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CHAPTER 2 Points, Lines, and Functions Chapter 2 Points, Lines, and Functions Section 2.1: An Introduction to the Coordinate Plane Points in the Coordinate Plane Points in the Coordinate Plane The Rectangular Coordinate System: 104 University of Houston Department of Mathematics SECTION 2.1 An Introduction to the Coordinate Plane Plotting Points in the Coordinate Plane: MATH 1300 Fundamentals of Mathematics 105 CHAPTER 2 Points, Lines, and Functions Example: Solution: 106 University of Houston Department of Mathematics SECTION 2.1 An Introduction to the Coordinate Plane Graphing Horizontal and Vertical Lines: Example: Solution: MATH 1300 Fundamentals of Mathematics 107 CHAPTER 2 Points, Lines, and Functions Graphing Other Lines: Example: Solution: 108 University of Houston Department of Mathematics SECTION 2.1 An Introduction to the Coordinate Plane Additional Example 1: Solution: MATH 1300 Fundamentals of Mathematics 109 CHAPTER 2 Points, Lines, and Functions Additional Example 2: 110 University of Houston Department of Mathematics SECTION 2.1 An Introduction to the Coordinate Plane Solution: Additional Example 3: MATH 1300 Fundamentals of Mathematics 111 CHAPTER 2 Points, Lines, and Functions Solution: Additional Example 4: Solution: 112 University of Houston Department of Mathematics SECTION 2.1 An Introduction to the Coordinate Plane (c) Draw a line through the points. MATH 1300 Fundamentals of Mathematics 113 CHAPTER 2 Points, Lines, and Functions Additional Example 5: Solution: 114 University of Houston Department of Mathematics SECTION 2.1 An Introduction to the Coordinate Plane MATH 1300 Fundamentals of Mathematics 115 CHAPTER 2 Points, Lines, and Functions 116 University of Houston Department of Mathematics Exercise Set 2.1: An Introduction to the Coordinate Plane 17. If the point (a, b) is in Quadrant I, identify the quadrant of each of the following points: (a) (-a, -b) (b) (-a, b) (c) (a, a) Plot the following points in a coordinate plane. 1. A(3, 4) 2. B(2, -5) 3. C(-3, -1) 4. D(-4, -6) 5. E(-5, 0) 6. F(0, -2) 18. If the point (a, b) is in Quadrant I, identify the quadrant of each of the following points: (a) (-b, a) (b) (b, b) (c) (-b, -a) 19. If the point (a, b) is in Quadrant II, then a 0 and b 0 . Identify the quadrant of each of the following points: (a) (-a, -b) (b) (b, a) (c) (a, -b) Write the coordinates of each of the points shown in the figure below. Then identify the quadrant or axis in which the point is located. y 7. G 8. H 9. I J 21. If the point (a, b) is in Quadrant IV, identify the quadrant of each of the following points: (a) (b, -b) (b) (-a, -a) (c) (b, a) I G 10. J H 20. If the point (a, b) is in Quadrant III, then a 0 and b 0 . Identify the quadrant of each of the following points: (a) (-a, b) (b) (b, a) (c) (-a, -b) x 11. K 22. If the point (a, b) is in Quadrant II, identify the quadrant of each of the following points: (a) (-a, b) (b) (b, b) (c) (a, -a) 12. L K L Plot each of the following sets of points in a coordinate plane. Then identify the quadrant or axis in which each point is located. 13. (a) (b) (c) (d) A(2, 5) B(-2, -5) C(2, -5) D(-2, 5) 14. (a) (b) (c) (d) A(4, -3) B(-4, -3) C(-4, 3) D(4, 3) 15. (a) (b) (c) (d) A(0, -2) B(-2, 0) C(2, 0) D(0, 2) 16. (a) (b) (c) (d) A(-3, 0) B(3, 0) (0, -3) D(0, 3) 23. If the point (a, b) is in Quadrant III, identify the axis on which each of the following points lies: (a) (a, 0) (b) (0, b) (c) (-b, 0) 24. If the point (a, b) is in Quadrant IV, identify the axis on which each of the following points lies: (a) (0, -b) (b) (-a, 0) (c) (b, 0) Answer True or False. 25. The point (0, 5) is on the x-axis. 26. The point (-4, 0) is in Quadrant II. 27. The point (1, -3) is in Quadrant IV. 28. The point (-2, -5) is in Quadrant III. 29. The point (0, 0) is in Quadrant I. MATH 1300 Fundamentals of Mathematics 30. The point (-6, 1) is in Quadrant IV. 31. If the point (a, b) is in Quadrant IV, then b 0 . 32. If the point (a, b) is in Quadrant II, then a 0 . 33. If the point (a, b) is in Quadrant I, then the point (b, a) is also in Quadrant I. 117 Exercise Set 2.1: An Introduction to the Coordinate Plane 34. If the point (a, b) is in Quadrant I, then the point (a, -b) is in Quadrant II. 35. If the point (a, b) is in Quadrant II, then the point (-a, -b) is in Quadrant III . 