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Berkeley City College Practice Problems Math 1 Precalculus - Final Exam Preparation Name__________________________________________ Please print your name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. List the intercepts for the graph of the equation. 1) 4x2 + y 2 = 4 1) Objective: (1.2) Find Intercepts from an Equation 2) y = x 2 + 13x + 40 2) Objective: (1.2) Find Intercepts from an Equation Write the standard form of the equation of the circle. 3) 3) y (3, 5) (7, 5) x Objective: (1.4) Write the Standard Form of the Equation of a Circle Write the standard form of the equation of the circle with radius r and center (h, k). 4) r = 6; (h, k) = (2, -3) 4) Objective: (1.4) Write the Standard Form of the Equation of a Circle Find the center (h, k) and radius r of the circle with the given equation. 5) 5(x + 6)2 + 5(y + 2)2 = 30 5) Objective: (1.4) Write the Standard Form of the Equation of a Circle Find the value for the function. 6) Find f(x - 1) when f(x) = 4x2 - 3x + 3. 6) Objective: (2.1) Find the Value of a Function Instructor: K Pernell 1 2 7) Find f(x + 1) when f(x) = x - 8 . x- 2 7) Objective: (2.1) Find the Value of a Function Find the domain of the function. 8) g(x) = 3x x2 - 25 8) Objective: (2.1) Find the Domain of a Function Defined by an Equation 9) h(x) = x-1 x3 - 81x 9) Objective: (2.1) Find the Domain of a Function Defined by an Equation 10) f(x) = 13 - x 10) Objective: (2.1) Find the Domain of a Function Defined by an Equation For the given functions f and g, find the requested function and state its domain. 11) f(x) = 7x - 4; g(x) = 9x - 9 Find f ∙ g. 11) Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions Solve the problem. 12) Find (fg)(4) when f(x) = x - 3 and g(x) = -5x2 + 12x - 4. 12) Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions 13) Find f (-2) when f(x) = 2x - 5 and g(x) = 3x2 + 14x + 4. g 13) Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions Find and simplify the difference quotient of f, f(x + h) - f(x) , h≠ 0, for the function. h 14) f(x) = 3x2 14) Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions 2 The graph of a function f is given. Use the graph to answer the question. 15) For what numbers x is f(x) > 0? 15) 100 100 -100 -100 Objective: (2.2) Obtain Information from or about the Graph of a Function 16) How often does the line y = 1 intersect the graph? 16) 5 5 -5 -5 Objective: (2.2) Obtain Information from or about the Graph of a Function Answer the question about the given function. 17) Given the function f(x) = 5x2 + 10x + 8, is the point (-1, 3) on the graph of f? 17) Objective: (2.2) Obtain Information from or about the Graph of a Function 18) Given the function f(x) = 4x2 + 8x - 6, is the point (-2, 2) on the graph of f? 18) Objective: (2.2) Obtain Information from or about the Graph of a Function Find the average rate of change for the function between the given values. 19) f(x) = x 2 + 7x; from 1 to 5 19) Objective: (2.3) Find the Average Rate of Change of a Function Write the equation of a sine function that has the given characteristics. 20) The graph of y = x2, shifted 6 units upward 20) Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts 21) The graph of y = x, shifted 7 units to the right 21) Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts 3 Graph the function by starting with the graph of the basic function and then using the techniques of shifting, compressing, stretching, and/or reflecting. 