College Algebra: Midterm Review


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1 College Algebra: A Missive from the Math Department Learning College Algebra takes effort on your part as the student. Here are some hints for studying that you may find useful. Work Problems If you do, and understand every exercise that is assigned (and others), then the exams will be mostly straightforward and familiar to you. Generally speaking, the exams in College Algebra focus on problem solving. The level of difficulty of the problems on examinations will usually vary from one problem to another  just as the difficulty of (practice) exercises vary. Some problems will be easy, while some others may be a little more challenging. In brief, the more you practice, the better you will become at problem solving and your understanding of the material will deepen and improve. Study For Understanding A primary goal of this course is for you to learn to solve problems. It is not enough to solve a problem by finding a similar example worked in the text or in your lecture notes and plugging in different numbers. After all, you won t have your text and notes available during exams! You should study the text and your notes until you think you understand the new ideas, and then test your understanding by working the exercises. Read Actively Read the material on each subject in the text before the material is covered in class; by doing so, you will be prepared to understand what your instructor says in each lecture. This makes it easier to take notes and to focus your attention on the more difficult material. Reread a portion of the book after some later material has had a chance to sink in. If you don t understand, mark it and be sure to keep coming back to try again until you do understand it. Read with your pencil in hand; work out the exercises in the margin as you go. Take Advantage Of Office Hours, and Tutoring We have free tutoring at HCC both online and in person. See the HCC website for information on tutoring. You can also discuss this with your instructor; they will be happy to help you. Whether visiting with your instructor or attending tutoring, being prepared with specific questions or problems can make your visit and study time more productive. The first midterm covers all the material from Chapter Sections 1.4, 1.5, 1.6, 1.7, 2.1, 2.2, 2., 2.4, 2.5, 2.6, 2.7, 2.8, and.1. You should be familiar with the concepts in these sections. The information on this sheet is intended to help you identify some of the key points that you should be comfortable knowing. It is not a substitute for lectures, homework assignments, and quizzes nor is it intended to reflect a comprehensive coverage of all of the course content up to the midpoint of the semester. Thus, it is possible that there will be questions on exams that are not completely covered or answered by the material on this review sheet. Work through each of these problems; only use the answers to check your work. If you have difficulty, then first look for similar problems in the textbook; then ask your instructor for guidance. In addition, working with the HCC tutors can be very beneficial and help improve your time spent studying. Best Wishes, The Math Department Created/Revised by Math Department 2/17/18 Page 1
2 Solving Equations and Inequalities Quadratic Equations You should be familiar with the following terms, concepts, and expressions: quadratic equation the standard form of a quadratic equation zero or root solution completing the square the quadratic formula discriminant Note 1. A quadratic equation in standard form is written as follows: ax 2 + bx+c = 0 where a, b, and c R with a 0. Note 2. You should be able to solve a quadratic equation (i.e. find the zeros of a quadratic function) by using any of the following techniques: factoring, completing the square using the quadratic formula. Note. You should be able to determine whether a quadratic equation has 0, 1, or 2 realnumber solutions by examining the sign of its discriminant, (i.e., b 2 4ac). Note 4. You should be able to solve an equation which can be reduced to a quadratic equation by a simple substitution or algebraic manipulation(s). Exercise 1. Completing the Square a) Determine the value of m that makes the quadratic trinomial a perfect square: x 2 +42x+m. For this value of m, write the trinomial as a perfect square. b) Determine the value of m that makes the quadratic trinomial a perfect square: 2x x+m. For this value of m, write the trinomial as a perfect square. Exercise 2. Given the equation x 2 = k, complete the following statements. a) If, then there will be exactly one real solution to the equation. b) If, then there will be two distinct real solutions to the equation. c) If, then there will be two imaginary solutions to the equation. Exercise. For each of the following quadratic equations, compute the discriminant and determine nature of the solutions (i.e., the number of real or imaginary solutions) for the equation. (Do not actually solve the equations.) a) 2x 2 5x+9 = 0 b) x 2 = x+2 c) 2t(2t ) = 1 d) 18 5 x x = 1 10 Created/Revised by Math Department 2/17/18 Page 2
