MAC 1105 Fall 2010 Sample Final Solve the equation by factoring. 1) 15x2 + 29x + 12 = 0 4 3 A) - , 3 5 B) - 4 1 ,15 4 C) 4 3 ,3 5 D) 4 3 , 3 5 Objective: (1.4) Solve Quadratic Equation by Factoring Solve the equation by completing the square. 2) x2 + 4x + 13 = 0 A) -2 - i 13, -2 + i 13 C) {2 - 3i, 2 + 3i} B) {1, -5} D) {-2 - 3i, -2 + 3i} Objective: (1.4) Solve Quadratic Equation by Completing the Square Solve the equation using the quadratic formula. 3) (2x - 1)(x + 1) = 1 1 + 17 1 - 17 , A) 4 4 C) B) -1 + 37 -1 - 37 , 2 2 D) 1+ 2 17 1 , 2 17 -1 + 17 -1 - 17 , 4 4 Objective: (1.4) Solve Quadratic Equation by Quadratic Formula Solve the problem. 4) A rectangular garden has dimensions of 17 feet by 14 feet. A gravel path of consistent width is to be built around the garden. How wide can the path be if there is enough gravel for 516 square feet? A) 8.5 ft B) 7 ft C) 6 ft D) 8 ft Objective: (1.4) Solve Apps: Quadratic Equations 5) A rock falls from a tower that is 288 feet high. As it is falling, its height is given by the formula h = 288 - 16t2 . How many seconds will it take for the rock to hit the ground (h=0)? Round your answer to the nearest tenth, if necessary. A) 4.2 sec B) 16.5 sec C) 17 sec D) 5184 sec Objective: (1.4) Solve Apps: Quadratic Equations Solve the equation by making an appropriate substitution. 6) (x + 2)2 + 7(x + 2) + 10 = 0 A) {-3, 0} B) {-7, -4} C) {4, 7} D) {0, 3} C) {9} D) {-1} Objective: (1.5) Solve Equations Using Substitution Solve the equation. 7) x - 3x - 2 = 4 A) {1, 2} B) {2, 9} Objective: (1.5) Solve Equations Containing Radicals 1 Solve the rational inequality. Write the solution in interval notation. x + 25 <2 8) x+4 A) - , 17 (4, ) B) -4, 17 C) (- , -4) 17, D) (- , -4) (4, ) Objective: (1.6) Solve Rational Inequality Solve the problem. 9) 5(x + 1) - 4 > 31 and 2(5x + 1) +4 < 14 4 , (6, ) A) B) (6, ) 5 C) , C) - 1 8 4 5 D) Objective: (1.6) Solve Compound And Inequality Solve the absolute value equation. 10) 8x - 8 + 8 = -1 1 17 A) - , 8 8 B) 17 1 , 8 8 D) Objective: (1.7) Solve Absolute Value Equation II Solve the inequality. 11) 5x - 7 2 A) - , - 9 5 [2, ) B) - , 1 9 , 5 C) 9 , 5 D) 1, C) 5 13 , 6 8 D) 9 5 Objective: (1.7) Solve Absolute Value Inequality Solve the equation. 12) 7x - 9 = x - 4 5 13 A) - , 6 8 B) 5 7 ,6 4 Objective: (1.7) Solve Absolute Value Equation III Test the equation for symmetry with respect to the x-axis, the y-axis, and the origin. 13) y = -9x4 + 6x - 6 A) origin only B) x-axis only C) no symmetry Objective: (2.2) Find Symmetries in Graph Specify the center and radius of the circle. 14) x2 - 16x + y2 + 16y + 92 = 0 A) center: (-8, 8); radius: 36 C) center: (8, -8); radius: 6 B) center: (8, -8); radius: 36 D) center: (-8, 8); radius: 6 Objective: (2.2) Find Center and Radius of Circle Find the standard form of the equation of a circle that satisfies the given conditions. 15) Center (-5, 3); passing through the point (-2, 7) A) (x + 3)2 + (y - 5)2 = 9 B) (x - 3)2 + (y + 5)2 = 9 C) (x + 5)2 + (y - 3)2 = 25 D) (x - 5)2 + (y + 3)2 = 25 Objective: (2.2) Find Equation of Circle 2 D) x-axis, origin Solve the problem. 16) Marty's Tee Shirt & Jacket Company is to produce a new line of jackets with an embroidery of a Great Pyrenees dog on the front. There are fixed costs of $690 to set up for production, and variable costs of $40 per jacket. Write an equation that can be used to determine the total cost, C, encountered by Marty's Company in producing x jackets. A) C = 690x + 40 B) C = (690 + 40) x C) C = 690 + 40x D) C = 690 - 40x Objective: (2.3) Solve Apps: Write Linear Equation Use the given conditions to find an equation in slope-intercept form of each of the nonvertical lines. Write vertical lines in the form x = h. 17) Parallel to -5x + 8y = -39; passing through (3, -5) 8 5 55 3 39 5 55 A) y = x + 1 B) y = x C) y = - x D) y = - x + 5 8 8 8 8 8 8 Objective: (2.3) Write Equation of Line Satisfying Conditions Find the function value. 