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Potomac Falls High School Precalculus Semester Exam Review 2013 – 2014 You must show work to receive credit! 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 at 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 height of the arch 10 feet from the center. 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? Precalculus Semester Exam Review 3 MATRICES State the dimensions of the following matrices. 3 2 7 9 11) 1 0 3 5 8 2 10 6 9 12) 6 5 Solve for the variable(s): 3 5 0 2 3 11 13) 3 25 2 x 4 15 14 Find the product. If the product is not defined, explain why. 1 0 14) 4 6 5 4 9 3 0 1 15) 0 2 4 2 5 2 3 7 16) 0 4 2 0 1 6 Precalculus Semester Exam Review 4 Questions 17-18: Write an inventory matrix and a cost per item matrix. Then use multiplication to write a total cost matrix. 17) A softball team needs to buy 12 bats at $21 each, 45 balls at $4 each, and 15 uniforms at $30 each. 18) 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 19-20: Find the inverse of each matrix, if it exists. No calculator. 2 6 19) 1 3 14 8 20) 6 4 Questions 21-22: Solve for X. 2x2 by hand, 3x3 with calculator. 5 4 10 21) X 3 2 16 2 1 0 5 22) 1 4 2 X 15 3 2 1 7 Precalculus Semester Exam Review 5 Questions 23-24: 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 23) 7 x 3 y 3 x 5 y 10 z 13 24) 2 x y 3 z 18 4 x 6 y 12 z 7 FUNCTIONS 25) 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. 26) Graph: y x 2 2 Precalculus Semester Exam Review 6 27) Given f ( x) x 2 1 , evaluate each of the following and simplify: b) f ( a b) a) f (2) 2 c) f x 28) 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: ____________ 29) Find the inverse of each function. Is the function one-to-one? a) f ( x) x 2 b) g ( x) 1 2x 3 30) 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: ____________ Precalculus Semester Exam Review 7 31) Determine whether each function is even, odd, or neither. Show your algebraic check. a) f ( x) 2 x 4 3 x 2 4 b) g ( x) 3x5 2 x3 x 2 1 32) Graph the reflection of each function in the given line. a) x-axis c) Line y x b) y-axis 33) 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 ________ Precalculus Semester Exam Review y ________ 8 34) 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 = ______ TRIGONOMETRY 11 counterclockwise rotation, find the measure of the angle in degrees. Round answer to three 9 decimal places. angle measure = _________ 35) For an 36) Express 218 18' 46" in decimal form. Round answer to nearest thousandth. 37) Express 7 in degrees. 5 _____________ 38) Express 318 in radians. Leave answer in terms of . 39) If cos _____________ _____________ 12 , and is in Quadrant IV, determine the exact value of sin . 13 sin ________ Precalculus Semester Exam Review 9 40) 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 _______ 41) Determine the exact value of tan 690 . _____________ 42) Evaluate sin 62.2 to four decimal places. _____________ 43) If 0 360 and cot 1.4176 , determine to the nearest tenth of a degree. _____________ 44) Find the reference angle of 125° _____________ 45) Find the measure of a coterminal angle ' for the angle Precalculus Semester Exam Review 10 7 5 _____________ For each function determine the amplitude, period, phase shift, vertical shift, and the equations for two vertical asymptotes, if applicable. Function 46) f ( x) 3cos Amp Period Phase Vertical shift Shift Domain Range 4 47) g ( x) sin(3 ) 1 48) h( x) tan 2 2 3 49) Write a sine function with an amplitude of 3, with a period of , and with a phase shift of 2 6 y = _________________________ 50) Write a tangent function with a vertical stretch of 3, a period of 2 , and a phase shift of . 4 y = _________________________ Precalculus Semester Exam Review 11 51) 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 52) 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 53) 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 12 For Questions # 54-55, graph the function given. As part of your work shown, label the x-axis and y-axis and the critical points on the graph. Also identify all the aspects of the function used to create the graph. 54) y cos 3 x 1 2 55) y 3 tan 3x 2 4 Amplitude: ______ Asymptotes: _____ and _____ Period: ______ Period: Unit: ______ Unit: ______ ______ Phase Shift: ______ Phase Shift: ______ Vertical Shift: ______ Vertical Shift: ______ Precalculus Semester Exam Review 13