slope & y-intercept class work -...

Pre-Calc Parent Functions ~1~ NJCTL.org

Slope & y-intercept – Class Work

Identify the slope and y-intercept for each equation

1. 𝑦 = 3𝑥 − 4 2. 𝑦 = −2𝑥 3. 𝑦 = 7

4. 𝑥 = −5 5. 𝑦 = 0 6. 𝑦 − 3 = 4(𝑥 + 6)

7. 𝑦 + 2 = −1

2(𝑥 + 6) 8. 2𝑥 + 3𝑦 = 9 9. 4𝑥 − 7𝑦 = 14

Write the equation of the given line in slope-intercept form and match with the correct alternate form.

10. A

11. B

12. C

13. D

14. E

15. F

16. Write an equation: Cal C. drives past mile marker 27 at 11am and mile marker 145 at 1pm.

𝒎 = 𝟑

𝒃 = −𝟒

𝒎 = −𝟐

𝒃 = 𝟎

𝒎 = 𝟎

𝒃 = 𝟕

𝒎 = 𝒖𝒏𝒅𝒆𝒇

𝒃 = 𝒏𝒐𝒏𝒆

𝒎 = 𝟎

𝒃 = 𝟎

𝒎 = 𝟒

𝒃 = 𝟐𝟕

𝒎 = −𝟏

𝟐

𝒃 = −𝟓

𝒎 = −𝟐

𝟑

𝒃 = 𝟑

𝒎 =𝟒

𝟕

𝒃 = −𝟐

I. 𝑦 + 4 = −1

2(𝑥 − 2)

II. 𝑁𝑜 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑒 𝐹𝑜𝑟𝑚

(𝑆𝑙𝑜𝑝𝑒 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑)

III. 5𝑥 + 7𝑦 = 35

IV. 𝑦 − 6 = 3(𝑥 − 4)

V. 2𝑥 − 7𝑦 = 14

VI. 𝑦 − 6 = 0(𝑥 + 2)

𝒚 = −𝟓

𝟕𝒙 + 𝟓 III

𝒚 = 𝟎𝒙 + 𝟔 VI

𝒚 = −𝟏

𝟐𝒙 − 𝟑 I

𝒚 =𝟐

𝟕𝒙 − 𝟐 V

𝒚 = 𝟑𝒙 − 𝟔 IV

𝒙 = 𝟖 II

𝒚 = 𝟓𝟗𝒙 + 𝟐𝟕


Slope & y-intercept – Homework

Identify the slope and y-intercept for each equation

17. 𝑦 = −5𝑥 − 2 18. 𝑦 = 3𝑥 19. 𝑦 = −2

20. 𝑥 = 10 21. 𝑥 = 0 22. 𝑦 − 4 = 2(𝑥 − 8)

23. 𝑦 + 3 = −2

5(𝑥 + 10) 24. 3𝑥 + 4𝑦 = 12 25. 2𝑥 − 6𝑦 = 12

Write the equation of the given line.

26. A

27. B

28. C

29. D

30. E

31. F

32. Write an equation: Cal C. drives past mile marker 45 at 11am and mile marker 225 at 2pm.

I. 𝑦 + 4 = −4(𝑥 − 3)

II. 𝑥 + 3𝑦 = −1

III. 𝑦 + 6 = 0(𝑥 − 1)

IV. 𝑁𝑜 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑒 𝐹𝑜𝑟𝑚

(𝑆𝑙𝑜𝑝𝑒 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑)

