Chapter 2



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Chapter 2

Points, Lines, and Functions

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Section 2.1: An Introduction to the Coordinate Plane

➢ Points in the Coordinate Plane

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Points in the Coordinate Plane

The Rectangular Coordinate System:

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Plotting Points in the Coordinate Plane:

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Example:

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Solution:

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Graphing Horizontal and Vertical Lines:

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Example:

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Solution:

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Graphing Other Lines:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Additional Example 4:

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Solution:

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(c) Draw a line through the points.

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Additional Example 5:

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Solution:

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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)

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.

7. G

8. H

9. I

10. J

11. K

12. 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) A(2, 5)

(b) B(-2, -5)

(c) C(2, -5)

(d) D(-2, 5)

14. (a) A(4, -3)

(b) B(-4, -3)

(c) C(-4, 3)

(d) D(4, 3)

15. (a) A(0, -2)

(b) B(-2, 0)

(c) C(2, 0)

(d) D(0, 2)

16. (a) A(-3, 0)

(b) B(3, 0)

(c) (0, -3)

(d) D(0, 3)

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)

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 [pic] and [pic]. Identify the quadrant of each of the following points:

(a) (-a, -b) (b) (b, a) (c) (a, -b)

20. If the point (a, b) is in Quadrant III, then [pic] and [pic]. Identify the quadrant of each of the following points:

(a) (-a, b) (b) (b, a) (c) (-a, -b)

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)

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)

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.

30. The point (-6, 1) is in Quadrant IV.

31. If the point (a, b) is in Quadrant IV, then [pic].

32. If the point (a, b) is in Quadrant II, then [pic].

33. If the point (a, b) is in Quadrant I, then the point (b, a) is also in Quadrant I.

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 .

36. If the point (a, b) is in Quadrant IV, then the point (-b, a) is in Quadrant I.

37. If the point (a, b) is in Quadrant III, then [pic].

38. If the point (a, b) is on the y-axis, then [pic].

39. If the point (a, b) is on the y-axis, then [pic].

40. If the point (a, b) is on the y-axis, then [pic].

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) Plot the above points on a coordinate plane.

b) What do the above points have in common?

c) Draw a line through the above points.

d) What is the equation of the line drawn in part (c)?

44. Given the following points:

A(-3, 4), B(0, 4), C(1, 4), D(3, 4)

a) Plot the above points on a coordinate plane.

b) What do the above points have in common?

c) Draw a line through the above points.

d) 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.

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.

47. Graph the line [pic].

48. Graph the line [pic].

49. Graph the line [pic].

50. Graph the line [pic].

51. On the same set of axes, graph the lines [pic] and [pic].

52. On the same set of axes, graph the lines [pic] and [pic].

53. On the same set of axes, graph the lines [pic] and [pic].

54. On the same set of axes, graph the lines [pic] and [pic].

Graph the following lines by first completing the table and then plotting the points on a coordinate plane.

|x |y |

|-2 | |

|-1 | |

|0 | |

|1 | |

|2 | |

55. [pic]

|x |y |

|-2 | |

|-1 | |

|0 | |

|1 | |

|2 | |

56. [pic]

|x |y |

|0 | |

|[pic] | |

| |-5 |

| |2 |

|[pic] | |

57. [pic]

58. [pic]

|x |y |

|2 | |

| |-1 |

|[pic] | |

| |-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).

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Section 2.2: The Distance and Midpoint Formulas

➢ The Distance Formula

➢ The Midpoint Formula

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The Distance Formula

Finding the Distance Between Two Points:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Additional Example 4:

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Solution:

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Additional Example 5:

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Solution:

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Additional Example 6:

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Solution:

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Additional Example 7:

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Solution:

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Use the Pythagorean Theorem to determine c.

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The Midpoint Formula

Finding the Midpoint of a Line Segment:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Additional Example 4:

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Solution:

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Additional Example 5:

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Solution:

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Use the Pythagorean Theorem to find the missing side of each of the following triangles.

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.

1.

63.

64.

65.

Answer the following.