47. Graph the line x 2 . 48. Graph the line y 5 . 49. Graph the line y 4 . 36. If the point (a, b) is in Quadrant IV, then the point (-b, a) is in Quadrant I. 50. Graph the line x 3 . 37. If the point (a, b) is in Quadrant III, then b 0 . 51. On the same set of axes, graph the lines x 1 and y 3 . 38. If the point (a, b) is on the y-axis, then a 0 . 39. If the point (a, b) is on the y-axis, then b 0 . 52. On the same set of axes, graph the lines x 5 and y 2 . 40. If the point (a, b) is on the y-axis, then a 0 . 53. On the same set of axes, graph the lines x 41. If the point (a, b) is on the y-axis, then the point (b, a) is on the x-axis. 42. If the point (a, b) is on the x-axis, then the point (a, 3) lies in Quadrant I . Answer the following. 43. Given the following points: A(3, 5), B(3, 1), C(3, 0), D(3, -2) (a) (b) (c) (d) and y 0 . 54. On the same set of axes, graph the lines x 0 and y 52 . Graph the following lines by first completing the table and then plotting the points on a coordinate plane. 55. y 3x 2 Plot the above points on a coordinate plane. What do the above points have in common? Draw a line through the above points. What is the equation of the line drawn in part (c)? Plot the above points on a coordinate plane. What do the above points have in common? Draw a line through the above points. What is the equation of the line drawn in part (c)? 45. (a) List four points that are on the x-axis. (b) Analyze the coordinates of the points you have listed. What do they have in common? (c) Give the equation of the x-axis. x y -2 -1 0 1 44. Given the following points: A(-3, 4), B(0, 4), C(1, 4), D(3, 4) (a) (b) (c) (d) 7 2 2 56. y 2x 5 x y -2 -1 0 1 46. (a) List four points that are on the y-axis. (b) Analyze the coordinates of the points you have listed. What do they have in common? (c) Give the equation of the y-axis. 118 2 University of Houston Department of Mathematics Exercise Set 2.1: An Introduction to the Coordinate Plane 57. y 4x 7 x y 0 1 4 -5 2 32 58. y 5x 1 x y 2 -1 3 5 -6 0 Answer the following. 59. Graph the line segment with endpoints (-7, 0) and (0, 7). 60. Graph the line segment with endpoints (3, 5) and and (-5, -3). 61. Graph the line segment with endpoints (1, -4) and (-1, 4) 62. Graph the line segment with endpoints (-2, 6) and (6, 2). MATH 1300 Fundamentals of Mathematics 119 CHAPTER 2 Points, Lines, and Functions Section 2.2: The Distance and Midpoint Formulas The Distance Formula The Midpoint Formula The Distance Formula Finding the Distance Between Two Points: Example: 120 University of Houston Department of Mathematics SECTION 2.2 The Distance and Midpoint Formulas Solution: Additional Example 1: Solution: Additional Example 2: Solution: MATH 1300 Fundamentals of Mathematics 121 CHAPTER 2 Points, Lines, and Functions Additional Example 3: Solution: Additional Example 4: Solution: 122 University of Houston Department of Mathematics SECTION 2.2 The Distance and Midpoint Formulas MATH 1300 Fundamentals of Mathematics 123 CHAPTER 2 Points, Lines, and Functions Additional Example 5: Solution: 124 University of Houston Department of Mathematics SECTION 2.2 The Distance and Midpoint Formulas MATH 1300 Fundamentals of Mathematics 125 CHAPTER 2 Points, Lines, and Functions Additional Example 6: Solution: 126 University of Houston Department of Mathematics SECTION 2.2 The Distance and Midpoint Formulas Additional Example 7: Solution: MATH 1300 Fundamentals of Mathematics 127 CHAPTER 2 Points, Lines, and Functions Use the Pythagorean Theorem to determine c. 128 University of Houston Department of Mathematics SECTION 2.2 The Distance and Midpoint Formulas The Midpoint Formula Finding the Midpoint of a Line Segment: Example: Solution: Additional Example 1: MATH 1300 Fundamentals of Mathematics 129 CHAPTER 2 Points, Lines, and Functions Solution: Additional Example 2: Solution: Additional Example 3: 130 University of Houston Department of Mathematics SECTION 2.2 The Distance and Midpoint Formulas Solution: Additional Example 4: Solution: MATH 1300 Fundamentals of Mathematics 131 CHAPTER 2 Points, Lines, and Functions Additional Example 5: Solution: 132 University of Houston Department of Mathematics SECTION 2.2 The Distance and Midpoint Formulas MATH 1300 Fundamentals of Mathematics 133 Exercise Set 2.2: The Distance and Midpoint Formulas Use the Pythagorean Theorem to find the missing side of each of the following triangles. 