3 Exercise 4. Solve the following quadratic equations by using any valid method of your choosing. a) (t+) 2 +t 2 9t = 8 b) 18 5 x x = 1 c) (w 5) 2 = d) x 2 +2x+5 = 0 e) (t+1) 2 = t 2 f) (x 1)(x+4) = (x+2)(x+1) Exercise 5. Solve each of the following word problems. a) The hypotenuse of a right triangle is 40 feet long. One leg of the triangle is ft longer than the other leg. Find the lengths of the legs of the triangle. b) Nolan throws a baseball straight up in the air from a cliff that is 52 feet high. The initial velocity is 72 ft sec. The height, in feet, of the object after t sec is given by h = 16t 2 +72t+52. Find the time at which the height of the object is 124 feet above the ground. Exercise 6. Multiple Choice Solve each quadratic equation by any valid method of your choosing. However, for these multiple choice problems, choose the answer choice that represents the sum of the squares of the solutions to the given equation. a) Solve 2x 2 +x 6 = 0. The sum of the squares of the solutions is. A. 1 2 B C. 1 4 D. 7 4 E. None of A. through D. is correct b) Solve y 2 8y +2 = 0. The sum of the squares of the solutions is. A. 5 B. 5 9 C. 7 9 D. 1 9 E. None of A. through D. is correct Exercise 7. Let m R. Consider the equation x 2 2mx 5 = 0. a) Which of the following represents solutions to this equation? A. x = 2m ± m C. x = m ± 2 m E. x = m ± m 2 15 B. x = m ± m D. x = m ± m F. None of A. through E. are correct b) Discuss the nature of the solutions to this equation (i.e., are the solutions real or imaginary?) c) Suppose you are told that x = 1 is a solution to this equation. i. Find m. ii. Find the other solution to the equation. Exercise 8. Let a R with a 0. Consider the equation ax 2 2x+ = 0. a) For this quadratic equation (in the variable x), find the discriminant. b) If this equation has one exactly real solution, then what can we say about a? c) What are the values of a, if any, that result in the equation having imaginary solutions? Created/Revised by Math Department 2/17/18 Page
4 Other Types Of Equations Exercise 9. Quadratic In Form a) Solve the equation (2x 2 5) 2 16(2x 2 5)+9 = 0. b) Solve the equation p 2/ +8 = 2p 1/. Exercise 10. Solve each of the cubic equations by factoring. a) Solve the equation x +9x 2 = 16(x+9). b) Solve the equation 75x +x 2 100x 4 = 0. Exercise 11. Solve each radical equation. a) Solve the equation 9x + 28 = x+4. The sum of the squares of the solutions is. A. 4 B. 25 C. 16 D. 1 E. None of A. through D. is correct b) Solve the equation 9 x x + 4 = 1. Exercise 12. Solve each absolute value equation. a) Solve the equation 2 = x b) Solve the equation 4 x = 2x+1. The sum of the squares of the solutions is. A. 17 B. 14 C. D. 16 E. None of A. through D. is correct Exercise 1. Solve each rational equation. a) Solve the equation b) Solve the equation x x x 4 = 2x2 14x x 2 2x 8. 2 (x + 2) 2 x + 2 = 5. Inequalities Exercise 14. Solve the following linear inequalities by using any valid method of your choosing. Write your answer in interval notation. a) 4(x + 2) 6 + 4(2x + 1) b) 2 5 (2x 1) > 10 c) 1 2 (x 9) 4 (x 1) 4 (x ) 2 d) 0 < x+2 19 e) 5 x f) 6+x 2x > 6+x Exercise 15. Solve the following linear inequalities by using any valid method of your choosing. Write your answer in interval notation. a) x b) 2 8 x +1 < 19 c) t+4 d) 10 < 5x+4 +2 e) x < x f) 5x+1 > 16 or 7x > 46 Created/Revised by Math Department 2/17/18 Page 4