66. Given the following points:

[pic] and [pic]

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).

67. Given the following points:

[pic] and [pic]

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).

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.)

68. [pic] and [pic]

69. [pic] and [pic]

70. [pic] and [pic]

71. [pic] and [pic]

72. [pic] and [pic]

73. [pic] and [pic]

74. [pic] and [pic]

75. [pic] and [pic]

Find the midpoint of the line segment joining points A and B.

76. [pic] and [pic]

77. [pic] and [pic]

78. [pic] and [pic]

79. [pic] and [pic]

80. [pic] and [pic]

81. [pic] and [pic]

82. [pic] and [pic]

83. [pic] and [pic]

Answer the following.

84. (a) Graph the line segment with endpoints [pic] and [pic].

(b) Find the distance from A to B.

(c) Find the midpoint of [pic].

85. (a) Graph the line segment with endpoints [pic] and [pic].

(b) Find the distance from A to B.

(c) Find the midpoint of [pic].

86. If [pic] is the midpoint of the line segment joining points A and B, and A has coordinates [pic], find the coordinates of B.

87. If [pic] is the midpoint of the line segment joining points A and B, and A has coordinates [pic], find the coordinates of B.

88. If [pic] is the midpoint of the line segment joining points A and B, and B has coordinates [pic],

(a) Find the coordinates of A.

(b) Find the length of [pic].

89. If [pic] is the midpoint of the line segment joining points A and B, and B has coordinates [pic],

(a) Find the coordinates of A.

(b) Find the length of [pic].

90. Determine which of the following points is closer to the origin: [pic] or [pic]?

91. Determine which of the following points is closer to the point [pic]: [pic] or [pic]?

92. A circle has a diameter with endpoints [pic] and [pic].

a) Find the coordinates of the center of the circle.

b) Find the length of the radius of the circle.

93. A circle has a diameter with endpoints [pic] and [pic].

a) Find the coordinates of the center of the circle.

b) Find the length of the radius of the circle.

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Section 2.3: Slope and Intercepts of Lines

➢ The Slope of a Line

➢ Intercepts of Lines

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The Slope of a Line

Finding the Slope of a Line:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Additional Example 4:

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Solution:

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Intercepts of Lines

Finding Intercepts of Lines:

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Horizontal Lines:

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Vertical Lines:

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Example:

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Solution:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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State whether the slope of each of the following lines is positive, negative, zero, or undefined.

1. p

94. q

95. r

96. s

97. t

98. w

Find the slope of the line that passes through the following points. If undefined, state ‘Undefined.’

99. [pic]

100. [pic]

101. [pic]

102. [pic]

103. [pic]

104. [pic]

105. [pic]

106. [pic]

107. [pic]

108. [pic]

109. [pic]

110. [pic]

111. [pic] and [pic]

112. [pic] and [pic]

113. [pic] and [pic]

114. [pic] and [pic]

Find the slope of each of the following lines. If undefined, state ‘Undefined.’

115. c

116. d

117. e

118. f

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.

|x |y |

|0 | |

|[pic] | |

| |[pic] |

| |0 |

|[pic] | |

119. [pic]

|x |y |

|2 | |

| |2 |

| |4 |

|[pic] | |

|[pic] | |

120. [pic]

|x |y |

| |[pic] |

|5 | |

|9 | |

| |[pic] |

|[pic] | |

121. [pic]

122. [pic]

|x |y |

|[pic] | |

|0 | |

| |7 |

|8 | |

| |0 |

Answer the following.

123. 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?

124. 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) [pic]

(b) [pic]

(c) [pic]

(d) [pic]

For each of the following graphs:

(a) State the x-intercept.

(b) State the y-intercept.

(c) State the coordinates of the x-intercept.

(d) State the coordinates of the y-intercept.

(e) Find the slope of the line.

125.

126.

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.