6. Pythagorean Theorem: In a right triangle, if a and b are the measures of the legs, and c is the measure of the hypotenuse, then a2 + b2 = c2. (a) Plot the above points on a coordinate plane. (b) Draw segment AB. This will be the hypotenuse of triangle ABC. (c) Find a point C such that triangle ABC is a right triangle. Draw triangle ABC. (d) Use the Pythagorean theorem to find the distance between A and B (the length of the hypotenuse of the triangle). c a b 1. c Use the distance formula to find the distance between the two given points. (You can also use the method from the previous two problems to double-check your answer.) 5 12 2. Given the following points: A(3, 1) and B(1, 5) 7 a 7. (3, 6) and (5, 9) 8. (4, 7) and (2, 3) 9. (5, 0) and (2, 6) 10. (9, 4) and (2, 3) 5 3. 11. (4, 0) and (0, 7) 6 2 b 4. 12. (4, 8) and (10, 1) 13. 5, 12 and 3, 56 14. 32 , 1 and 34 , 0 c 6 8 Find the midpoint of the line segment joining points A and B. 15. A(7, 6) and B(3, 8) Answer the following. 5. Given the following points: A(1, 2) and B(4, 7) 16. A(5, 9) and B(1, 3) 17. A(7, 0) and B(4, 8) 18. A(7, 5) and B(4, 3) (a) Plot the above points on a coordinate plane. (b) Draw segment AB. This will be the hypotenuse of triangle ABC. (c) Find a point C such that triangle ABC is a right triangle. Draw triangle ABC. (d) Use the Pythagorean theorem to find the distance between A and B (the length of the hypotenuse of the triangle). 134 19. A(3, 0) and B(0, 9) 20. A(6, 7) and B(10, 6) 21. A 13 , 5 and B 53 , 7 22. A 3, 12 and B 8, 56 University of Houston Department of Mathematics Exercise Set 2.2: The Distance and Midpoint Formulas Answer the following. 23. (a) Graph the line segment with endpoints A(2, 6) and B(5, 4) . (b) Find the distance from A to B. (c) Find the midpoint of AB . 24. (a) Graph the line segment with endpoints A(4, 0) and B(2, 5) . (b) Find the distance from A to B. (c) Find the midpoint of AB . 25. If M (4, 7) is the midpoint of the line segment joining points A and B, and A has coordinates (2, 3) , find the coordinates of B. 26. If M (5, 3) is the midpoint of the line segment joining points A and B, and A has coordinates (1, 6) , find the coordinates of B. 27. If M (3, 5) is the midpoint of the line segment joining points A and B, and B has coordinates (1, 2) , (a) Find the coordinates of A. (b) Find the length of AB . 28. If M (2, 1) is the midpoint of the line segment joining points A and B, and B has coordinates (5, 3) , (a) Find the coordinates of A. (b) Find the length of AB . 29. Determine which of the following points is closer to the origin: A(5, 6) or B(3, 7) ? 30. Determine which of the following points is closer to the point (4, 1) : A(2, 3) or B(6, 6) ? 31. A circle has a diameter with endpoints A(5, 9) and B(3, 5) . (a) Find the coordinates of the center of the circle. (b) Find the length of the radius of the circle. 32. A circle has a diameter with endpoints A(2, 7) and B(8, 1) . (a) Find the coordinates of the center of the circle. (b) Find the length of the radius of the circle. MATH 1300 Fundamentals of Mathematics 135 CHAPTER 2 Points, Lines, and Functions Section 2.3: Slope and Intercepts of Lines The Slope of a Line Intercepts of Lines The Slope of a Line Finding the Slope of a Line: 136 University of Houston Department of Mathematics SECTION 2.3 Slope and Intercepts of Lines Example: Solution: MATH 1300 Fundamentals of Mathematics 137 CHAPTER 2 Points, Lines, and Functions 138 University of Houston Department of Mathematics SECTION 2.3 Slope and Intercepts of Lines Additional Example 1: Solution: Additional Example 2: Solution: MATH 1300 Fundamentals of Mathematics 139 CHAPTER 2 Points, Lines, and Functions Additional Example 3: Solution: Additional Example 4: Solution: 140 University of Houston Department of Mathematics SECTION 2.3 Slope and Intercepts of Lines MATH 1300 Fundamentals of Mathematics 141 CHAPTER 2 Points, Lines, and Functions