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Pre-Calculus H Summer Assignment Part 1: Functions (0, 4) (–3,0) (4, –1) 1. For which of the following functions is f (3) f (3) ? (D) f ( x) 4 x2 f ( x) 4 4 f ( x) x f ( x) 4 x3 (E) f ( x) x4 4 (A) (B) (C) (6,0) 3. The graph of y f ( x) is shown above. If 3 x 6 , for how many values of x does f ( x) 2 ? (A) (B) (C) (D) (E) 0 1 2 3 more than 3 y y 6 4 2 –6 –4 –2 O 2 4 6 x –2 –4 x –6 2. The figure above shows the graph of the function h. Which of the following is closest to h(5)? (A) (B) (C) (D) (E) 1 2 3 4 5 4. The figure above shows the graph of a quadratic function f that has a minimum at the point (1, 1). If f (b) f (3) , which of the following could be the value of b? (A) (B) (C) (D) (E) –3 –2 –1 1 5 y (7, 6) (–6, 0) (6, 0) O x 5. Based on the graph of the function f above, what are the values of x for which f ( x) is negative? (A) (B) (C) (D) (E) 6 x 0 0 x6 6 x7 6 0 6 6 x 0 and 6 x 7 7. The quadratic function g is given by g ( x) ax2 bx c , where a and c are negative constants. Which of the following could be the graph of g? (A) x O (B) x O 3 2 x2 for all nonzero x, then x what is the value of f (2) ? 6. If f ( x) (A) (B) 11 2 7 2 1 2 5 2 (C) (D) (E) 7 (C) x O (D) x O (E) O x x 0 1 2 f(x) a 24 b 8. The table above shows some values for the function f . If f is a linear function, what is the value of a b ? (A) (B) (C) (D) (E) 24 36 48 72 It cannot be determined from the information given. 9. Let the function f be defined by f ( x) x 1 . If 2 f ( p) 20 , what is the value of f (3 p) ? Questions 10-11 refer to the following functions g and h. g (n) n2 n h(n) n2 n 10. g (5) h(4) (A) (B) (C) (D) (E) 0 8 10 18 32 11. Which of the following is equivalent to h(m 1) ? (A) (B) (C) (D) (E) g (m) g (m) 1 g (m) 1 h(m) 1 h(m) 1 y g 4 3 f 2 1 O 12. If the function f is defined by f ( x) x2 bx c , where b and c are positive constants, which of the following could be the graph of f? (A) O x 1 2 3 x 4 13. The graphs of the functions f and g are lines, as shown above. What is the value of f (3) g (3) ? (A) (B) (C) (D) (E) 1.5 2 3 4 5.5 (B) O x yx 2 Q P yax (C) O x 2 O 14. The figure above shows the graphs of y x2 and y a x2 for some constant (D) a. If the length of PQ is equal to 6, what is the value of a ? O x (A) (B) (C) (D) (E) (E) O x 6 9 12 15 18 15. Let the function h be defined by h(t ) 2(t 3 3) . When h(t ) 60 , what is the value of 2 3t ? (A) (B) (C) (D) (E) 35 11 7 –7 –11 17. Let the operation be defined a b aa bb for all numbers a and b where a b . If 1 2 2 x , what is the value of x? (A) (B) (C) (D) (E) 4 3 2 1 0 y B (– 12 , b) C ( 12 , c) y O 4 A (– 12 , a) x 4 16. The shaded region in the figure above is bounded by the x-axis, the line x 4 , and the graph of y f ( x) . If the point (a, b) lies in the shaded region, which of the following must be true? I. a 4 II. b a III. b f (a) (A) (B) (C) (D) (E) x y = f(x) I only III only I and II only I and III only I, II, and III D ( 12 , d) Note: Figure not drawn to scale. 18. In the figure above, ABCD is a rectangle. Points A and C lie on the graph of y px3 , where p is a constant. If the area of ABCD is 4, what is the value of p? y f ( x) y g ( x) (–1, 3) (2, 1) x O x O (1, –3) (4, –5) 19. Let the function f be defined by f ( x) 2 x 1 . If 12 f ( t ) 4 , what is the value of t? (A) (B) (C) (D) (E) 3 2 7 2 9 2 49 4 81 4 (A) (B) (C) (D) (E) 20. For all positive integers w and y, where w y , let the operation be defined w y 2 . For how many 2w y positive integers w is w 1 equal to 4? by w y (A) (B) (C) (D) (E) 21. The figures above show the graphs of the functions f and g. The function f is defined by f ( x) x3 4x . The function g is defined by g ( x) f ( x h) k , where h and k are constants. What is the value of hk? None One Two Four More than four –6 –3 –2 3 6 22. Let f be the function defined by f ( x) x2 18 . If m is a positive number such that f (2m) 2 f (m) , what is the value of m ? Pre-Calculus H Summer Assignment Part 2: Complex Numbers 1. Expanding Complex Numbers A. What is 1 1 1 75 ? 