5 Relations and Functions You should be familiar with the following terms, concepts, and/or expressions: the standard form of the equation of a circle center and radius function graph xintercept yintercept domain range VerticalLine Test functional evaluation function addition, subtraction, multiplication, and division function composition piecewise defined functions even and odd functions function transformations (reflection, translation, and scaling) Note 1. You will need to recall the technique of completing the square. Circles Exercise 16. Which of the following equations defines a circle? I. 2(x 7) 2 + (y + 5) 2 = 4 II. (x 7) 2 (y + 5) 2 = 2 2 III. (x 7) 2 + (y 5) 2 = 2 2 IV. x 2 + 2x + y 2 + 4y + 4 = 0 A. Only I. B. Only II. C. Only III. D. Only IV E. Only I and III. F. Only II and III. G. Only II and IV. H. Only III and IV. Exercise 17. Identify the the center and the radius of the circle whose equation is (x 2) 2 +(y+) 2 = 25. Exercise 18. Give the standard form of the equation of the circle that has its center at ( 5, 8) and a diameter of 12. Exercise 19. Write the standard form of the equation of the circle that has its center at (4, ) with the point ( 1, 9) on the circle. Exercise 20. Write the standard form of the equation of the circle given by x 2 6x+y 2 +8y+12 = 6. Identify the center and radius of this circle. Created/Revised by Math Department 2/17/18 Page 5
6 Functions Exercise 21. Given the function defined by f(x) = x x 2, evaluate and simplify each of the following: a) f( ) b) f(2a) where a is a real number (i.e., a R) c) f( x) d) f( + h) f( ) where h is a real number (i.e., h R) f( + h) f( ) e) where h R and h 0 h f(x + h) f( x) f) where h R and h 0 h Exercise 22. State the domain of each of functions defined by each of the following: a) p(x) = b) g(x) = x + 2 c) h(x) = x + 2 x + 2 d) f(t) = e) q(x) = x+2 f) r(x) = x t + 2 x 2 4 Exercise 2. Your grandfather has a blue spruce tree on his property that drops a lot of cones during the onset of winter. He will pay you fifteen cents (i.e., $0.15) for each cone you pick up and bag. As a further enticement to assist with his work, he also gives you $5 to show up to work no matter how many cones you ultimately bag. a) Write an expression that represents the total amount of money (in dollars) the you could earn as a function of the number of cones you pick up and bag. b) Identify the independent and dependent variables of your function. c) If you pick up and bag 250 cones today, how much money does Grandpa need to pay you for your work? Exercise 24. Let f(x) = x 2 x+2. If f(x) = 22, find x. Linear Equations in Two Variables and Linear Functions You should be familiar with the following terms, concepts, and expressions: midpoint slope xintercept yintercept quadrant slopeintercept form pointslope form horizontal line vertical line perpendicular parallel Created/Revised by Math Department 2/17/18 Page 6
7 Exercise 25. Let P = (2,) and Q = (4,6) and R = (2, 6). a) Find the distance between P and Q. b) Find the midpoint between R and Q. c) Find the slope of the line containing P and Q. d) Find the slope of the line containing P and R. e) Find the equation of the line containing P and Q. f) Find the equation of the line containing P and R. Exercise 26. Determine which of the following ordered pairs are solutions to the equation 2x+y = 8. P = (0, 4) Q = ( 4, 0) R = ( 10, 4) T = (10, 4) A. Only P. B. Only Q. C. Only R. D. Only T. E. Only P and R. F. Only Q and T. G. Only Q and R. H. Only R and T. Exercise 27. Let L be the line defined by 2x+4y = 8. a) Find the xintercepts and yintercepts of L. b) Find the slope of L. c) Write L in slopeintercept form. Exercise 28. Let L be the line defined by x 2 y = 1. a) Find the xintercepts and yintercepts of L. b) Find the slope of L. c) Write L in slopeintercept form. Exercise 29. Let P = ( 2,1+a) and Q = (,6+2a). a) Find the slope of the line L containing P and Q. b) If the L is parallel to 8x 2y = 6, find a. Exercise 0. Suppose that L 1 and L 2 are lines such that: L 1 contains P = (2, 5) and Q = (4, 9) and L 2 contains S = ( 1, 4) and T = (, 2). a) Let m 1 be the slope of L 1. Find m 1. b) Let m 2 be the slope of L 2. Find m 2. c) Without graphing the lines, determine whether L 1 and L 2 are parallel, perpendicular, or neither. Exercise 1. Which of the following represents the pointslope formula for the line with slope m that contains the point P = (x 1,y 1 )? A. y = mx + b B. y 1 y = m (x x 1 ) C. y y 1 = m (x x 1 ) D. m = y 2 y 1 x 2 x 1 Created/Revised by Math Department 2/17/18 Page 7