Intercepts of Lines Finding Intercepts of Lines: 142 University of Houston Department of Mathematics SECTION 2.3 Slope and Intercepts of Lines Horizontal Lines: Vertical Lines: MATH 1300 Fundamentals of Mathematics 143 CHAPTER 2 Points, Lines, and Functions Example: Solution: 144 University of Houston Department of Mathematics SECTION 2.3 Slope and Intercepts of Lines Example: Solution: Additional Example 1: Solution: MATH 1300 Fundamentals of Mathematics 145 CHAPTER 2 Points, Lines, and Functions Additional Example 2: Solution: 146 University of Houston Department of Mathematics SECTION 2.3 Slope and Intercepts of Lines Additional Example 3: MATH 1300 Fundamentals of Mathematics 147 CHAPTER 2 Points, Lines, and Functions Solution: 148 University of Houston Department of Mathematics Exercise Set 2.3: Slope and Intercepts of Lines State whether the slope of each of the following lines is positive, negative, zero, or undefined. 1. 2. p q 3. r 4. s 5. 6. y t x w 25. e 26. f x w c 24. d y e 23. c r t Find the slope of each of the following lines. If undefined, state ‘Undefined.’ f p q s Find the slope of the line that passes through the following points. If undefined, state ‘Undefined.’ 7. (0, 0) and (3, 7) 8. (8, 0) and (3, 6) 9. (2, 5) and (4, 10) d For each of the following: (a) Complete the given table. (b) Plot the points on a coordinate plane and graph the line. (c) Use two points from the table to find the slope of the line. 27. y 4x 1 10. (7, 3) and (5, 9) x 11. (6, 4) and (2, 4) 0 12. (5, 1) and (5, 8) 2 y 13. (2, 3) and (6, 7) 3 14. (2, 6) and (5, 10) 0 12 15. (3, 8) and (3, 4) 16. (8, 7) and (1, 7) 17. (2, 8) and (0, 3) 18. (1, 4) and (7, 2) 28. y 3x 2 19. 12 , 1 and 32 , 16 20. 2, 34 and 15 , 85 21. 22. 53 , 107 and 14 , 87 2, 7 4 9 and 56 , x y 2 2 4 1 2 MATH 1300 Fundamentals of Mathematics 34 3 149 Exercise Set 2.3: Slope and Intercepts of Lines 29. y 32 x 4 x y 4 5 9 8 3 2 For each of the following graphs: (a) (b) (c) (d) (e) State the x-intercept. State the y-intercept. State the coordinates of the x-intercept. State the coordinates of the y-intercept. Find the slope of the line. y 33. 30. y 53 x 6 x y x 5 0 7 8 0 y 34. Answer the following. 31. Examine the relationship in numbers 27-30 between each of the equations and the corresponding slope that you found for each line. Do you see any pattern? Can you determine the slope of the line from simply looking at its equation? x 32. Based on the pattern found in the previous problem, state the slope of the following lines without graphing the line or performing any calculations: (a) y 2 x 9 (b) y 7 x 5 (c) y 54 x 2 (d) y 73 x 4 For each of the following equations: (a) Find the x- and y-intercepts of the line. (b) State the coordinates of the intercepts. (c) Plot the x- and y-intercepts on a coordinate plane. (d) Graph the line, based on the intercepts. 35. y 2x 8 36. y 3x 6 37. y 4 x 5 38. y 3x 7 39. 5x 2 y 20 150 University of Houston Department of Mathematics Exercise Set 2.3: Slope and Intercepts of Lines 40. 2x 3 y 18 Answer the following. 41. 3x 5 y 30 55. Examine the relationship in numbers 53 and 54 between each of the equations and the corresponding y-intercept that you found for each line. Do you see any pattern? Can you determine the y-intercept of the line from simply looking at its equation? 42. 3x 24 4 y 43. 2x 3 y 10 44. 4x 6 y 9 45. 5x 3 y 21 0 56. Based on the pattern found in the previous problem, state the y-intercept of the following lines without graphing the line or performing any calculations: (a) y 2 x 9 (b) y 7 x 5 46. 4x 7 y 8 0 47. 2 x 2 y 7 48. 3x 15 49. 4 y 12 (c) y 54 x 2 50. 4x 4 y 15 (d) y 73 x 4 51. 6 x 24 52. 2 y 14 For each of the following: (a) Complete the given table. (b) Plot the points on a coordinate plane and graph the line. (c) Find the x- and y-intercepts of the line. (d) Find the slope of the line. 53. y 2 x 8 x y 0 0 2 6 0.5 54. y x 3 x y 0 0 3 1.5 2 MATH 1300 Fundamentals of Mathematics 151 CHAPTER 2 Points, Lines, and Functions Section 2.4: Equations of Lines Writing Equations of Lines Writing Equations of Lines Different Forms for Equations of Lines: Example: Solution: 152 University of Houston Department of Mathematics SECTION 2.4 Equations of Lines Example: Solution: MATH 1300 Fundamentals of Mathematics 153 CHAPTER 2 Points, Lines, and Functions Example: Solution: 154 University of Houston Department of Mathematics SECTION 2.4 Equations of Lines Example: Solution: Additional Example 1: Solution: MATH 1300 Fundamentals of Mathematics 155 CHAPTER 2 Points, Lines, and Functions Additional Example 2: Solution: To sketch the graph, begin by using the y-intercept to plot the point 0,1 . 