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Potomac Falls High School Precalculus Semester Exam Review 2012 – 2013 This review covers the major topics in the material that will be tested on the semester exam. It is not necessarily all inclusive and additional study and problem solving practice may be required to fully prepare for the semester exam. Place answers in the blanks, when provided. Use additional paper, if necessary. Calculators may be used; however, the semester exam will have non-calculator portions. Therefore, prudence suggests you prepare with and without a calculator so you can handle any contingency. Name:___________________ Date:_______________ Period:______ Teacher:____________ Precalculus Semester Exam Review 1 CONICS Questions 1 – 4: Write the equation of a circle, in standard form, with the given conditions. 1) r = 5, (h, k) = (2, –3) 2) Center at the point (–4, 5) and tangent to the y-axis. 3) Center at the point (4, –7) and containing the point (–2, 6). 4) With endpoints af a diameter at the points (5, 7) and (–2, –2). Questions 5 – 10: Convert to standard form. Identify the features specified for each. 5) y 2 4 y 4 x 4 0 Vertex: ________ Focus: _________ Directrix: _______ Length of the latus rectum: ______ 6) 2 x 2 2 y 2 12 x 8 y 24 0 Center: ________ Radius: ________ Precalculus Semester Exam Review 2 7) y 2 4 x 2 16 x 2 y 19 0 Center: ________ Transverse axis:: _________ (equation) Vertices: ________________ Foci: ___________________ Equations of asymptotes: __________________________ 8) 2 x 2 3 y 2 8x 6 y 5 0 Center: ________ Major axis: ____________ (equation) Minor axis: ____________ (equation) Vertices: ________________ Foci: ___________________ 9) An arch in the form of a semi-ellipse is 60 feet wide and 20 feet high at the center. Find the heightof the arch 10 feet from the center. Precalculus Semester Exam Review 3 10) The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cables are 500 feet apart and 120 feet high. If the cables are at a height of 15 feet midway between the towers, what is the height of the cables at a point 100 feet from the center of the bridge? 11) A sealed-beam headlight is in the shape of a paraboloid of revolution. The bulb, which is placed at the focus, is 1 inch from the vertex. If the depth is to be 2 inches, what is the diameter of the headlight at its opening? 12) A whispering gallery is 80 feet wide. The height of the ceiling in the center is 25 feet. Where are the foci in relation to the walls of the gallery? Precalculus Semester Exam Review 4 CONICS State the dimensions of the following matrices. 3 2 7 9 13) 1 0 3 5 8 2 10 6 3 2 7 9 14) 1 0 3 5 8 2 10 6 9 15) 6 5 8 17 6 7 5 15 16) 1 143 2 11 13 3 Solve for the variable(s): 3 5 0 2 3 11 17) 3 25 2 x 4 15 14 5 6 0 1 12.5 10 7 1 2 23 18) 5 8 x 4 2 3 2 16 1 6 4 11 y 7 8 5 3 27.5 13 13.5 100 37 14 Find the product. If the product is not defined, explain why. 5 19) 3 1 7 5 2 3 7 22) 0 4 2 0 1 6 Precalculus Semester Exam Review 1 0 20) 4 6 5 4 9 1 1 3 0 23) 4 3 2 12 4 2 4 5 9 3 0 1 21) 0 2 4 2 Solve for the variables. 2 1 2 1 6 24) 3 2 4 x 19 0 2 4 3 y 9 2 4 1 3 y 4 25) 2 1 13 8 2 x 1 1 1 Questions 26 – 27: Write an inventory matrix and a cost per item matrix. Then use multiplication to write a total cost matrix. 26) A softball team needs to buy 12 bats at $21 each, 45 balls at $4 each, and 15 uniforms at $30 each. 27) A teacher is buying supplies for two art classes. For class 1, the teacher buys 24 tubes of paint, 12 brushes, and 17 canvasses. For class 2, the teacher buys 20 tubes of paint, 14 brushes, and 15 canvasses. Each tube of paint costs #3.35, each brush costs $1.75, and each canvass costs $4.50. Questions 28 – 29: Find the inverse of each matrix, if it exists. No calculator. 2 6 28) 1 3 14 8 29) 6 4 Precalculus Semester Exam Review 6 Questions 30 – 31: Solve for X. 2x2 by hand, 3x3 with calculator. 5 4 10 30) X 3 2 16 2 1 0 5 31) 1 4 2 X 15 3 2 1 7 Questions 32 – 33: Solve the following systems using matrices. Check your answers. All 2x2 systems must be solved by hand. Systems 3x3 and larger may be solved with a calculator. However, show the matrix equation for all systems. 