c) ", 16 d) Solve: " = 1 a), 3 b) " c),3 d), Solve: 5x = 5. a) 4 b) 100 c) d) Solve: 3x 4 = 17. a) -39,63 b) ", 7 c) " d) 7
7 Solve Inequalities linear, compound, and absolute value 24. Solve: 15 + x > 2. a) x 17 or x 13 b) 17 < x < 13 c) x < 13 or x > 17 d) x < 17 or x > Solve: 2 < 2x a) x < b) 4 < x 3 c) < x d) < x Properties of Polynomials and patterns in graphs of polynomials 26. Determine the leading coefficient of the polynomial P(x) = 5x + 4x 2 6x 3 + x 4 a) 10 b) 4 c) 1 d) Determine the degree of the polynomial P(x) = 3x 4 2x a) 3 b) 1 c) 6 d) At most, how many turning points will the following polynomial have? P(x) = x 3 5x 4 9x +12x a) 3 b) 9 c) 11 d) 4
8 29. Use the rational zeros theorem to determine which number cannot be a possible zero of P(x) = 4x 4 + 3x 2 + x 3 a) 4 3 b) 3 4 c) 1 4 d) 3 Factor Theorem, Remainder Theorem, and Synthetic division 30. Use synthetic division to find the quotient and remainder when 9x 3 6x 2 + 4x 3 is divided by x + 1 a) 9x 2 + 3x + 7, R4 b) 9x 2 3x + 7, R( 4) c) 9x 2 3x + 7, R( 10) d) 9x 2 15x +19, R( 22) 31. Use synthetic division to determine which number is a zero of P(x) = x 4 37x 2 24x a) 5 b) -6 c) -2 d) Use synthetic division to find P(-3) for P(x) = 2x 4 + 3x 3 5x a) -286 b) -124 c) -34 d) -100
9 Finding Zeros of a polynomial 33. Find all the zeros of the polynomial function P(x) = x 4 2x 3 5x 2 + 4x + 6 a) 2, 3,± 2 b) 1,3,± 2 c) 1, 3,± 2 d) 1,3,± 5 Writing Polynomial functions given the zeros, Irrational Roots Theorem, and Complex Root Theorem 34. Find a polynomial function of lowest degree with rational coefficients and -5 and 7 as some of its zeros. a) f (x) = x 2 2x 35 b) f (x) = x 3 + 5x 2 7x 35 c) f (x) = x 3 5x 2 7x + 35 d) f (x) = x 2 + 7x + 5x Properties of inverses { ( )}. 35. Find the inverse of the relation ( 3, 2 ),( 4, 0), ( 5, 1), 1, 6 { } a) ( 3, 2), ( 4, 0), ( 5,1), ( 1, 6 ) b) {( 2, 3), ( 0, 4), ( 1, 5), ( 6,1) } c) ( 2, 3), ( 0, 4), ( 1, 5 ),( 6, 1) { } (" 1 d) 3, 1 % " 1 $ ', $ # 2 & # 4, 0 % & ', " $ 1 # 5, 1 % & ', " 1, 1 % + ) $ # 6 ', * &-
10 36. Determine which of the following is a one to one function. Finding Inverse Functions 37. Find a formula for the inverse of the function f (x) = 1 2 x 3 a) f 1 (x) = 2(x + 3) b) f 1 (x) = 2x + 3 c) f 1 (x) = x + 6 d) f 1 (x) = 1 2 x + 3
11 Exponential Functions, Graphs and Natural Base 38. Which of the following graphs illustrates the graph of g(x) = e x Evaluate: e 4 e 3 a) b) c) d) Convert to an exponential equation: log 5 x =10 a) x =10 5 b) x 10 = 5 c) 10 x = 5 d) x = 5 10
12 Logarithmic Functions and Graphs 41. Which of the following describes the transformation of the graph of f (x) = ln(x + 4) from its parent function. a) Vertical Stretch by a factor of 4 b) Horizontal Shrink by a factor of 4 c) Horizontal Shift of 4 units to the right d) Horizontal Shift of 4 units to the left 42. Convert to a logarithmic equation: 2 x =10 a) x = log 2 20 b) x = log10 c) 2 = log x 20 d) x = log Using your calculator, find log 2 10 a) 1 b) c) d) Logarithmic Function Properties 44. Express in terms of sums and differences of logarithms: log x 4 a) 2 log x log y log z b) 1 log x log y log z 2 yz c) 2 log x 1 2 log y log z d) 2log x 1 2 log y 1 2 log z
13 45. Given that log a 3 = and log a 12 =1.3869, find log a 9 a) b) c) d) Simplify: lne 4t a) 4t b) 4t c) 4 d) e 4t 47. Solve: log = x a) 25 b) 3 c) -3 d) 1 3 Solving Exponential and Logarithmic Equations 48. Solve: log 3 (3x + 6) log 3 (x 6) = 2 a) 10 b) 3 2 c) 14 d) Does not exist 49. Solve: 4 x+5 = 64 x a) 5 b) 1 3 c) 5 2 d) No solution 50. Solve: e 2 x = 0.4 a) b) c) d)
14 Open Ended Problems Complete the free response questions below. You must show work for each problem. Please put your final answer on the line provided. 51. Factor: m Factor: 5n Factor and Solve: 3y 14y 24 = Find the zeros of the function: f (x) = 4x 2 4x The director of the Glen Island Zoo wants to use 170 m of fencing to enclose a petting area of 1750 m 2. Find the dimensions of the petting area. 55.
15 56. Using synthetic division, determine whether the numbers are zeros of the polynomial function. i, 2i, 4 ; g(x) = x 3 4x 2 + 4x Solve for all the zeros of the polynomial. Provide exact answers. g(x) = x 3 4x 2 + 2x Find a polynomial function of lowest degree with rational coefficients and 3 and 4i as some of its zeros Suppose that $750 is invested at 7% interest, compounded semiannually. Find the amount of money in the account when t = 6 and when t = Solve the following logarithmic equation. Check for extraneous solutions. ln(x +8)+ ln(x 1) = 2 ln x 60.
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