156 University of Houston Department of Mathematics SECTION 2.4 Equations of Lines Additional Example 3: Solution: MATH 1300 Fundamentals of Mathematics 157 CHAPTER 2 Points, Lines, and Functions Additional Example 4: Solution: 158 University of Houston Department of Mathematics SECTION 2.4 Equations of Lines MATH 1300 Fundamentals of Mathematics 159 Exercise Set 2.4: Equations of Lines Write an equation in slope-intercept form for each of the following lines. y 1. x For each of the following equations, (a) Write the equation in slope-intercept form. (b) Identify the slope and the y-intercept of the line. (c) Graph the line. 5. 2x y 5 6. y 4x 0 7. 5x y 1 8. 3x y 6 9. x 4y 0 10. x 3 y 9 y 2. 11. 5x 4 y 12 x 12. 2x 5 y 10 13. 5 y 2x 30 0 14. 3x 2 y 8 0 15. 3. 5 4 x 12 y 1 y 16. 23 x 12 y 1 x Each set of conditions below describes the properties of a particular line. Using these conditions, (a) Graph the line. (b) Write an equation for the line in point-slope form. (c) Write an equation for the line in slopeintercept form. (Do this algebraically, and then check to see if your result matches your graph.) 4. y 17. Slope x 160 2 ; passes through 6, 4 3 18. Slope 5 ; passes through 4, 3 2 19. Passes through 8, 2 20. Passes through 4, 7 and 4, 7 and 1, 3 University of Houston Department of Mathematics Exercise Set 2.4: Equations of Lines Write an equation in slope-intercept form for the line that satisfies the given conditions. 4 21. Slope ; y-intercept 3 7 22. Slope 4 ; y-intercept 5 23. Slope 4 ; passes through 5, 3 5 3 24. Slope ; passes through 12, 5 4 25. Slope 26. Slope 2 ; passes through 3, 2 9 1 ; passes through 4, 2 5 27. Passes through 10, 2 and 5, 7 28. Passes through 6, 1 and 9, 4 29. Passes through 4, 5 and 1, 2 30. Passes through 7, 0 and 3, 5 31. x-intercept 7 ; y-intercept 5 32. x-intercept 2 ; y-intercept 6 33. Slope 34. Slope 3 ; x-intercept 4 2 1 ; x-intercept 6 5 Answer the following, assuming that each situation can be modeled by a linear equation. 35. If a company can make 21 computers for $23,000, and can make 40 computers for $38,200, write an equation that represents the cost C of x computers. 36. A certain electrician charges a $40 traveling fee, and then charges $55 per hour of labor. Write an equation that represents the cost C of a job that takes x hours. MATH 1300 Fundamentals of Mathematics 161 CHAPTER 2 Points, Lines, and Functions Section 2.5: Parallel and Perpendicular Lines Pairs of Lines – Parallel and Perpendicular Lines Pairs of Lines - Parallel and Perpendicular Lines Parallel Lines: Perpendicular Lines: Two lines with slopes m1 and m2 perpendicular if and only if m1m2 1 . 162 University of Houston Department of Mathematics SECTION 2.5 Parallel and Perpendicular Lines Example: Solution: MATH 1300 Fundamentals of Mathematics 163 CHAPTER 2 Points, Lines, and Functions Example: Solution: 164 University of Houston Department of Mathematics SECTION 2.5 Parallel and Perpendicular Lines Additional Example 1: Solution: Additional Example 2: Solution: MATH 1300 Fundamentals of Mathematics 165 CHAPTER 2 Points, Lines, and Functions Additional Example 3: Solution: Additional Example 4: Solution: 166 University of Houston Department of Mathematics SECTION 2.5 Parallel and Perpendicular Lines MATH 1300 Fundamentals of Mathematics 167 Exercise Set 2.5: Parallel and Perpendicular Lines State whether the following pairs of lines are parallel, perpendicular, or neither. 1. y 3x 5 y 3x 7 2. y 52 x 1 3. y 73 x 5 18. x 5 x 5 y x7 2 3 5. y 2x 5 y 2x 5 6. y 5x 7 y 15 x 3 8. 9. 16. x 3 y 3 17. y 2 x0 y 32 x 5 7. y 14 y 52 x 3 y 73 x 4 4. 