V. 3𝑥 − 2𝑦 = 8

VI. 2𝑥 − 9𝑦 = 2

𝒎 = 𝟎

𝒃 = −𝟐

𝒎 = 𝟑

𝒃 = 𝟎

𝒎 = −𝟓

𝒃 = −𝟐





𝒎 = 𝟐

𝒃 = −𝟏𝟐

𝒎 = −𝟐

𝟓

𝒃 = −𝟕

𝒎 = −𝟑

𝟒

𝒃 = 𝟑

𝒎 =𝟏

𝟑

𝒃 = −𝟐

𝒚 =𝟑

𝟐𝒙 − 𝟒 V

𝒚 = −𝟒𝒙 + 𝟖 I

𝒙 = −𝟔 IV

𝒚 = −𝟏

𝟑𝒙 − 𝟏 II

𝒚 =𝟐

𝟗𝒙 − 𝟐 VI

𝒚 = 𝟎𝒙 − 𝟔 III

𝒚 = 𝟔𝟎𝒙 + 𝟒𝟓


Standard Form – Class Work

Find the x and y intercepts for each equation.

33. 2𝑥 + 3𝑦 = 12 34. 4𝑥 + 5𝑦 = 10

35. 𝑥 − 3𝑦 = 10 36. 4𝑥 = 9

37. 𝑦 = 0

Standard Form – Homework

Find the x and y intercepts for each equation.

38. 3𝑥 − 5𝑦 = 15 39. 7𝑥 + 2𝑦 = 14

40. 𝑥 − 𝑦 = 9 41. 𝑦 = 7

42. 𝑥 = 0

𝒙: (𝟔, 𝟎)

𝒚: (𝟎, 𝟒) 𝒙: (

𝟓

𝟐, 𝟎)

𝒚: (𝟎, 𝟐)

𝒙: (𝟏𝟎, 𝟎)

𝒚: (𝟎, −𝟏𝟎

𝟑)

𝒙: (𝟗

𝟒, 𝟎)

𝒚: 𝒏𝒐𝒏𝒆

𝒙: 𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒆

𝒚: (𝟎, 𝟎)

𝒙: (𝟓, 𝟎)

𝒚: (𝟎, −𝟑)

𝒙: (𝟐, 𝟎)

𝒚: (𝟎, 𝟕)

𝒙: (𝟗, 𝟎)

𝒚: (𝟎, −𝟗)

𝒙: 𝒏𝒐𝒏𝒆

𝒚: (𝟎, 𝟕)

𝒙: (𝟎, 𝟎)

𝒚: 𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒆


Horizontal and Vertical Lines – Class Work

Write the equation for the described line

43. vertical through (1,3) 44. horizontal through (1,3)

45. vertical through (-2, 4) 46. horizontal through (-2, 4)

Horizontal and Vertical Lines – Homework

Write the equation for the described line

47. vertical through (4,7) 48. horizontal through (8,-10)

49. vertical through (8, -10) 50. horizontal through (4, 7)

𝒙 = 𝟏 𝒚 = 𝟑

𝒙 = −𝟐 𝒚 = 𝟒

𝒙 = 𝟒 𝒚 = −𝟏𝟎

𝒙 = 𝟖 𝒚 = 𝟕


Parallel and Perpendicular Lines – Class Work

Write the equation for the described line in slope intercept form

51. Parallel to 𝑦 = 3𝑥 + 4 through (3,1) 52. Perpendicular to 𝑦 = 3𝑥 + 4 through (3,1)

53. Parallel to 𝑦 = −1

2𝑥 + 6 through (4, −2) 54. Perpendicular to 𝑦 = −

1

2𝑥 + 6 through (4, −2)

55. Parallel to 𝑦 = 5 through (−1, −8) 56. Perpendicular to 𝑦 = 5 through (−1, −8)

Parallel and Perpendicular Lines – Homework

Write the equation for the described line in slope intercept form

57. Parallel to 𝑦 = −2𝑥 + 1 through (−6,1) 58. Perpendicular to 𝑦 = −2𝑥 + 1 through (−6,1)

59. Parallel to 𝑦 =1

3𝑥 − 5 through (−6,0) 60. Perpendicular to 𝑦 =

1

3𝑥 − 5 through (−6,0)

