MATH 101.00 Final Review - Fall 2018 PDF

MATH 101.00 - Final review 1. Expressing an interval in set-builder notation and graphing it on a number line. (Module 1) (a) Example: Write in set-builder notation and graph the interval (−3, 4]. 2. Evaluating expressions with absolute values. (Module 1) (a) Example: Evaluate −| − 32...

MATH 101.00 - Final review 1. Expressing an interval in set-builder notation and graphing it on a number line. (Module 1) (a) Example: Write in set-builder notation and graph the interval (−3, 4]. 2. Evaluating expressions with absolute values. (Module 1) (a) Example: Evaluate −| − 32 |. 3. Completing a table of values for an equation and graphing on rectangular coor- dinate axes. (Module 1) (a) Example: Make a table and graph the equation y = (x − 1)2 + 2, for x = −2, −1, 0, 1, 2. 4. Solving linear equations. (Module 1) 5 1 1 (a) Example: Solve x + = 2x −. 4 2 2 5. Solving an application problem using a linear equation. (Module 1) (a) Example: An average score of 90 for 5 exams is needed for a final grade of A. Jill’s first 4 exams were: 95, 89, 78, and 91. What grade does Jill need on her 5th exam to get an A? 6. Simplifying an expression using rules of exponents, leaving no negative exponents or repeated bases. (Module 2)  5 −1 6 −2 2x y z (a) Example: Simplify: 5x−3 y −1 z 2 7. Converting between scientific notation and decimal notation. (Module 2) (a) Example: Convert to scientific notation: (7, 000, 000)(0.0034). 5.2 × 105 (b) Example: Convert to decimal notation:. 2.6 × 103 8. Identifying whether a relation is a function, and finding its domain and range. (Module 3) (a) Example: Does the relation represent a function? Find domain and range. {(−1, 1), (−2, 2), (−3, 1), (−1, 2)} 9. Evaluating a given function f (x) at a given value for x. (Module 3) (a) Example: Evaluate f (x) = −x3 − 3x2 − 1 at x = −2. 10. Identifying function values, domain and range from a graph. (Module 3)  (a) Example: What is the domain and range of the function f (x) below? f (−2)? f (2)? 8 y 7 6 5 4 3 2 1 x −8−7−6−5−4−3−2−1 −1 1 2 3 4 5 6 7 8 −2 −3 −4 −5 −6 −7 −8 11. Combining different functions, using addition, subtraction, multiplication or di- vision. (Module 3) (a) Example: Find (f − g)(x) if f (x) = 4x2 − 2 and g(x) = −3x2 − 5x − 2. 12. Rewriting an equation of a line in slope-intercept format and graphing. (Module 4) (a) Example: Rewrite in slope-intercept format and graph 2x − 3y = 12. 13. Using the point-slope form of a line to find its equation from the two points it goes through. (Module 4) (a) Example: Find the line going through (−3, 2) and (1, 4). Rewrite in slope-intercept format. 14. Finding a line going through a point and is parallel/perpendicular to another line. (Module 4) (a) Example: Find a line going through the point (1, −1) parallel to the line 2x − y = 5. (b) Example: Find a line going through the point (−4, 2) perpendicular to the line −x+3y = 9. 15. Solving a system of linear equations, using graphing, substitution or addition method. (Module 5) (a) Example: Solve the system:  4x + 3y = 14 2x + 3y = 4 16. Using a system of equations to solve a problem. (Module 5) (a) Example: A flour mill mixes a 40% whole wheat flour with a 70% whole wheat flour to obtain a 60% whole wheat mixture. If 45kg of the final product was obtained, how much of each type of flour was used? 17. Solving an inequality in one-variable and graphing the solution on the number line. (Module 6) 1 (a) Example: Solve and graph the solution on a number line: 7(2x + 6) < x − 6(3 − x) 2 18. Solving a compound inequalities and graphing the solution on the number line. (Module 6) (a) Example: Solve and graph the solution, 5x − 2 ≤ 3 and 4x + 7 > 3. (b) Example: Solve and graph the solution, x − 5 > 0 or 3(x − 31 ) < 8. 