15. y 4 2x 5 y 7 5x 2 y 6 3x 4 y 8 3x 4 y 8 2x 3 y 5 4x 6 y 11 10. x y 5 0 x y 2 11. The line passing through (2, 5) and (7, 9) The line passing through (2, 6) and (2, 1) 12. The line passing through (4, 7) and (0, 5) The line passing through (3, 8) and (5, 9) 13. The line passing through (6, 0) and (4, 10) The line passing through (3, 7) and (7, 11) 14. The line passing through (1, 7) and (2, 5) The line passing through (6, 6) and (2, 5) 19. The line passing through (4, 5) and (1, 5) The line passing through (2, 3) and (0, 3) 20. The line passing through (2, 6) and (2, 8) The line passing through (3, 4) and (5, 4) Each set of conditions below describes a particular line. Using these conditions, write an equation for each line in the following two forms: (a) Point-slope form (b) Slope-intercept form 21. Passes through (4, 7) ; parallel to the line y 2x 5 22. Passes through (4, 7) ; perpendicular to the line y 2x 5 23. Passes through (12, 5) ; perpendicular to the line y 6x 1 24. Passes through (12, 5) ; parallel to the line y 6x 1 25. Passes through (3, 7) ; parallel to the line y 54 x 2 26. Passes through (3, 7) ; perpendicular to the line y 54 x 2 27. Passes through (1, 6) ; perpendicular to the line 2x 3 y 7 28. Passes through (1, 6) ; parallel to the line 2x 3 y 7 168 University of Houston Department of Mathematics Exercise Set 2.5: Parallel and Perpendicular Lines Write an equation for the line that satisfies the given conditions. With the exception of vertical lines, write all equations in slope-intercept form. 29. Passes through (1, 4) ; parallel to the x-axis 30. Passes through (1, 4) ; parallel to the y-axis 31. Passes through (2, 6) ; parallel to the line x4 32. Passes through (2, 6) ; parallel to the line y4 33. Passes through (2, 3) ; and is (a) parallel to the line y 23 x 5 (b) perpendicular to the line y 23 x 5 34. Passes through (20, 2) ; and is (a) parallel to the line y 5x 3 (b) perpendicular to the line y 5x 3 35. Passes through (2, 3) ; parallel to the line 5x 2 y 6 36. Passes through (1, 5) ; parallel to the line 4x 3 y 8 37. Passes through (2, 3) ; perpendicular to the line 5x 2 y 6 38. Passes through (1, 5) ; perpendicular to the line 4x 3 y 8 39. Passes through (4, 6) ; parallel to the line containing (3, 5) and (2, 1) 40. Passes through (8, 3) ; parallel to the line containing (2, 3) and (4, 6) 41. Perpendicular to the line containing (3, 5) and (7, 1) ; passes through the midpoint of the line segment connecting these points 42. Perpendicular to the line containing (4, 2) and (10, 4) ; passes through the midpoint of the line segment connecting these points MATH 1300 Fundamentals of Mathematics 169 CHAPTER 2 Points, Lines, and Functions Section 2.6: An Introduction to Functions Definition of a Function Domain of a Function Definition of a Function Definition: 170 University of Houston Department of Mathematics SECTION 2.6 An Introduction to Functions Defining a Function by an Equation in the Variables x and y: The Function Notation: Example: Solution: MATH 1300 Fundamentals of Mathematics 171 CHAPTER 2 Points, Lines, and Functions Example: Solution: 172 University of Houston Department of Mathematics SECTION 2.6 An Introduction to Functions Additional Example 1: Solution: Additional Example 2: Solution: MATH 1300 Fundamentals of Mathematics 173 CHAPTER 2 Points, Lines, and Functions Additional Example 3: Solution: Additional Example 4: 174 University of Houston Department of Mathematics SECTION 2.6 An Introduction to Functions Solution: MATH 1300 Fundamentals of Mathematics 175 CHAPTER 2 Points, Lines, and Functions Additional Example 5: Solution: 176 University of Houston Department of Mathematics SECTION 2.6 An Introduction to Functions Domain of a Function Finding the Domain of a Function: Example: Solution: Example: Solution: MATH 1300 Fundamentals of Mathematics 177 CHAPTER 2 Points, Lines, and Functions Additional Example 1: Solution: 178 University of Houston Department of Mathematics SECTION 2.6 An Introduction to Functions Additional Example 2: Solution: Additional Example 3: Solution: MATH 1300 Fundamentals of Mathematics 179 CHAPTER 2 Points, Lines, and Functions 180 University of Houston Department of Mathematics Exercise Set 2.6: An Introduction to Functions For each of the examples below, determine whether the mapping makes sense within the context of the given situation, and then state whether or not the mapping represents a function. 