22) f(x) = (x - 3)2 - 4 22) y 10 5 -10 -5 5 10 x -5 -10 Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts 23) f(x) = (x + 6)3 + 7 23) y 10 5 -10 -5 5 10 x -5 -10 Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts Use the accompanying graph of y = f(x) to sketch the graph of the indicated equation. 24) y = - 2f(x + 5) + 4 y y 10 y = f(x) 10 5 5 -10 -5 5 10 x -10 -5 5 -5 -5 -10 -10 Objective: (2.5) Graph Functions Using Compressions and Stretches 4 10 x 24) Determine the slope and y-intercept of the function. 25) h(x) = -5x - 3 25) Objective: (3.1) Graph Linear Functions Use the slope and y-intercept to graph the linear function. 26) g(x) = -2x + 1 26) y 5 -5 5 x -5 Objective: (3.1) Graph Linear Functions Find the vertex and axis of symmetry of the graph of the function. 27) f(x) = -x2 - 6x + 5 27) Objective: (3.3) Identify the Vertex and Axis of Symmetry of a Quadratic Function Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value. 28) f(x) = x 2 + 2x - 2 28) Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function 29) f(x) = -x2 - 2x - 6 29) Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function Solve the problem. 30) You have 220 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. 30) Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function 31) You have 324 feet of fencing to enclose a rectangular region. What is the maximum area? 31) Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function 32) The quadratic function f(x) = 0.0038x2 - 0.45x + 36.90 models the median, or average, age, y, at which U.S. men were first married x years after 1900. In which year was this average age at a minimum? (Round to the nearest year.) What was the average age at first marriage for that year? (Round to the nearest tenth.) Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function 5 32) Use the figure to solve the inequality. 33) f(x) < 0 16 33) y 12 8 4 (-4, 0) -16 -12 -8 (3, 0) -4 4 8 12 16 x -4 -8 -12 -16 Objective: (3.5) Solve Inequalities Involving a Quadratic Function Solve the inequality. 34) x2 - 8x ≥ 0 34) Objective: (3.5) Solve Inequalities Involving a Quadratic Function 35) x2 - 64 ≤ 0 35) Objective: (3.5) Solve Inequalities Involving a Quadratic Function Form a polynomial whose zeros and degree are given. 36) Zeros: -3, -2, 2; degree 3 36) Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity 37) Zeros: -4, -2, -1, 1; degree 4 37) Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. 38) f(x) = 2(x - 7)(x + 5)3 38) Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity 39) f(x) = 2(x2 + 3)(x + 2)2 39) Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity 40) f(x) = 1 x(x2 - 5) 3 40) Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity Find the x- and y-intercepts of f. 41) f(x) = (x + 1)(x - 6)(x - 1)2 41) Objective: (4.1) Analyze the Graph of a Polynomial Function 6 42) f(x) = -x2(x + 6)(x2 + 1) 42) Objective: (4.1) Analyze the Graph of a Polynomial Function Find the power function that the graph of f resembles for large values of |x|. 43) f(x) = 7x - x3 43) Objective: (4.1) Analyze the Graph of a Polynomial Function Find the vertical asymptotes of the rational function. 3x 44) f(x) = (x - 4)(x - 8) 44) Objective: (4.2) Find the Vertical Asymptotes of a Rational Function 45) f(x) = x-4 45) 16x - x 3 Objective: (4.2) Find the Vertical Asymptotes of a Rational Function Give the equation of the horizontal asymptote, if any, of the function. 