3x 7 y 7 32) 7 x 3 y 3 x 5 y 10 z 13 33) 2 x y 3 z 18 4 x 6 y 12 z 7 34) A flower farmer wants to plant three types of bulbs: gladiolas, irises, and tulips. The gladiolas cost $75per acre to plant, the irises cost $100 per acre to plant, and the tulips cost $50 per acre to plant. The farmer wants to plant 200 acres of bulbs and spend a total of $15,000. The farmer decides to plant twice as many gladiolas as irises. Write a system of equations and use a matrix to find the total number of acres of each type of flower. Precalculus Semester Exam Review 7 FUNCTIONS 35) The cost of having a carpet installed is $25.00 for delivery and $1.50 per square yard for the actual installation. a) Find a linear equation that models the cost of having a carpet delivered and installed. b) Find the number of square yards of carpet installed if the bill for delivery and installation is $60.25 without the tax. 36) Graph: y x 2 2 37) Given f ( x) x 2 1 , evaluate each of the following and simplify: a) f (2) b) f ( a b) 2 c) f x 38) Find the domain and range of each function: a) g ( x) x 1.5 D: ___________ b) h( x) R: ____________ 3 x x 20 2 D: ___________ R: ____________ Precalculus Semester Exam Review 8 39) Find the inverse of each function. Is the function one-to-one? a) f ( x) x 2 b) g ( x) 1 2x 3 40) Graph each function and determine its domain and range. x 1, x 0 a) f ( x) x 1, x 0 b) g ( x) x 3 D: ____________ D: ____________ R: ____________ R: ____________ 41) Determine whether each function is even, odd, or neither. Show your algebraic check. a) f ( x) 2 x 4 3 x 2 4 Precalculus Semester Exam Review b) g ( x) 3x5 2 x3 x 2 1 9 42) Graph the reflection of each function in the given line. a) x-axis c) Line y x b) y-axis 43) Graph each equation by determining the basic function and using transformations. Write the “basic” function from which each is derived. a) y b) y 2 x 3 1 x2 y ________ y ________ 44) The height, s, in feet, of a ball thrown into the air is given by s 16t 2 96t 112 , where t is the time in seconds. Find the maximum height and the time the ball takes to reach this height. maximum height = ____________ time to reach max height = ______ Precalculus Semester Exam Review 10 TRIGONOMETRY 11 counterclockwise rotation, find the measure of the angle in degrees. Round answer to three 9 decimal places. angle measure = _________ 45) For an 46) Express 218 18' 46" in decimal form. Round answer to nearest thousandth. 47) Express 7 in degrees. 5 _____________ 48) Express 318 in radians. Leave answer in terms of . 49) If cos _____________ _____________ 12 , and is in Quadrant IV, determine the exact value of sin . 13 sin ________ 50) The terminal side of an angle in standard position passes through the point ( 7, 24) . Determine the exact values of the six trigonometric functions. sin _______ csc _______ cos _______ sec _______ tan _______ cot _______ Precalculus Semester Exam Review 11 51) Determine the exact value of tan 690 . _____________ 52) Evaluate sin 62.2 to four decimal places. _____________ 53) If 0 360 and cot 1.4176 , determine to the nearest tenth of a degree. _____________ For each function determine the amplitude, period, phase shift, vertical shift, and the equations for two vertical asymptotes, if applicable. Function 54) f ( x) 3cos Amp Period Phase Vertical shift Shift 4 55) g ( x) sin(3 ) 1 56) h( x) tan 2 2 3 Precalculus Semester Exam Review 12 Domain Range 57) Write a sine function with an amplitude of 3, with a period of , and with a phase shift of 2 6 y = _________________________ 58) Write a tangent function with a vertical stretch of 3, a period of 2 , and a phase shift of . 4 y = _________________________ 59) Using your knowledge of the Unit Circle, find the exact values: a) sin 3 9 e) sec 2 7 4 2 b) tan 3 c) csc 7 f) cos 6 15 g) sin 4 d) cot 13 6 17 h) tan 6 60) Given a point on the terminal side of an angle, find the exact value of the requested trig function. a) P 8, 17 sin b) 1 1 P , c) 3 4 tan P 3, 5 cos 61) You are riding a bicycle with wheels that have a radius of 25 inches. If you are traveling at 18 miles per hour, what is the rotational speed of the tires in revolutions per second? What is the angular speed of the wheels in radians per second? Precalculus Semester Exam Review 13