1. Erik conducts a science experiment and maps the temperature outside his kitchen window at various times during the morning. Express each of the following rules in function notation. (For example, “Subtract 3, then square” would be written as f ( x ) ( x 3)2 .) 7. (a) Divide by 7, then add 4 (b) Add 4, then divide by 7 8. (a) Multiply by 2, then square (b) Square, then multiply by 2 9. (a) Take the square root, then subtract 6 squared (b) Take the square root, subtract 6, then square 57 9 62 10 65 Temp. (oF) Time 2. Dr. Kim counts the number of people in attendance at various times during his lecture this afternoon. 1 85 2 10. (a) Add 4, square, then subtract 2 (b) Subtract 2, square, then add 4 Complete the table for each of the following functions. 11. f ( x) x3 5 f ( x) 2 87 3 Time x 1 # of People 0 1 State whether or not each of the following mappings represents a function. 2 3. 7 9 -3 A 0 5 4 12. g ( x) ( x 4)2 1 f ( x) 3 B 1 4. -6 8 4 -7 A B 9 5. 1 4 6 9 -2 -6 1 A Find the domain of each of the following functions. Write the domain first as an inequality, and then express it in interval notation. B 13. f ( x) 6. x 1 x 0 8 2 4 A 14. f ( x ) 4 x B MATH 1300 Fundamentals of Mathematics 181 Exercise Set 2.6: An Introduction to Functions 5 x 3 15. f ( x) 16. f ( x) 7 x8 17. f ( x) x6 x4 18. h( x) x4 x6 8 19. f (t ) 2t 5 20. h(t ) 2 3t 4 35. h( x) 2 x 9 36. h(t ) 3t 2 37. g ( x) 1 5x 38. f ( x) 4 x 39. f ( x) 8 5 2 x 40. f ( x) 2 7 x 4 41. H ( x) x2 x6 3 x x 21. g ( x) 4x 1 4x 9 42. G( x) 22. f ( x ) 5x 7 3x 7 43. f (t ) 3 t 1 23. g ( x) x 1 x2 9 x2 24. h( x) 2 x 25 25. f ( x) x 2 2 x 24 26. f ( x) 7 2 x 27. g ( x) 3x 5 28. h( x) x2 16 29. f (t ) t 30. h( x) 3 x 31. f ( x) x 5 32. g ( x) x 7 33. f ( x) 3 x 5 182 34. g ( x) 3 x 7 44. g ( x) 3 2x 9 45. h(t ) 3 46. f ( x) t 1 t 5 3 2x 9 4x 7 47. h( x) 5 x 48. h( x) 4 x 49. g ( x) 6 3x 5 50. g ( x) 5 2 x 7 51. f ( x) x 52. g ( x) x 2 53. H ( x) 2x 6 54. f ( x) 3x 5 University of Houston Department of Mathematics Exercise Set 2.6: An Introduction to Functions 55. f ( x) 56. f ( x ) 2 x7 62. If g ( x) x 7 , (a) Find g (0) (b) Find x when g ( x) 0 5 x (c) Find g 2 (d) Find x when g ( x) 2 57. f ( x) x 3 x4 58. f ( x) x9 x 1 (e) Find g 3 (f) Find x when g ( x) 3 63. If h( x) x 2 , find (a) h(7) (b) h(25) (c) h Evaluate the following. 59. If f ( x) 5 x 4 , 64. If h( x) x 2 , find (a) Find f (3) (b) Find x when f ( x) 3 (c) Find f 12 (a) h(7) (b) h(25) (c) h (d) Find x when f ( x) 12 (e) Find f 0 60. If f ( x) 3x 1 , 34 (e) Find f 0 (a) f (16) (b) f (12) (c) (a) Find f (5) (b) Find x when f ( x) 5 (d) Find x when f ( x) 14 65. If f ( x) x 3 , find (f) Find x when f ( x) 0 (c) Find f 14 f 9 66. If f ( x) x 3 , find (a) 3 4 (f) Find x when f ( x) 0 61. If h( x) x 3 , f (16) (b) f (12) (c) f 9 67. If g ( x) x2 5x 6 , (a) Find g (3) (a) Find h(1) (b) Find g 4 (b) Find x when h( x) 1 (c) Find g 12 (c) Find h 2 (d) Find x when h( x) 2 (e) Find h 7 (f) Find x when h( x) 7 (d) Find g 0 68. If h(t ) t 2 2t 15 , (a) Find h(0) (b) Find h(6) (c) Find h 5 (d) Find h 32 MATH 1300 Fundamentals of Mathematics 183 Exercise Set 2.6: An Introduction to Functions 69. If f ( x) 2 x , x 3 (a) Find f (7) (b) Find f (0) (c) Find f 5 (d) Find f 3 (e) Find f 2 70. If g ( x) 5 2x , x4 (a) Find g (2) (b) Find g (4) (c) Find g 52 (d) Find g 3 (e) Find g (0) 184 University of Houston Department of Mathematics SECTION 2.7 Functions and Graphs Section 2.7: Functions and Graphs Graphing a Function Graphing a Function The Graph of a Function: The Vertical Line Test: MATH 1300 Fundamentals of Mathematics 185 CHAPTER 2 Points, Lines, and Functions Example: Solution: \ 186 University of Houston Department of Mathematics SECTION 2.7 Functions and Graphs Example: Solution: MATH 1300 Fundamentals of Mathematics 187 CHAPTER 2 Points, Lines, and Functions Example: Solution: 188 University of Houston Department of Mathematics SECTION 2.7 Functions and Graphs Additional Example 1: Solution: MATH 1300 Fundamentals of Mathematics 189 CHAPTER 2 Points, Lines, and Functions Additional Example 2: The graph of y f x is shown below. (a) Find the domain of f. (b) Find the range of f. (c) Find the following function values: f 3 ; f 1 ; f 0 ; f 