8 Exercise 2. Matching: Match the equation to the form of the line that it represents. You may assume that k, m, b, A, and B are constants and that A and B are not both equal to zero. Form Standard Form Horizontal Line Equation x = k y = mx + b Vertical Line y y 1 = m (x x 1 ) SlopeIntercept Form PointSlope Formula Ax + By = C y = k Exercise. The speed of sound is approximately 1125 ft (i.e., 1100 feet per second). Thus, for every sec onesecond difference between seeing lightning and hearing the associated thunder clap, we can estimate that a storm is approximately 1100 feet away. Let d represent the distance (in feet) that a storm is from you (the observer). Let t represent the difference in time between seeing lightning and hearing thunder (in seconds). a) Construct the linear model that represents the approximate distance (d) the lightning strike was is from the observer based on the time delay in hearing the hearing thunderclap (t). In this case, we know that t 0. b) Use the linear model to determine how far away a lightning strike is for the following differences in time between seeing lightning strike and hearing the associated thunder clap. i) 4 seconds, ii) 12 seconds, iii) 16 seconds. c) If the lightning strike is known to be 4.5 miles (i.e., 2, 760 feet) away, then how many seconds will pass between seeing lightning and hearing thunder? Exercise 4. The graph of the line L with equation ax+by = 1 is given below in Figure 1. Which of the following must be true? A. a > 0 and b < 0 B. a > 0 and b > 0 C. a < 0 and b < 0 D. a < 0 and b > 0 E. a = 0 and b > 0 F. None of A. through E. is true L y x Transformations of Functions and PiecewiseDefined Functions Exercise 5. The graph of f(x) = x 2 experiences the following successive transformations: (1) a reflection about the yaxis, (2) a translation 2 unit down, () a reflection about the xaxis. Identify the function that represents the resulting curve. A. g(x) = x B. g(x) = x C. g(x) = x 2 2 D. g(x) = x 2 2 E. None of A. through D. is correct Figure 1 Created/Revised by Math Department 2/17/18 Page 8
9 Exercise 6. Let g(x) = 2 f(x )+5. Which of the following successive transformations is a description of g? A. The graph of f shifted units to the left, reflected about the yaxis, stretched vertically by a factor of 2, and then shifted up 5 units. B. The graph of f shifted units to the right, reflected about the yaxis, stretched vertically by a factor of 2, and then shifted up 5 units. C. The graph of f shifted units to the left, reflected about the xaxis, stretched vertically by a factor of 2, and then shifted up 5 units. E. None of A. through D. is correct D. The graph of f shifted units to the right, reflected about the xaxis, stretched vertically by a factor of 2, and then shifted up 5 units. Exercise 7. Use the concepts of transformations to sketch the graphs of a) h(x) = (x+2) 2 b) g(x) = x + c) m(x) = 1 x 2 Exercise 8. Suppose that h is an odd function and h( 2) =, h(6) = 11, and h(1) = 5. Find h(2). x 7, for x 4 x Exercise 9. Let f(x) = 2 +, for 4 < x < 1. Evaluate each of the following: 7x + 2, for 1 < x 5 2 x 1, for x > 5 a) f(1) b)f( 6) c)f(4) d)f(7) e)f(2) f)f(a) where 2 < a.25 The Algebra of Functions Exercise 40. For f(x) = x + 6 and g(x) = 2x 2 7, find: a) (f + g)(10) b)f() g() c)(fg)( 2) d)( f g )( 5) e)( g f )( 5) f)( f g )(x) Exercise 41. For f(x) = x 2 x+2 and g(x) = 6x 7, find: a) f(g(1)) b)(f g)(0) c)(g f)(0) d)(g f)(1) e)(f g)(x) f)(g f)(0) Exercise 42. For f(x) = x 2 and g(x) = 2x 2 x 2, state the domain of h(x) = f(x) g(x). Write your answer using interval notation. Exercise 4. For f(x) = 1 x 2 and g(x) = 2 x, state the domain of h(x) = (f g)(x). Write your answer using interval notation. Created/Revised by Math Department 2/17/18 Page 9
10 Quadratic Functions Exercise 44. Find the vertex of f(x) = 6(x 4) Exercise 45. For f(x) = x 2 +6x+5, answer each of the following. a) Determine the yintercept. b) Determine the xintercept(s), if any. c) Identify the vertex. d) Determine the axis of symmetry. e) Write the range of the function in interval notation. f) Sketch the graph of the function. Exercise 46. The axis of symmetry of the graph of f(x) = ax 2 +8x+9 is x = 2. Find a. Exercise 47. If f(x) = 8x 2 +4x 1 is expressed in the form f(x) = a(x h) 2 +k where a, h, and k are constants (with a 0), what is the value of k? Exercise 48. Suppose that f is a quadratic function with axis of symmetry given by x = 5. If f(1) = 9, f(5) = 1, f(7) =, and f(9) = 9, then answer the following questions. a) Find the vertex of the parabola. b) Is the parabola open up or open down? c) Are there any xintercepts of f? d) What is f()? e) (Bonus) Find constants a, b, and c such that f(x) = ax 2 + bx + c. Exercise 49. Find the points of intersection between the line L given by y x = 5 and the parabola given by y = x 2 2x 5. (See Figure 2.) y y = x 2 2x 5 y x = 5 x Figure 2 Created/Revised by Math Department 2/17/18 Page 10
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