2 46) h(x) = 8x - 5x - 2 5x2 - 4x + 8 46) Objective: (4.2) Find the Horizontal or Oblique Asymptotes of a Rational Function 3 47) h(x) = 3x - 4x - 7 2x + 2 47) Objective: (4.2) Find the Horizontal or Oblique Asymptotes of a Rational Function 48) g(x) = x + 8 x2 - 49 48) Objective: (4.2) Find the Horizontal or Oblique Asymptotes of a Rational Function Find the indicated intercept(s) of the graph of the function. 49) y-intercept of f(x) = (5x - 15)(x - 3) x2 + 9x- 19 49) Objective: (4.3) Analyze the Graph of a Rational Function 50) x-intercepts of f(x) = (x - 2)(2x + 9) x2 + 5x - 5 50) Objective: (4.3) Analyze the Graph of a Rational Function 7 Graph the function. 51) f(x) = 2x (x - 3)(x - 5) 51) y 40 20 -8 -4 4 8 x -20 -40 Objective: (4.3) Analyze the Graph of a Rational Function 2 52) f(x) = x + x - 30 x2 - x - 20 52) y 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 x -4 -6 -8 -10 Objective: (4.3) Analyze the Graph of a Rational Function Solve the inequality. 53) 2x2 + 3x < 20 53) Objective: (4.4) Solve Polynomial Inequalities 54) x- 7 < 0 x+9 54) Objective: (4.4) Solve Rational Inequalities 55) x + 18 < 9 x 55) Objective: (4.4) Solve Rational Inequalities 8 56) 8x ≥ 4x 7-x 56) Objective: (4.4) Solve Rational Inequalities Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. 57) f(x) = x4 - 24x2 - 25 57) Objective: (4.5) Find the Real Zeros of a Polynomial Function 58) f(x) = x3 + 3x2 - 4x - 12 58) Objective: (4.5) Find the Real Zeros of a Polynomial Function 59) f(x) = 4x3 - 3x2 + 16x - 12 59) Objective: (4.5) Find the Real Zeros of a Polynomial Function Find the intercepts of the function f(x). 60) f(x) = x3 + 2x2 - 5x - 6 60) Objective: (4.5) Find the Real Zeros of a Polynomial Function 61) f(x) = -x2(x + 6)(x2 + 1) 61) Objective: (4.5) Find the Real Zeros of a Polynomial Function Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. 62) Degree 4; zeros: 4 - 5i, 8i 62) Objective: (4.6) Use the Conjugate Pairs Theorem Form a polynomial f(x) with real coefficients having the given degree and zeros. 63) Degree: 3; zeros: -2 and 3 + i. 63) Objective: (4.6) Find a Polynomial Function with Specified Zeros 64) Degree: 4; zeros: -1, 2, and 1 - 2i. 64) Objective: (4.6) Find a Polynomial Function with Specified Zeros For the given functions f and g, find the requested composite function value. 65) f(x) = x + 5, g(x) = 2x; Find (f ∘ g)(0). 65) Objective: (5.1) Form a Composite Function 66) f(x) = 2x + 4, g(x) = 2x2 + 1; Find (g ∘ g)(1). 66) Objective: (5.1) Form a Composite Function 67) f(x) = 2x + 7, g(x) = -2/x; Find (g ∘ f)(3). 67) Objective: (5.1) Form a Composite Function For the given functions f and g, find the requested composite function. 68) f(x) = 7x + 6, g(x) = 5x - 1; Find (f ∘ g)(x). Objective: (5.1) Form a Composite Function 9 68) 69) f(x) = 3 , g(x) = 8 ; x- 1 3x Find (f ∘ g)(x). 69) Objective: (5.1) Form a Composite Function 70) f(x) = x + 4, g(x) = 8x - 8; Find (f ∘ g)(x). 70) Objective: (5.1) Form a Composite Function Indicate whether the function is one-to-one. 71) {(5, -3), (6, -3), (7, -7), (8, 9)} 71) Objective: (5.2) Determine Whether a Function Is One-to-One 72) {(4, 5), (-5, -4), (8, -3), (-8, 3)} 72) Objective: (5.2) Determine Whether a Function Is One-to-One Decide whether or not the functions are inverses of each other. 