61. Parallel to 𝑥 = 5 through (−3,7) 62. Perpendicular to 𝑥 = 5 through (−3,7)

𝒚 = 𝟑𝒙 − 𝟖 𝒚 = −𝟏

𝟑𝒙 + 𝟐

𝒚 = −𝟏

𝟐𝒙 𝒚 = 𝟐𝒙 − 𝟏𝟎

𝒚 = −𝟖 𝒙 = −𝟏

𝒚 = −𝟐𝒙 − 𝟏𝟏 𝒚 =𝟏

𝟐𝒙 + 𝟒

𝒚 =𝟏

𝟑𝒙 + 𝟐 𝒚 = −𝟑𝒙 − 𝟏𝟖

𝒙 = −𝟑 𝒚 = 𝟕


Point-Slope Form – Class Work

Write the equation for the described line in point slope form.

63. Slope of 6 through (5,1) 64. Slope of -2 through (−4,3)

65. Slope of 1 through (8,0) 66. Perpendicular to 𝑦 = −2𝑥 + 1 through (1, −6)

Convert the following equations to slope-intercept form and standard form

67. 𝑦 − 4 = 5(𝑥 + 3) 68. 𝑦 = −2(𝑥 − 1) 69. 𝑦 + 7 =1

5(𝑥 − 10)

Point-Slope Form – Homework

Write the equation for the described line in point slope form.

70. Slope of -4 through (4, −2) 71. Slope of 3 through (0, −9)

72. Slope of 1/4 through (6,0) 73. Perpendicular to 𝑦 = −1

2𝑥 + 6 through (5, −2)

Convert the following equations to slope-intercept form and standard form

74. 𝑦 − 3 = 7(𝑥 − 2) 75. 𝑦 + 1 = −4(𝑥 − 7) 76. 𝑦 + 3 =1

6(𝑥 − 18)

𝒚 − 𝟏 = 𝟔(𝒙 − 𝟓) 𝒚 − 𝟑 = −𝟐(𝒙 + 𝟒)

𝒚 − 𝟎 = 𝟏(𝒙 − 𝟖) 𝒚 + 𝟔 =𝟏

𝟐(𝒙 − 𝟏)

𝒚 = 𝟓𝒙 + 𝟏𝟗

𝟓𝒙 − 𝒚 = −𝟏𝟗

𝒚 = −𝟐𝒙 + 𝟐

𝟐𝒙 + 𝒚 = 𝟐 𝒚 =

𝟏

𝟓𝒙 − 𝟗

𝒙 − 𝟓𝒚 = 𝟒𝟓

𝒚 + 𝟐 = −𝟒(𝒙 − 𝟒) 𝒚 + 𝟗 = 𝟑(𝒙 − 𝟎)

𝒚 − 𝟎 =𝟏

𝟒(𝒙 − 𝟔) 𝒚 + 𝟐 = 𝟐(𝒙 − 𝟓)

𝒚 = 𝟕𝒙 − 𝟏𝟏

𝟕𝒙 − 𝒚 = 𝟏𝟏

𝒚 = −𝟒𝒙 + 𝟐𝟕

𝟒𝒙 + 𝒚 = 𝟐𝟕 𝒚 =

𝟏

𝟔𝒙 − 𝟔

𝒙 − 𝟔𝒚 = 𝟑𝟔


Writing Linear Equations – Class Work

Write an equation based on the given information in slope intercept form.

77. A line through (7, −1) and (−3,4) 78. A line through (8,2) and (6, −2)

79. A line perpendicular to 𝑦 − 7 =1

2(𝑥 + 2) through (−1, −8)

80. A line parallel to 4𝑥 − 6𝑦 = 10 through (9,7)

81. A function with constant increase passing through (2,3) and (8,9)

82. The cost of a 3.8 mile taxi ride cost $5.50 and the cost of a 4 mile ride costs $5.70

83. A valet parking services charges $45 for 2 hours and $55 for 3 hours

𝒚 = −𝟏

𝟐𝒙 +

𝟓

𝟐 𝒚 = 𝟐𝒙 − 𝟏𝟒

𝒚 = −𝟐𝒙 − 𝟏𝟎

𝒚 =𝟐

𝟑𝒙 + 𝟏

𝒚 = 𝒙 + 𝟏

𝒚 = 𝟏. 𝟎𝟎𝒙 + 𝟏. 𝟕𝟎

𝒚 = 𝟏𝟎𝒙 + 𝟐𝟓


Writing Linear Equations – Homework

Write an equation based on the given information in slope intercept form.