19. Solving equations and inequalities with absolute value signs. (Module 6) (a) Example: Solve 2|3 − 4x| = 10 (b) Example: Solve and graph the solution |5 − 2x| ≤ 1 20. Solving and graphing a linear inequality in two variables. (Module 6) (a) Example: Solve and graph 5x + 2y > −8 21. Solving and graphing a system of linear inequalities in two variables. (Module 6) (a) Example: Solve and graph the solution  x−y >5 2x − y ≤ 4 22. Adding, subtracting, and multiplying polynomials. (Module 7) (a) Example: Perform the indicated operation. (4x3 − 2x + 8) − (−x3 + 3x2 + 6) (b) Example: Perform the indicated operation. (2x2 + 6)(−x2 + 2x − 1) 23. Factoring a polynomial completely by trial and error, grouping or recognizing special forms. (Module 7) (a) Example: Factor completely. 2x3 − x2 + 4x − 2 (b) Example: Factor completely. x6 − 64 24. Solving quadratic equations by factoring. (Module 7) (a) Example: Solve 2x2 − 9x − 5 = 0 25. Solving application problems using quadratic equations. (Module 7) (a) Example: An object is thrown upward the ground has height h = −16t2 + 64t (where t is measured in seconds). How long will it be before the object hits the ground? 26. Finding the domain of a rational function. (Module 8) x−2 (a) Example: Find the domain of f (x) =. x2 + 5x − 6 27. Adding, subtracting, multiplying, dividing rational expressions, and simplifying the answer. (Module 8) 3x x+3 (a) Example: Perform the following operation and simplify + 2 x2 + x − 6 x − 4x + 4 x4 − 9x2 x4 + 2x3 − 8x2 (b) Example: Perform the following operation and simplify 2 ÷ x − 4x + 3 x2 − 16 28. Simplifying complex rational expression. (Module 8) x x+3 (a) Example. Simplify x x+3 +x 29. Dividing polynomials using short or long division of polynomials. (Module 8) (a) Example: Divide (12x4 y 8 − 9x3 y 5 + 3x2 y 2 − 3xy) ÷ (3x2 y) (b) Example: Divide (x3 − 12x2 − 42) ÷ (x − 3) 30. Solving rational equations. (Module 8) x 3 7x (a) Example: Solve + = 2 x−5 x+2 x − 3x − 10 31. Solving a formula for the specified variable. (Module 8) E (a) Example: Solve I = for r. R+r 32. Using rational equations to solve application problems with average cost, motion and work. (Module 8) (a) Example: A laptop manufacturer spends $6,000 for facilities per month and it costs $50 to produce each laptop. How many laptops must the manufacturer produce each month to have an average cost per unit of $80? (b) Example: Jane can clean the house in 8 hours. When she works together with Mike, they do the job in 6 hours. How long will it it take Mike to clean the house by himself? (c) Example: Gary rides the bicycles 7 mph slower than Carrie. In the time it takes Gary to bicycle 45 miles, Carrie can travel 66 miles. How fast does Gary bike? 33. Solving direct and inverse variation problems. (Module 8) (a) Example: The amount of gasoline a car uses is directly proportional to how much it travels. If a car uses 8 gallons of gasoline to travel 290 miles, how much gasoline will it use to travel 435 miles? (b) Example: 3 friends want to go camping and they pack enough water to last for 6 days. Just as they are about to leave they are joined by 3 more people. How long will their water last now? The total days the water supply will last is inversely proportional to how many people go on the camping trip. 34. Evaluating radical expressions. (Module 9)  r (−4)2 (a) Example: Evaluate 49 35. Finding the domain and evaluating radical functions. (Module 9) √ (a) Example: Find the domain of f (x) = 4 5x − 4 and evaluate at x = 17. 