73) f(x) = 8x - 5, g(x) = x + 8 5 73) Objective: (5.2) Find the Inverse of a Function Defined by an Equation 74) f(x) = 2x - 2, g(x) = 1 x + 1 2 74) Objective: (5.2) Find the Inverse of a Function Defined by an Equation 75) f(x) = 2x2 + 1, g(x) = x-1 2 75) Objective: (5.2) Find the Inverse of a Function Defined by an Equation The function f is one-to-one. Find its inverse. 76) f(x) = x2 + 4, x ≥ 0 76) Objective: (5.2) Find the Inverse of a Function Defined by an Equation 77) f(x) = 5x - 7 3 77) Objective: (5.2) Find the Inverse of a Function Defined by an Equation 78) f(x) = 3 x+7 78) Objective: (5.2) Find the Inverse of a Function Defined by an Equation Solve the equation. 79) 27 - 3x = 1 4 79) Objective: (5.3) Solve Exponential Equations 10 2 80) 2x - 3 = 64 80) Objective: (5.3) Solve Exponential Equations Change the exponential expression to an equivalent expression involving a logarithm. 81) 73 = 343 81) Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements to Exponential Statements 82) 52 = x 82) Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements to Exponential Statements Change the logarithmic expression to an equivalent expression involving an exponent. 83) log 1 = -3 2 8 83) Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements to Exponential Statements 84) ln x = 4 84) Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements to Exponential Statements 85) ln 1 = -5 e5 85) Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements to Exponential Statements Find the exact value of the logarithmic expression. 86) log4 1 64 86) Objective: (5.4) Evaluate Logarithmic Expressions 87) log 5 5 87) Objective: (5.4) Evaluate Logarithmic Expressions Solve the equation. 88) log3 (x2 - 2x) = 1 88) Objective: (5.4) Solve Logarithmic Equations 89) 7 + 9 ln x = 4 89) Objective: (5.4) Solve Logarithmic Equations 90) ln x +5 = 3 90) Objective: (5.4) Solve Logarithmic Equations 11 91) e x +7 =5 91) Objective: (5.4) Solve Logarithmic Equations Write as the sum and/or difference of logarithms. Express powers as factors. x3 92) log 4 y8 92) Objective: (5.5) Write a Logarithmic Expression as a Sum or Difference of Logarithms 7 93) log 16 3 q 2p 93) Objective: (5.5) Write a Logarithmic Expression as a Sum or Difference of Logarithms 94) log 4 mn 19 94) Objective: (5.5) Write a Logarithmic Expression as a Sum or Difference of Logarithms Express as a single logarithm. 95) 3 loga (2x + 1) - 2 loga (2x - 1) + 2 95) Objective: (5.5) Write a Logarithmic Expression as a Single Logarithm Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round your answer to two decimal places. 96) log8.7 7.6 96) Objective: (5.5) Evaluate Logarithms Whose Base Is Neither 10 Nor e Solve the equation. 97) log (3 + x) - log (x - 5) = log 3 97) Objective: (5.6) Solve Logarithmic Equations 98) log3 x + log3(x - 24) = 4 98) Objective: (5.6) Solve Logarithmic Equations Solve the equation. Express irrational answers in exact form and as a decimal rounded to 3 decimal places. 99) 3 x = 41 - x 99) Objective: (5.6) Solve Exponential Equations Convert the angle in degrees to radians. Express the answer as multiple of !. 100) 144° 100) Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees 101) 87° 101) Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees 12 Convert the angle in radians to degrees. 102) - 11! 6 102) Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees 103) 34 ! 9 103) Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees Find the exact value. Do not use a calculator. 104) cos 2! 104) Objective: (6.2) Find the Exact Values of the Trigonometric Functions of Quadrantal Angles 105) tan (19!) 