18) Let f(x) = 6x2 - 4x + 6. Find f(-x). A) -6x2 + 4x - 6 B) 5x2 + 5x + 5 C) 6x2 + 4x + 6 D) -6x2 + 5x + 6 Objective: (2.4) Evaluate Function Given Equation Find the domain of the function. x4 - 7x3 + 5 19) f(x) = 3x2 - 7x - 20 A) (- , 4) C) - , - B) (- , -4) (4, ) 5 3 - 5 , 3 D) - , - Objective: (2.4) Find Domain of Function 3 5 3 -4, 5 3 - 5 ,4 3 5 , 3 (4, ) Locate relative maximum and relative minimum points on the graph. State whether each relative extremum point is a turning point. 20) A) (-2, 3), (1, 2), and (4, 4) are relative maxima points and turning points. (2, -2) is a relative minimum point and a turning point. B) (-2, 3), (1, 2), and (4, 4) are relative maxima points. (0, 0) and (2, -2) are relative minima points. C) (-2, 3), (1, 2), and (4, 4) are relative maxima points and turning points. (0, 0) and (2, -2) are relative minima points and turning points. D) (4, 4) is a relative maximum point and a turning point. (2, -2) is a relative minimum point and a turning point. Objective: (2.5) Find Relative Maxima Find the requested value. 21) Find f(0) for f(x) = x - 8, if x < 5 7 - x, if x 5 A) 7 B) -3 C) 2 Objective: (2.6) Evaluate Piecewise Function Graph the function. x if x 0 22) f(x) = x2 if 0 < x 3 Objective: (2.6) Graph Piecewise Function 4 D) -8 Write an equation for a function whose graph fits the given description. 23) The graph of f(x) = x2 is shifted 2 units to the left. This graph is then vertically shrunk by a factor of reflected across the x-axis. Finally, the graph is shifted 7 units downward. 1 1 1 A) y = - (x - 2)2 - 7 B) y = - (x - 2)2 + 7 C) y = - (x + 2)2 - 7 6 6 6 D) y = 1 and 6 1 (x - 2)2 - 7 6 Objective: (2.7) Write Equation for Description of Function Graph the function by starting with a function from the library of functions and then using the techniques of shifting, compressing, stretching, and/or reflecting. 24) g(x) = - 2(x + 5)2 + 3 Objective: (2.7) Graph Transformation Graph the function y = g(x), given the graph of y = f(x). 1 25) g(x) = - f(x) 2 Objective: (2.7) Graph Transformation of Function Given Graph Find the given value. f (-5) when f(x) = 3x - 2 and g(x) = 5x2 + 14x + 3. 26) Find g A) - 17 58 B) 3 58 C) 5 13 Objective: (2.8) Evaluate Sum/Difference/Product/Quotient of Function 5 D) 5 58 Find the composite function for the given functions. 27) f g for f(x) = x + 4 and g(x) = 8x - 8 A) 2 2x - 1 B) 2 2x + 1 C) 8 x - 4 D) 8 x + 4 - 8 C) 2302 D) 4606 Objective: (2.8) Perform Composition of Functions Evaluate the expression. 28) (g g)(5) when f(x) = 4x + 4 and g(x) = 2x2 - 2 A) 5410 B) 94 Objective: (2.8) Evaluate Composition of Functions Determine whether the given function is one-to-one. If it is one-to-one, find its inverse. 9 29) f(x) = x-6 A) Not one-to-one B) f-1 (x) = -6 + 9x x C) f-1 (x) = 6x + 9 x D) f-1 (x) = x -6 + 9x Objective: (2.9) Find Inverse of Function (Equation) The graph of a function f is given. On the same axes, sketch the graph of f-1. 30) Objective: (2.9) Graph Inverse of Function Given Graph Find the quadratic function y = f(x) that has the given vertex and whose graph passes through the given point. 31) vertex: (-1, -4) passing through: (-3, -8) A) y = -x2 - 2x - 5 B) y = -3x2 + 2x + 5 C) y = x2 + 2x - 4 D) y = -x2 + 1x - 4 Objective: (3.1) Write Quadratic Function Given Vertex and Point Determine whether there is a maximum or minimum value for the given function, and find that value. 32) f(x) = - 7 -4x2 - 8x A) Minimum: 0 B) Minimum: 3 C) Maximum: 3 D) Maximum: -3 Objective: (3.1) Find Max/Min Value of Quadratic Function Solve the problem. 33) A farmer has 1400 feet of fence with which to fence a rectangular plot of land. The plot lies along a river so that only three sides need to be fenced. Estimate the largest area that can be fenced. A) 245,000 ft2 B) 196,000 ft2 C) 367,500 ft2 D) 294,000 ft2 Objective: (3.1) Solve Apps: Quadratic Functions 6 Find the vertical asymptote(s), if any, of the graph of the rational function. x 2 + 4x 34) f(x) = x2 - 3x - 28 A) x = -7, x = 4 C) no vertical asymptote B) x = 7, x = -4 D) x = 7 Objective: (3.6) Determine Vertical Asymptote of Function Use the graph of the rational function f(x) to complete the statement. . 