127. [pic]

128. [pic]

129. [pic]

130. [pic]

131. [pic]

132. [pic]

133. [pic]

134. [pic]

135. [pic]

136. [pic]

137. [pic]

138. [pic]

139. [pic]

140. [pic]

141. [pic]

142. [pic]

143. [pic]

144. [pic]

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.

|x |y |

|0 | |

| |0 |

|2 | |

| |6 |

|[pic] | |

145. [pic]

|x |y |

| |0 |

|0 | |

| |[pic] |

|1.5 | |

|[pic] | |

146. [pic]

Answer the following.

147. 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?

148. 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) [pic]

(b) [pic]

(c) [pic]

(d) [pic]

[pic]

Section 2.4: Equations of Lines

➢ Writing Equations of Lines

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Writing Equations of Lines

Different Forms for Equations of Lines:

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Example:

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Solution:

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Example:

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Solution:

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Example:

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Solution:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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To sketch the graph, begin by using

the y-intercept to plot the point [pic].

[pic]

[pic]

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Additional Example 3:

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Solution:

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Additional Example 4:

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Solution:

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Write an equation in slope-intercept form for each of the following lines.

1.

149.

150.

151.

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.

152. [pic]

153. [pic]

154. [pic]

155. [pic]

156. [pic]

157. [pic]

158. [pic]

159. [pic]

160. [pic]

161. [pic]

162. [pic]

163. [pic]

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 slope-intercept form. (Do this algebraically, and then check to see if your result matches your graph.)

164. Slope [pic]; passes through [pic]

165. Slope [pic]; passes through [pic]

166. Passes through [pic] and [pic]

167. Passes through [pic] and [pic]

Write an equation in slope-intercept form for the line that satisfies the given conditions.

168. Slope [pic]; y-intercept [pic]

169. Slope [pic]; y-intercept [pic]

170. Slope [pic]; passes through [pic]

171. Slope [pic]; passes through [pic]

172. Slope [pic]; passes through [pic]

173. Slope [pic]; passes through [pic]

174. Passes through [pic] and [pic]

175. Passes through [pic] and [pic]

176. Passes through [pic] and [pic]

177. Passes through [pic] and [pic]

178. x-intercept [pic]; y-intercept [pic]

179. x-intercept [pic]; y-intercept [pic]

180. Slope [pic]; x-intercept [pic]

181. Slope [pic]; x-intercept [pic]

Answer the following, assuming that each situation can be modeled by a linear equation.

182. 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.

183. 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.

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Section 2.5: Parallel and Perpendicular Lines

➢ Pairs of Lines – Parallel and Perpendicular Lines

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Pairs of Lines - Parallel and Perpendicular Lines

Parallel Lines:

[pic]

[pic]

Perpendicular Lines:

Two lines with slopes [pic] and [pic] perpendicular if and only if [pic].

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Example:

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Solution:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Additional Example 4:

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Solution:

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State whether the following pairs of lines are parallel, perpendicular, or neither.

1. [pic]

[pic]

184. [pic]

[pic]

185. [pic]

[pic]

186. [pic]

[pic]

187. [pic]

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188. [pic]

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189. [pic]

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190. [pic]

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191. [pic]

[pic]

192. [pic]

[pic]

193. The line passing through [pic] and [pic]

The line passing through [pic] and [pic]

194. The line passing through [pic] and [pic]

The line passing through [pic] and [pic]

195. The line passing through [pic] and [pic]

The line passing through [pic] and [pic]

196. The line passing through [pic] and [pic]

The line passing through [pic] and [pic]

197. [pic]

[pic]

198. [pic]

[pic]

199. [pic]

[pic]

200. [pic]

[pic]

201. The line passing through [pic] and [pic]

The line passing through [pic] and [pic]

202. The line passing through [pic] and [pic]

The line passing through [pic] and [pic]

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

203. Passes through [pic]; parallel to the line [pic]

204. Passes through [pic]; perpendicular to the line [pic]

205. Passes through [pic]; perpendicular to the line [pic]

206. Passes through [pic]; parallel to the line [pic]

207. Passes through [pic]; parallel to the line [pic]

208. Passes through [pic]; perpendicular to the line [pic]

209. Passes through [pic]; perpendicular to the line [pic]

210. Passes through [pic]; parallel to the line [pic]

Write an equation for the line that satisfies the given conditions. With the exception of vertical lines, write all equations in slope-intercept form.