105) Objective: (6.2) Find the Exact Values of the Trigonometric Functions of Quadrantal Angles 106) cos 16! 3 106) Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60° 107) sec 19! 4 107) Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60° Find the exact value of the expression. Do not use a calculator. 108) tan 7! + tan 5! 4 4 108) Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60° 109) sin 135° - sin 270° 109) Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60° 110) tan 150° cos 210° 110) Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60° Name the quadrant in which the angle θ lies. 111) cot θ < 0, cos θ > 0 111) Objective: (6.3) Determine the Signs of the Trigonometric Functions in a Given Quadrant In the problem, sin θ and cos θ are given. Find the exact value of the indicated trigonometric function. 112) sin θ = 1 , cos θ = 4 15 4 Find cot θ. 112) Objective: (6.3) Find the Values of the Trigonometric Functions Using Fundamental Identities 13 Find the exact value of the indicated trigonometric function of θ. 113) tan θ = - 8 , θ in quadrant II Find cos θ. 5 113) Objective: (6.3) Find Exact Values of the Trig Functions of an Angle Given One of the Functions and the Quadrant of the Angle Without graphing the function, determine its amplitude or period as requested. 114) y = -2 sin 1 x Find the amplitude. 3 114) Objective: (6.4) Determine the Amplitude and Period of Sinusoidal Functions 115) y = -3 cos 1 x 4 Find the period. 115) Objective: (6.4) Determine the Amplitude and Period of Sinusoidal Functions Match the given function to its graph. 116) 1) y = sin 2x 2) y = 2 cos x 3) y = 2 sin x 4) y = cos 2x A 3 -2π 116) B y 3 2 2 1 1 -π π 2π x -2π -π -1 -1 -2 -2 -3 -3 C y π 2π π 2π x D y -2π 3 3 2 2 1 1 -π π 2π x -2π -π -1 -1 -2 -2 -3 -3 Objective: (6.4) Graph Sinusoidal Functions Using Key Points 14 y x MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. 117) Which one of the equations below matches the graph? A) y = 4 cos 2x B) y = 2 cos 1 x 4 C) y = 4 sin 1 x 2 117) D) y = 4 cos 1 x 2 Objective: (6.4) Graph Sinusoidal Functions Using Key Points 118) Which one of the equations below matches the graph? A) y = 2 cos 3x B) y = 2 sin 1 x 3 C) y = -2 sin 1 x 3 D) y = 2 cos 1 x 3 118) Objective: (6.4) Graph Sinusoidal Functions Using Key Points SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 119) What is the y-intercept of y = csc x? 119) Objective: (6.5) Graph Functions of the Form y = A tan(ωx) + B and y = A cot(ωx) + B Find the exact value of the expression. 120) cos-1 3 120) 2 Objective: (7.1) Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function 15 121) cos-1 - 3 121) 2 Objective: (7.1) Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function Find the inverse function f-1 of the function f. 122) f(x) = 3 cos x + 6 122) Objective: (7.1) Find the Inverse Function of a Trigonometric Function 123) f(x) = 2 tan(10x - 3) 123) Objective: (7.1) Find the Inverse Function of a Trigonometric Function Find the exact solution of the equation. 124) 4 cos-1 x = ! 124) Objective: (7.1) Solve Equations Involving Inverse Trigonometric Functions Find the exact value of the expression. 1 125) cos sin-1 2 125) Objective: (7.2) Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions 4 126) sec sin-1 9 126) Objective: (7.2) Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions Solve the equation on the interval 0 ≤ θ < 2". 