1 . (d) For what value(s) of x is f x 2 ? Solution: Part (a): 190 University of Houston Department of Mathematics SECTION 2.7 Functions and Graphs Part (b): Part (c): MATH 1300 Fundamentals of Mathematics 191 CHAPTER 2 Points, Lines, and Functions Part (d): Additional Example 3: Solution: 192 University of Houston Department of Mathematics SECTION 2.7 Functions and Graphs Additional Example 4: Solution: MATH 1300 Fundamentals of Mathematics 193 CHAPTER 2 Points, Lines, and Functions 194 University of Houston Department of Mathematics SECTION 2.7 Functions and Graphs Additional Example 5: Solution: MATH 1300 Fundamentals of Mathematics 195 CHAPTER 2 Points, Lines, and Functions Additional Example 6: Solution: 196 University of Houston Department of Mathematics SECTION 2.7 Functions and Graphs MATH 1300 Fundamentals of Mathematics 197 CHAPTER 2 Points, Lines, and Functions Additional Example 7: Solution: 198 University of Houston Department of Mathematics SECTION 2.7 Functions and Graphs MATH 1300 Fundamentals of Mathematics 199 Exercise Set 2.7: Functions and Graphs Determine whether or not each of the following graphs represents a function. 9. y x 1. y x y 10. 2. y x x 3. y x 4. y x 5. For each set of points, (a) Graph the set of points. (b) Determine whether or not the set of points represents a function. Justify your answer. 11. (1, 5), (2, 4), (3, 4), (2, 1), (3, 6) 12. (3, 2), (1, 2), (0, 3), (2, 1), (2, 1) 13. (2, 0), (4, 1), (6, 0), (3, 1), (5, 2) 14. (1, 4), (2, 3), (4, 1), (4, 2), (2, 3) y x Answer the following. 6. y x 7. y x 15. Analyze the coordinates in each of the sets above. Describe a method of determining whether or not the set of points represents a function without graphing the points. 16. Determine whether or not each set of points represents a function without graphing the points. Justify each answer. (a) (7, 3), (3, 7), (1, 5), (5, 1), (2, 1) (b) 8. (c) y (d) (6, 3), (4, 3), (2, 3), (3, 3), (5, 3) (3, 6), (3, 4), (3, 2), (3, 3), (3, 5) (2, 5), (5, 2), (2, 5), (5, 2), (5, 2) x 200 University of Houston Department of Mathematics Exercise Set 2.7: Functions and Graphs Answer the following. 17. The graph of y f (x) is shown below. 19. The graph of y g (x) is shown below. (a) Find the domain of the function. Write your answer in interval notation. (a) Find the domain of the function. Write your answer in interval notation. (b) Find the range of the function. Write your answer in interval notation. (b) Find the range of the function. Write your answer in interval notation. (c) Find the following function values: (c) Find the following function values: g (2); g (0); g (2); g (4); g (6) f (2); f (0); f (4); f (6) (d) For what value(s) of x is f ( x) 9 ? y (d) Which is greater, g (2) or g (3) ? y f g x x 18. The graph of y g (x) is shown below. 20. The graph of y f (x) is shown below. (a) Find the domain of the function. Write your answer in interval notation. (a) Find the domain of the function. Write your answer in interval notation. (b) Find the range of the function. Write your answer in interval notation. (b) Find the range of the function. Write your answer in interval notation. (c) Find the following function values: g (2); g (0); g (1); g (3); g (6) (c) Find the following function values: (d) For what value(s) of x is g ( x) 2 ? (d) Which is smaller, f (0) or f (3) ? f (3); f (2); f (1); f (1); f (4) y y f g x MATH 1300 Fundamentals of Mathematics x 201 Exercise Set 2.7: Functions and Graphs For each of the following functions: (a) State the domain of the function. Write your answer in interval notation. (b) Choose x-values corresponding to the domain of the function, calculate the corresponding yvalues, plot the points, and draw the graph of the function. 38. x 3y 4 39. 2 y 5 x 7 0 40. 3x 4 y 8 0 21. f ( x) 32 x 6 22. f ( x) 23 x 4 23. h( x) 3x 5, 1 x 3 24. h( x) 2x, 3 x 2 25. g ( x) x 3 26. g ( x) x 4 27. f ( x) x 3 28. f ( x) 5 x 29. F ( x) x 2 4 x 30. G ( x) ( x 3)2 1 For each of the following equations, (a) Solve for y. (b) Determine whether the equation defines y as a function of x. (Do not graph.) 31. 3 y 5x 8 32. 2 x 9 6 y 2 33. 2 y 3x2 7 34. y 2 1 5x 35. x 3 y 2 36. x 2 y 3 37. 202 y 2 x University of Houston Department of Mathematics