211. Passes through [pic]; parallel to the x-axis

212. Passes through [pic]; parallel to the y-axis

213. Passes through [pic]; parallel to the line [pic]

214. Passes through [pic]; parallel to the line [pic]

215. Passes through [pic]; and is

(a) parallel to the line [pic]

(b) perpendicular to the line [pic]

216. Passes through [pic]; and is

(a) parallel to the line [pic]

(b) perpendicular to the line [pic]

217. Passes through [pic]; parallel to the line [pic]

218. Passes through [pic]; parallel to the line [pic]

219. Passes through [pic]; perpendicular to the line [pic]

220. Passes through [pic]; perpendicular to the line [pic]

221. Passes through [pic]; parallel to the line containing [pic] and [pic]

222. Passes through [pic]; parallel to the line containing [pic] and [pic]

223. Perpendicular to the line containing [pic] and

[pic]; passes through the midpoint of the line segment connecting these points

224. Perpendicular to the line containing [pic] and [pic]; passes through the midpoint of the line segment connecting these points

[pic]

Section 2.6: An Introduction to Functions

➢ Definition of a Function

➢ Domain of a Function

[pic]

Definition of a Function

Definition:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Defining a Function by an Equation in the Variables x and y:

[pic]

[pic]

The Function Notation:

[pic]

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

Additional Example 1:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Additional Example 2:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

Additional Example 3:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Additional Example 4:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Additional Example 5:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Domain of a Function

Finding the Domain of a Function:

[pic]

Example:

[pic]

Solution:

[pic]

Example:

[pic]

Solution:

[pic]

Additional Example 1:

[pic]

Solution:

[pic]

[pic]

[pic]

Additional Example 2:

[pic]

Solution:

[pic]

[pic]

[pic]

Additional Example 3:

[pic]

Solution:

[pic]

[pic]

[pic]

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.

225. Dr. Kim counts the number of people in attendance at various times during his lecture this afternoon.

State whether or not each of the following mappings represents a function.

226.

227.

228.

229.

Express each of the following rules in function notation. (For example, “Subtract 3, then square” would be written as[pic].)

230. (a) Divide by 7, then add 4

(b) Add 4, then divide by 7

231. (a) Multiply by 2, then square

(b) Square, then multiply by 2

232. (a) Take the square root, then subtract 6 squared

(b) Take the square root, subtract 6, then square

233. (a) Add 4, square, then subtract 2

(b) Subtract 2, square, then add 4

Complete the table for each of the following functions.

|x |[pic] |

|[pic] | |

|[pic] | |

|0 | |

|1 | |

|2 | |

234. [pic]

|x |[pic] |

|[pic] | |

|[pic] | |

|1 | |

|4 | |

|6 | |

235. [pic]

Find the domain of each of the following functions. Write the domain first as an inequality, and then express it in interval notation.

236. [pic]

237. [pic]

238. [pic]

239. [pic]

240. [pic]

241. [pic]

242. [pic]

243. [pic]

244. [pic]

245. [pic]

246. [pic]

247. [pic]

248. [pic]

249. [pic]

250. [pic]

251. [pic]

252. [pic]

253. [pic]

254. [pic]

255. [pic]

256. [pic]

257. [pic]

258. [pic]

259. [pic]

260. [pic]

261. [pic]

262. [pic]

263. [pic]

264. [pic]

265. [pic]

266. [pic]

267. [pic]

268. [pic]

269. [pic]

270. [pic]

271. [pic]

272. [pic]

273. [pic]

274. [pic]

275. [pic]

276. [pic]

277. [pic]

278. [pic]

279. [pic]

280. [pic]

281. [pic]

Evaluate the following.