127) 2 cos θ + 3 = 2 127) Objective: (7.3) Solve Equations Involving a Single Trigonometric Function Solve the equation. Give a general formula for all the solutions. 128) sin θ = 1 128) Objective: (7.3) Solve Equations Involving a Single Trigonometric Function Solve the equation on the interval 0 ≤ θ < 2". 129) cos2 θ + 2 cos θ + 1 = 0 129) Objective: (7.3) Solve Trigonometric Equations Quadratic in Form 130) 2 sin 2 θ = sin θ 130) Objective: (7.3) Solve Trigonometric Equations Quadratic in Form Simplify the trigonometric expression by following the indicated direction. 131) Rewrite in terms of sine and cosine: tan x ∙ cot x Objective: (7.4) Use Algebra to Simplify Trigonometric Expressions 16 131) 132) Multiply sin θ by 1 + cos θ 1 - cos θ 1 + cos θ 132) Objective: (7.4) Use Algebra to Simplify Trigonometric Expressions The polar coordinates of a point are given. Find the rectangular coordinates of the point. 2! 133) 7, 3 133) Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates 3! 134) -5, 4 134) Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates The rectangular coordinates of a point are given. Find polar coordinates for the point. 135) (0, -8) 135) Objective: (9.1) Convert from Rectangular Coordinates to Polar Coordinates 136) (- 3, -1) 136) Objective: (9.1) Convert from Rectangular Coordinates to Polar Coordinates The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). 137) x2 + 4y 2 = 4 137) Objective: (9.1) Transform Equations between Polar and Rectangular Forms 138) y 2 = 16x 138) Objective: (9.1) Transform Equations between Polar and Rectangular Forms Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. 139) r = 5 139) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6 r Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations 17 140) r = 2 sin θ 140) 6 5 4 3 2 1 1 2 3 4 5 6 r -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations 141) r sin θ = 5 141) 6 5 4 3 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 1 2 3 4 5 r Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. 142) 3 + i 142) Objective: (9.3) Convert a Complex Number between Rectangular Form and Polar Form 143) 2 + 2i 143) Objective: (9.3) Convert a Complex Number between Rectangular Form and Polar Form Find zw or z as specified. Leave your answer in polar form. w ! ! 144) z = 8 cos + i sin 6 6 144) ! ! w = 3 cos + i sin 2 2 Find zw. Objective: (9.3) Find Products and Quotients of Complex Numbers in Polar Form 18 145) z = 10(cos 45° + i sin 45°) w = 5(cos 15° + i sin 15°) Find z . w 145) Objective: (9.3) Find Products and Quotients of Complex Numbers in Polar Form Write the expression in the standard form a + bi. 146) 2(cos 15° + i sin 15°) 3 146) Objective: (9.3) Use De Moivre's Theorem 147) (- 3 + i)6 147) Objective: (9.3) Use De Moivre's Theorem MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the equation to the graph. 148) (y + 2)2 = 8(x + 1) 148) A) B) y -10 y 10 10 5 5 -5 5 10 x -10 -5 -5 -5 -10 -10 C) 5 10 x 5 10 x D) y -10 y 10 10 5 5 -5 5 10 x -10 -5 -5 -5 -10 -10 Objective: (10.2) Analyze Parabolas with Vertex at (h, k) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find an equation for the parabola described. 149) Vertex at (8, 7); focus at (8, 3) 149) Objective: (10.2) Analyze Parabolas with Vertex at (h, k) 19 Write an equation for the graph. 150) 150) y 5 -5 5 x (2, -1) -5 Objective: (10.3) Analyze Ellipses with Center at (h, k) Find the center, foci, and vertices of the ellipse. 