7 0 i (i 1) i B. What is (2 3i)5 ? 2. Graphing Complex Numbers A. Let z1 5 2i . In the space provided below, graph the following complex numbers: z1 , i z1 ( z1 ) 2 and z1 . B. Calculate z1 . 3. Equating Like Coefficients. Find the values of x and y in the following: A. 2 x 5i 4(2 yi) 12 B. (2 i)2 (2 yi) x 5i 4. Square Root of a Complex Number Example: What is the (complex) value of 4 3i ? Solution: We suppose that the answer is some number. We don’t know what the answer is, so we’ll call it a bi , since that can be any number you could think of. In other words, we need to find a and b such that: 4 3i a bi Now algebra happens. Square both sides and get 4 3i a 2 b2 2abi and get a system of two equations to solve by equating like coefficients: a 2 b2 4 and 2ab 3 . Solving that systems (steps omitted for brevity) yields a 3 22 and b 22 or a 3 2 2 and b 2 2 . Thus 4 3i 3 2 2 A. What is the value of 2 2 i or 3 4i ? 4 3i 3 2 2 2 2 i B. What is the value of 8 6i ? 5. Proofs (Verifications). Let z a bi . Prove: 1 A. Re( z ) ( z z ) 2 B. 4 z 2 z z z z 2 2 Pre-Calculus H Summer Assignment Part 3: Graphing Polynomials 1. If f ( x) 3x3 8x2 15x 4 has one root given by f (1) 0 , find the other roots of f . 3. A polynomial function g ( x) has exactly three x intercepts, at x 2 , x 1 and at x 5 . If g (2) 4 , give a possible equation for g ( x) . 2. If f ( x) 4x3 11x2 4x 20 has one root given by f (2) 0 , find the other roots of f . 4. A quadratic polynomial g ( x) ax2 bx c has rational values for a, b and c. If g (1 i) 0 and g (1) 2 , find the values of a, b and c. 5. Find the quotient and remainder when 4 x3 2 x 1 is divided by x 2 3 . 6. Find the value of k such that when x 4 3x3 kx 2 4 x 40 is divided by x 2 , the remainder is 56. 7. In the space provided below, sketch the graph of y ( x 3)( x 1)( x 4) . Your graph need not be to scale, but it must correctly indicate the nature of the graph at its intercepts, turning points, and end behavior. 8. In the space provided below, sketch the graph of y ( x 4)( x 2 x 4) . Your graph need not be to scale, but it must correctly indicate the nature of the graph at its intercepts, turning points, and end behavior. Pre-Calculus H Summer Assignment Part 4: Logarithms 1. If x 0 and y 0 , then log x2 ( y) (A) log x y 2 (B) log x (D) (log x y)2 (C) log x ( 12 y) y 2. Which of the following correctly solves for x in the equation, ln x 2 2 ? (A) e 2 2 (B) ln 2 e (D) 2 e 2 (C) 2e 2 3. Which of the following is the solution to the equation 3e2 x 2e x 1 ? 1 (A) x ln1 (B) x (C) x ln 3 ln 3 (D) x ln 23 4. Which of the following is the graph of G( x) log(2 x) (A) (B) (C) (D) 5. Which of the following is NOT true? (A) log a ( x y ) log a x log a y (B) ln e x x (C) log a 1 0 (D) log10 x log x x log x 10 6. Rewrite the expression as a single logarithm: log 7 2 2log 7 2 (A) ln 8 (B) 1 (C) log 8 7 (D) log 7 8 4 7. Which equation best represents the graph shown? 2 (A) (B) (C) (D) y log( x 2 ) y log1 x y log 5 x y log( x 1) -10 -5 5 -2 -4 8. Which of the following is a graph of h( x) 4 x (A) (B) (C) (D) 9. Identify the domain and range of the function h( x) 7 5x (A) Domain: (, ) ; Range: (, 7) (C) Domain: (5, ) ; Range: (7, ) (B) Domain: (, 7) ; Range: (, ) (D) Domain: (5, 5) ; Range: (7, 7) 10. State the asymptote of the function g ( x) 2 x 3 (A) y = 3 (C) y = –2 (B) y = 2 (D) y = –3 11. An object is placed in a hot oven until its temperature becomes 245 F. It is then taken out and left to cool in a room at a temperature of 72 F. Which of the following equations could represent the objects temperature as a function of time t, once it has been left out to