35) As x -2 +, f(x) A) 0 B) -2 C) - D) + Objective: (3.6) Determine Behaviors of Function Given Graph Find the horizontal asymptote(s), if any, of the graph of the rational function. 8x2 - 7x - 3 36) g(x) = 5x2 - 6x + 2 A) y = 7 6 B) y = 0 C) y = 8 5 D) no horizontal asymptote Objective: (3.6) Determine Horizontal Asymptote of Function Solve the problem. 37) Economists use what is called a Leffer curve to predict the government revenue for tax rates from 0% to 100%. Economists agree that the end points of the curve generate 0 revenue, but disagree on the tax rate that produces the maximum revenue. Suppose an economist produces this rational function, 10x(100 - x) R(x) = , where R is revenue in millions at a tax rate of x percent. Use a graphing calculator to graph 75 + x the function. What tax rate produces the maximum revenue? What is the maximum revenue? A) 34.9%; $207 million B) 35.8%; $209 million C) 39.6%; $209 million D) 37.5%; $210 million Objective: (3.6) Solve Apps: Rational Functions 7 Solve for the requested variable. 38) y varies directly as x and inversely as the square of z, and y = 48 when x = 96 and z = 4. Find y when x = 100 and z = 2. A) y =100 B) y = 200 C) y =400 D) y =12.5 Objective: (3.8) Solve Variation Equation (Use Constant of Variation k) Solve the problem. 39) The volume V of a gas at constant temperature varies inversely as the pressure P on it. The volume of a gas is 240 cm3 under a pressure of 18 kg/cm2 . What will be its volume under a pressure of 30 kg/cm2 ? [Assume that the temperature remains constant]. A) 130 cm3 B) 400 cm3 C) 420 cm3 D) 144 cm3 Objective: (3.8) Solve Apps: Variation Solve the equation for x by first rewriting both sides as powers of the same base. 1 6x + 2 = 9 x- 4 40) 3 A) 1 4 B) 3 4 C) 5 3 D) 2 7 Objective: (4.1) Solve Exponential Equation Find the exponential function of the given form that contains the given point(s). 41) Form: f(x) = c · a x Points: (0, 4) and (2, 100) A) f(x) = 5x B) f(x) = 4 · 5 x C) f(x) = 2 D) f(x) = 20x Objective: (4.1) Find Exponential Function Given Conditions Solve the problem. 42) Kimberly invested $5000 in her savings account for 4 years. When she withdrew it, she had $5570.24. Interest was compounded continuously. What was the interest rate on the account? A) 2.85% B) 2.6% C) 2.7% D) 2.8% Objective: (4.2) Solve Apps: Natural Exponential Functions Convert to a logarithmic equation. 43) 63 = 216 A) 3 = log216 6 B) 216 = log6 3 C) 3 = log6 216 D) 6 = log3 216 Objective: (4.3) Write Exponential Equation in Logarithmic Form Evaluate the expression without a calculator. 44) log5 (log9 9) A) 0 B) 5 C) 9 D) 1 C) 14 D) 4 Objective: (4.3) Evaluate Logarithm Solve the logarithmic equation. 1 45) log27 x - 5 = 3 A) 387,420,494 B) 145.296115 Objective: (4.3) Solve Logarithmic Equation 8 Write the expression in expanded form. 5 9x2 1 - x 46) log 2(x + 1)2 A) log 9 + log x2 + log(1 - x)1/5 - log 2 - log(x + 1)2 1 B) log 9 + 2 log x + log(1 - x) - log 2 + 2 log(x + 1) 5 C) log(9x2 5 1 - x) - log(2(x + 1)2 ) 1 D) log 9 + 2 log x + log(1 - x) - log 2 - 2 log(x + 1) 5 Objective: (4.4) Expand Logarithmic Expression Use the change-of-base formula and a calculator to evaluate each logarithm. 47) log3 7 + log7 10 A) 2.0959 B) 2.9545 C) 1.4097 D) 2.5303 Objective: (4.4) Use Change of Base Formula to Evaluate Log Find the value of the expression without using a calculator. 48) 62 log6 3 + log6 2 A) 18 B) 11 C) 6 D) 12 Objective: (4.4) Evaluate Expression Containing Logs (No Calculator) Solve the exponential equation and approximate the result, correct to three decimal places. 49) 2(3x - 1) = 22 A) 1.153 B) 1.133 C) 4.000 D) 1.820 Objective: (4.5) Solve Exponential Equation Approximately Solve the logarithmic equation. 50) log (x + 10) - log (x + 4) = log x A) 2 B) 6 C) 2, -5 Objective: (4.5) Solve Logarithmic Equation 9 D)