151) 3x2 + 4y 2 - 36x + 32y + 160 = 0 151) Objective: (10.3) Analyze Ellipses with Center at (h, k) Graph the equation. (x + 1)2 + (y - 2)2 = 1 152) 9 4 152) y 5 -10 -5 5 10 x -5 Objective: (10.3) Analyze Ellipses with Center at (h, k) Find an equation for the hyperbola described. 153) center at (2, 4); focus at (0, 4); vertex at (1, 4) 153) Objective: (10.4) Analyze Hyperbolas with Center at (h, k) Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola. 154) x2 - 25y 2 + 6x + 50y - 41 = 0 Objective: (10.4) Analyze Hyperbolas with Center at (h, k) 20 154) Graph the hyperbola. (y + 2)2 - (x - 2)2 = 1 155) 4 9 155) y 5 -5 5 x -5 Objective: (10.4) Analyze Hyperbolas with Center at (h, k) Solve the system of equations by substitution. 156) 5x - 2y = -1 x + 4y = 35 156) Objective: (11.1) Solve Systems of Equations by Substitution Solve the system of equations. 157) x - y + 3z = -16 4x + z = -5 x + 2y + z = -3 157) Objective: (11.1) Solve Systems of Three Equations Containing Three Variables Perform the row operation(s) on the given augmented matrix. 158) R 2 = -4r1 + r2 1 3 10 4 2 -8 158) Objective: (11.2) Perform Row Operations on a Matrix 159) (a) R 2 = -4r1 + r2 (b) R 3 = -2r1 + r3 (c) R 3 = 6r2 + r3 1 -3 -5 -2 4 -5 -4 5 2 5 4 6 159) Objective: (11.2) Perform Row Operations on a Matrix Write the partial fraction decomposition of the rational expression. x 160) (x - 4)(x - 5) Objective: (11.5) Decompose P/Q, Where Q Has Only Nonrepeated Linear Factors 21 160) 161) x +3 3 x - 2x2 + x 161) Objective: (11.5) Decompose P/Q, Where Q Has Repeated Linear Factors 162) 10x + 2 (x - 1)(x2 + x + 1) 162) Objective: (11.5) Decompose P/Q, Where Q Has a Nonrepeated Irreducible Quadratic Factor Graph the equations of the system. Then solve the system to find the points of intersection. 163) y = x2 - 8x + 16 163) y = -x + 6 y 10 5 -10 -5 5 10 x -5 -10 Objective: (11.6) Solve a System of Nonlinear Equations Using Substitution Solve the system of equations using substitution. 164) x2 + y 2 = 25 164) x + y = -7 Objective: (11.6) Solve a System of Nonlinear Equations Using Substitution 165) 165) ln x = 3ln y 3x = 27y Objective: (11.6) Solve a System of Nonlinear Equations Using Substitution Solve using elimination. 166) x2 + y 2 = 145 166) x2 - y 2 = 17 Objective: (11.6) Solve a System of Nonlinear Equations Using Elimination 22 167) 167) 2x 2 + y 2 = 17 3x2 - 2y 2 = -6 Objective: (11.6) Solve a System of Nonlinear Equations Using Elimination 23 Answer Key Testname: MATH1_FINAL_EXAM_PRACTICE 1) (-1, 0), (0, -2), (0, 2), (1, 0) 2) (-5, 0), (-8, 0), (0, 40) 3) (x - 5)2 + (y - 5)2 = 4 23) y 10 4) (x - 2)2 + (y + 3)2 = 36 5) (h, k) = (-6, -2); r = 6 6) 4x2 - 11x + 10 5 -10 x2 + 2x - 7 7) x- 1 10 x 5 10 x -10 24) y 10 5 14) 3(2x+h) 15) [-100, -60), (70, 100) 16) three times 17) Yes 18) No 19) 13 20) y = x2 + 6 -10 -5 -5 -10 x-7 25) m = -5; b = - 3 26) y y 10 5 5 -10 5 -5 8) {x|x ≠ -5, 5} 9) {x|x ≠ -9, 0, 9} 10) {x|x ≤ 13} 11) (f ∙ g)(x) = 63x2 - 99x + 36; all real numbers 12) -36 3 13) 4 21) y = 22) -5 -5 5 10 x -5 5 -5 -5 -10 27) (-3, 14) ; x = -3 28) minimum; - 3 29) maximum; - 5 30) 55 ft by 55 ft 31) 6561 square feet 32) 1959, 23.6 years old 33) {x|-4 < x < 3}; (-4, 3) 34) (-∞, 0] or [8, ∞) 35) [-8, 8] 36) f(x) = x3 + 3x2 - 4x - 12 for a = 1 24 x Answer Key Testname: MATH1_FINAL_EXAM_PRACTICE 37) x4 + 6x3 + 7x2 - 6x - 8 38) 7, multiplicity 1, crosses x-axis; -5, multiplicity 3, crosses x-axis 39) -2, multiplicity 2, touches x-axis 40) 0, multiplicity 1, crosses x-axis; 5, multiplicity 1, crosses