36. Simplifying rational exponential expression. (Module 9) −3/2 x1/3 y 2/3  (a) Example: Simplify x−5/3 y 37. Converting radical notation to rational exponential notation, and simplifying. (Module 9) √ 3 x7 (a) Example: Simplify √ 8 x3 38. Adding, subtracting, multiplying, dividing radical expression and simplifying. (Module 9) p p (a) Example: Perform the indicated operation and simplify 3 9x11 y · 3 24x7 y 4 √ √ √ (b) Example: Perform the indicated operation and simplify 6 147 + 4 108 − 6 75 39. Rationalizing the denominator in a radical expression. (Module 9) √ 1−5 7 (a) Example: Rationalize the denominator √ √ 2+ 8 r 3 5 (b) Example: Rationalize the denominator 16x 40. Solving radical equations. (Module 9) √ √ (a) Example: Solve x + 6 + 2 − x = 4 41. Adding, subtracting, multiplying and dividing complex numbers and writing the result in a + bi form. (Module 9) √ √ (a) Example: Perform the indicated operation (4 + i 2) · (−3 − i 3) 3 + 2i (b) Example: Perform the indicated operation 5 − 4i 42. Solving a quadratic equation using the square root property. (Module 10) (a) Example: Solve (x − 4)2 = 7 43. Solving a quadratic equation by completing the square. (Module 10) (a) Example: Solve 2x2 − 3x + 2 = 0 44. Solving a quadratic equation by using the quadratic formula. (Module 10)  (a) Example: Solve x + 1 = 3x2 45. Graphing a quadratic function, and identifying the vertex, axis of symmetry x− and y− intercepts. (Module 10) (a) Example: Graph f (x) = −x2 − 4x + 5 and identify the vertex, axis of symmetry, and intercepts. 46. Solving application problems using quadratic functions. (Module 10) (a) Example: The owner of a ranch decides to enclose a rectangular region with 140 feet of fencing. To help the fencing cover more land, he plans to use one side of his barn as part of the enclosed region. What is the maximum area the rancher can enclose? (b) Example: An object in launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. Its position function with respect to time is s(t) = −16t2 +64t+80. What will be the object’s maximum height? When will it attain this height? 47. Graphing functions using transformations: vertical and horizontal shifts, reflec- tions, vertical and horizontal stretching and shrinking. (Module 11). (a) Example: Use the graph of f (x) = x3 to graph g(x) = −(x − 1)3 − 2 48. Graphing a piece-wise function. (Module 11) (a) Example: Graph   2x if x < −2 f (x) = 3 if − 2 ≤ x < 1 6 − x if x ≥ 1  49. Graphing exponential functions and identifying domain and range. (Module 12) (a) Example: Graph f (x) = 2x+1 and identify domain and range.  r nt 50. Using the compound interest formulas A = P 1 + or A = P ert to find the n accumulated value of an investment. (Module 12) (a) Example: Find the accumulated value of an investment of $2,500 at 3.5% compounded monthly for 3 years. (b) Example: Find the accumulated value of an investment of $10,000 at 2.7% compounded continuously for 7 years. 51. Finding the composition of two functions. (Module 12) √ (a) Example Find (f ◦ g) if f (x) = x + 7 and g(x) = 3x − 5. 52. Verifying that a pair of functions are inverses. (Module 12) 3 3 − 5x (a) Example Verify the two functions are invereses f (x) = 4x+5 and g(x) = 4x 53. Converting between exponential and logarithmic form of an expression. (Module 12) 1 1 (a) Example: Rewrite in exponential form log4 = − 2 2  2 1 1 (b) Example: Rewrite in logarithmic form = 3 9 54. Evaluating a logarithmic expression. (Module 12) 1 (a) Example: log2 √ 3 4 55. Graphing logarithmic functions the following functions and identifying the domain and range. (Module 12) (a) Example: Graph f (x) = log2 (x − 1) + 1 and identify the domain and range. 56. Finding the distance between two points. (Module 13) (a) Example: Find the distance between (−2, 4) and (3, −8) 57. Finding the midpoint between two points. (Module 13) (a) Example: Find the midpoint between (2, −9) and (−2, −4) 58. Finding the equation of a circle in standard form. (Module 13) (a) Example: Find the standard form of the equation of the circle with center (−1, 2) and radius 3.