282. If [pic],

(a) Find [pic]

(b) Find x when [pic]

(c) Find [pic]

(d) Find x when [pic]

(e) Find [pic]

(f) Find x when [pic]

283. If [pic],

(a) Find [pic]

(b) Find x when [pic]

(c) Find [pic]

(d) Find x when [pic]

(e) Find [pic]

(f) Find x when [pic]

284. If [pic],

(a) Find [pic]

(b) Find x when [pic]

(c) Find [pic]

(d) Find x when [pic]

(e) Find [pic]

(f) Find x when [pic]

285. If [pic],

(a) Find [pic]

(b) Find x when [pic]

(c) Find [pic]

(d) Find x when [pic]

(e) Find [pic]

(f) Find x when [pic]

286. If [pic], find

(a) [pic]

(b) [pic]

(c) [pic]

287. If [pic], find

(a) [pic]

(b) [pic]

(c) [pic]

288. If [pic], find

(a) [pic]

(b) [pic]

(c) [pic]

289. If [pic], find

(a) [pic]

(b) [pic]

(c) [pic]

290. If [pic],

(a) Find [pic]

(b) Find [pic]

(c) Find [pic]

(d) Find [pic]

291. If [pic],

(a) Find [pic]

(b) Find [pic]

(c) Find [pic]

(d) Find [pic]

292. If [pic],

(a) Find [pic]

(b) Find [pic]

(c) Find [pic]

(d) Find [pic]

(e) Find [pic]

293. If [pic],

(a) Find [pic]

(b) Find [pic]

(c) Find [pic]

(d) Find [pic]

(e) Find [pic]

[pic]

Section 2.7: Functions and Graphs

➢ Graphing a Function

[pic]

Graphing a Function

The Graph of a Function:

[pic]

[pic]

The Vertical Line Test:

[pic]

[pic]

[pic]

Example:

[pic]

Solution:

[pic]\

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

Additional Example 1:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

Additional Example 2:

The graph of [pic] is shown below.

(a) Find the domain of f.

(b) Find the range of f.

(c) Find the following function values: [pic].

(d) For what value(s) of x is [pic]?

[pic]

Solution:

Part (a):

[pic]

[pic]

[pic]

Part (b):

[pic]

[pic]

[pic]

Part (c):

[pic]

[pic]

[pic]

[pic]

Part (d):

[pic]

[pic]

[pic]

[pic]

Additional Example 3:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Additional Example 4:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Additional Example 5:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Additional Example 6:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Additional Example 7:

[pic]

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Determine whether or not each of the following graphs represents a function.

1.

294.

295.

296.

297.

298.

299.

300.

301.

302.

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.

303. [pic]

304. [pic]

305. [pic]

306. [pic]

Answer the following.

307. 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.

308. Determine whether or not each set of points represents a function without graphing the points. Justify each answer.

(a) [pic]

(b) [pic]

(c) [pic]

(d) [pic]

Answer the following.

309. The graph of [pic] is shown below.

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.

c) Find the following function values:

[pic]

d) For what value(s) of x is [pic]?

310. The graph of [pic] is shown below.

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.

c) Find the following function values:

[pic]

d) For what value(s) of x is [pic]?

311. The graph of [pic] is shown below.

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.

c) Find the following function values:

[pic]

d) Which is greater, [pic] or [pic]?

312. The graph of [pic] is shown below.

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.

c) Find the following function values:

[pic]

d) Which is smaller, [pic] or [pic]?

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 y-values, plot the points, and draw the graph of the function.

313. [pic]

314. [pic]

315. [pic]

316. [pic]

317. [pic]

318. [pic]

319. [pic]

320. [pic]

321. [pic]

322. [pic]

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.)

323. [pic]

324. [pic]

325. [pic]

326. [pic]

327. [pic]

328. [pic]

329. [pic]

330. [pic]

331. [pic]

332. [pic]

-----------------------

c

a

b

c

5

12

a

7

5

2

6

b

c

6

8

2

0

8

4

B

A

9

-6

1

-2

B

A

8

4

-7

9

-6

B

A

0

5

4

7

9

-3

B

A

# of People

85

87

1

2

3

Time

Temp. (oF)

Time

57

62

65

9

10

................
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