x-axis; - 5, multiplicity 1, crosses x-axis 41) x-intercepts: -1, 1, 6; y-intercept: -6 42) x-intercepts: -6, 0; y-intercept: 0 43) y = -x3 53) -4 , 54) (-9, 7) 55) (-∞, 0) or (3, 6) 56) (-∞, 0] or [5, 7) 57) -5, 5; f(x) = (x - 5)(x + 5)(x2 + 1) 58) -3, -2, 2; f(x) = (x + 3)(x + 2)(x - 2) 3 59) ; f(x) = (4x - 3)(x2 + 4) 4 60) x-intercepts: -3, -1, 2; y-intercept: -6 61) x-intercepts: -6, 0; y-intercept: 0 62) 4 + 5i, -8i 63) f(x) = x3 - 4x2 - 2x + 20 44) x = 4, x = 8 45) x = 0, x = -4 8 46) y = 5 64) f(x) = x4 - 3x3 + 5x2 - x - 10 65) 5 66) 19 2 67) 13 47) no horizontal asymptotes 48) y = 0 45 49) 0, 19 9 50) (2, 0), - , 0 2 68) 35x - 1 9x 69) 8 - 3x 51) y 70) 2 2x - 1 71) No 72) Yes 73) No 74) Yes 75) Yes; Exclude the interval (-∞, 1) 76) f-1(x) = x - 4, x ≥ 4 40 20 -8 -4 4 8 x -20 77) f-1(x) = 3x + 7 5 -40 52) 12 78) f-1(x) = x3 - 7 79) {3} 80) {3, -3} 81) log 343 = 3 7 82) log x = 2 5 -3 83) 2 = 1 8 y 10 8 6 4 2 -10 -8 -6 -4 5 2 -2 -2 2 4 6 8 10 84) e4 = x x 85) e-5 = 1 e5 -4 -6 86) -3 1 87) 2 -8 -10 -12 88) {3, -1} 25 Answer Key Testname: MATH1_FINAL_EXAM_PRACTICE 89) {e -1/3 } 121) 90) {e6 - 5} 91) {ln 5 - 7} 92) 3 log x - 8 log y 4 4 1 log 16 - 2 log q - log p 93) 3 3 3 7 94) 5! 6 122) f-1(x) = cos-1 x 123) f-1(x) = 1 tan-1 +3 2 10 1 log m + 1 log n - 1 log 19 4 4 4 2 2 2 2 124) 2 3 95) loga a (2x + 1) (2x - 1)2 125) 96) 0.94 97) {9} 98) {27} 2 3 2 126) 9 127) ln 4 ≈ 0.558 99) ln 3 + ln 4 65 65 2! , 4! 3 3 128) θ|θ = ! + 2k! 2 4! 100) 5 129) {!} 29! 101) 60 130) 0, !, ! , 5! 6 6 102) -330° 103) 680° 104) 1 105) 0 106) - 1 2 131) 1 1 + cos θ 132) sin θ 107) 108) 0 134) 109) 7 7 3 133) - , 2 2 2 2 +2 2 110) - 5 5 2 2 , -5 2 2 ! 135) 8, 2 5! 136) 2, 6 3 6 137) r2(cos2 θ + 4 sin2 θ) = 4 138) r sin 2 θ = 16 cos θ 111) IV 112) 15 113) - 5 x-6 3 89 89 114) 2 115) 8! 116) 1B, 2D, 3C, 4A 117) A 118) B 119) none ! 120) 6 26 Answer Key Testname: MATH1_FINAL_EXAM_PRACTICE 139) 145) 2(cos 30° + i sin 30°) 146) 4 2 + 4 2i 147) -64 148) D 149) (x - 8)2 = -16(y - 7) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6 r 150) (x - 2)2 + (y + 1)2 = 1 16 9 151) (x - 6)2 + (y + 4)2 = 1 4 3 center: (6, -4); foci: (7, -4), (5, -4); vertices: (8, -4), (4, -4) x2 + y 2 = 25; circle, radius 5, center at pole 152) y 140) 5 6 5 4 3 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -10 5 10 x -5 1 2 3 4 5 r 2 153) (x - 2)2 - (y - 4) = 1 3 154) center at (-3, 1) transverse axis is parallel to x-axis vertices at (-8, 1) and (2, 1) foci at (-3 - 26, 1) and (-3 + 26, 1) 1 1 asymptotes of y - 1 = - (x + 3) and y - 1 = (x + 3) 5 5 x2 + (y - 1)2 = 1; circle, radius 1, center at (0, 1) in rectangular coordinates 141) 5 4 3 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -5 155) y 1 2 3 4 5 r 5 -5 5 -5 y = 5; horizontal line 5 units above the pole 142) 2(cos 30° + i sin 30°) 143) 2 2(cos 45° + i sin 45°) 2! + i sin 2! 144) 24 cos 3 3 156) x = 3, y = 8; (3, 8) 157) x = 0, y = 1, z = -5; (0, 1, -5) 3 10 158) 1 0 -10 -48 27 x Answer Key Testname: MATH1_FINAL_EXAM_PRACTICE 1 -3 -5 -2 159) 0 7 16 13 0 53 110 88 -4 + 5 160) x-4 x-5 161) 3 + -3 + 4 x x - 1 (x - 1)2 162) 4 + -4x + 2 x - 1 x2 + x + 1 163) y 10 5 -10 -5 5 10 x -5 -10 (2, 4), (5, 1) 164) x = -3, y = -4; x = -4, y = -3 or (-3, -4), (-4, -3) 165) x = 3 3, y = 3 or (3 3, 3) 166) x = 9, y = 8; x = -9, y = 8; x = 9, y = -8; x = -9, y = -8 or (9, 8), (-9, 8), (9, -8), (-9, -8) 167) x = 2, y = 3; x = 2, y = -3; x = -2, y = 3; x = -2, y = -3 or (2, 3), (2, -3), (-2, 3), (-2, -3) 28