cool? (A) T (t ) 245 72e0.14t (B) T (t ) 72 173e0.14t (C) T (t ) 72 173e0.14t (D) T (t ) 245 72e0.14t 12. Which of the following is equivalent to e2 x ln a ? (A) e 2 a x (B) a x 2 (C) ea x (D) e2 a x 10 13. Solve the following. Give exact (calculator ready) answers and if your answers are in terms of logarithmic functions, use natural log ( ln ). A. e7 x 34 x 1 B. log3 2 x 1 2 log3 x 9 C. 2e x 3e x 5 D. 3ex 2 4e 2 x 14. Population growth is more realistically modeled logistically, rather than exponentially. In an experiment with the protozoan Paramecium, a biologist determines a system’s carrying capacity to be 48 (in units of 10,000 Paramecium) and its growth constant to be 0.556. Growth is measured over days and the population’s initial value is 6. (A) Write an equation that gives the population of the protozoan at a time t. (B) What will the population be after 4 days ? (C) When will the population be 50 ? Pre-Calculus H Summer Assignment Part 5: Conic Sections 1. Find the length of segment AB , and the coordinates of the midpoint of segment AB whose endpoints are A( –4, 7) and B( 6, 1). 4. Calculate the eccentricity of the conic x2 y 2 1 section given by 36 20 2. Give the equation of a circle with center (–2, 3) with radius 5. 5. Consider the parabola with vertex (–2, –1) and focus (–2, 2). Give the equation of the parabola. 3. A hyperbola has vertices at ( 2, 0) and at ( 2, 0) . Its asymptotes are given by 6. The equations of a hyperbola and a line are given by x2 2 y 2 7 and x 2 y 1 respectively. Find the coordinates of all points of intersection of the their graphs. y 12 x and y 12 x . Give the equation of the hyperbola. 7. Sketch the region bound by the curves y 9 x 2 and x 2 y 6 . Find the coordinates of all points of intersection of the curves. 8. Give an expression for A( x) the area of a rectangle as a function of an x coordinate, with one side on the x-axis and inscribed inside the top half of the ellipse given by y 9 94 x 2 . The rectangle is symmetric about the y axis. (x, y) Pre-Calculus H Summer Assignment Part 6: Trigonometry I. First and foremost, students must know the values of all six primary trigonometric functions of common angles on the unit circle. This knowledge must be automatic and accurate. Students will be expected to demonstrate their knowledge of the unit circle on the first day of class under timed circumstances. You should search for “the Unit Circle” online for blank worksheet templates to practice with and work out the values of all six primary trigonometric functions of common angles on [0, 2 ) . You will be expected to give the values of six randomly selected angles in a 90 second time period. Name: Give the values of the indicated trigonometric function for each corresponding highlighted angle. Date: Period: Seat: cos sin tan π 0 sec csc cot II. Additional Problems: 1. If is an angle such that cos 0 and tan 0 , state the quadrant that contains . 2. For the diagram shown below, give the value of csc . (–6, –2) x 3. If is an angle in the second quadrant such that tan 5 , find the value cos . 5. If sin 230 k , give another angle between 180 and 180 such that sin k . 4. If x 3cos , simplify the expression 6. Simplify: cot 1 1 tan 9 x2 in terms of functions of . x 7. Simplify: 3sin x 2sin x 2sin x csc x 6cos x cos x 8. The arc length of a sector is radius. 6 . The area of the same sector is . Find the sector’s angle and 4 1.2 2.00 2.25 1.00 0.75 0.6 2.75 0.50 0.4 0.2 3.00 1.2 3.25 1.25 0.8 2.50 9. In the figure at right, the values along the circle are angles measured in radians. The values along the axes are x and y coordinates respectively. Use the figure to approximate sin(5.75) to the nearest tenth. 1.50 1.75 -0.8 -0.6 -0.4 -0.2 -0.2 0.25 0.2 0.4 0.6 0.8 -0.4 3.50 -0.6 3.75 5.75 -0.8 4.00 4.25 4.50 5.50 4.75 5.00 1.2 1.2 6.25 6.00 5.25