84. A line through (4,9) and (−5, −9) 85. A line through (−8,2) and (8,2)

86. A line perpendicular to 4𝑥 − 6𝑦 = 10 through (2,2)

87. A line parallel to 𝑦 − 7 =1

2(𝑥 + 2) through (2,2)

88. A function with constant decrease passing through (2,3) and (8, −9)

89. The cost of a 3.8 mile taxi ride cost $8.25 and the cost of a 4 mile ride costs $8.75

90. A valet parking services charges $55 for 2 hours and $75 for 4 hours

𝒚 = 𝟐𝒙 + 𝟏 𝒚 = 𝟎𝒙 + 𝟐

𝒚 = −𝟑

𝟐𝒙 + 𝟓

𝒚 =𝟏

𝟐𝒙 + 𝟏

𝒚 = −𝟐𝒙 + 𝟕

𝒚 = 𝟐. 𝟓𝟎𝒙 − 𝟏. 𝟐𝟓

𝒚 = 𝟏𝟎𝒙 + 𝟑𝟓


Absolute Value Functions – Class Work

Graph each function. For each, state the domain and range.

91. 𝑦 = |𝑥 − 4| 92. 𝑦 = |2𝑥|

93. 𝑦 = |2𝑥 − 4| 94. 𝑦 = 2|𝑥|

95. 𝑦 = |𝑥| + 3 96. 𝑦 = 2|𝑥| + 3

97. 𝑦 = −|𝑥 − 3| 98. 𝑦 = −1

2|𝑥| − 3

𝑫: ℝ 𝑹: [𝟎, ∞) 𝑫: ℝ 𝑹: [𝟎, ∞)

𝑫: ℝ 𝑹: [𝟎, ∞) 𝑫: ℝ 𝑹: [𝟎, ∞)

𝑫: ℝ 𝑹: [𝟑, ∞) 𝑫: ℝ 𝑹: [𝟑, ∞)

𝑫: ℝ 𝑹: (−∞, 𝟎] 𝑫: ℝ 𝑹: (−∞, −𝟑]


Absolute Value Functions – Homework


99. 𝑦 = |𝑥 + 2| 100. 𝑦 = |3𝑥|

101. 𝑦 = |3𝑥 + 2| 102. 𝑦 =2

3|𝑥|

103. 𝑦 = |𝑥| − 1 104. 𝑦 =2

3|𝑥| − 1

105. 𝑦 = −|𝑥 + 4| 106. 𝑦 = −1

4|𝑥| + 4

𝑫: ℝ 𝑹: [𝟎, ∞) 𝑫: ℝ 𝑹: [𝟎, ∞)

𝑫: ℝ 𝑹: [𝟎, ∞) 𝑫: ℝ 𝑹: [𝟎, ∞)

𝑫: ℝ 𝑹: [−𝟏, ∞) 𝑫: ℝ 𝑹: [−𝟏, ∞)

𝑫: ℝ 𝑹: (−∞, 𝟎] 𝑫: ℝ 𝑹: (−∞, 𝟒]


Greatest Integer Functions – Class Work


107. 𝑦 = [𝑥 − 4] 108. 𝑦 = [2𝑥]

109. 𝑦 = [2𝑥 − 4] 110. 𝑦 = 2[𝑥]

111. 𝑦 = [𝑥] + 3 112. 𝑦 = 2[𝑥] + 3

113. 𝑦 = −[𝑥 − 3] 114. 𝑦 = −1

2[𝑥] − 3

𝑫: ℝ 𝑹: ℤ 𝑫: ℝ 𝑹: ℤ

𝑫: ℝ 𝑹: ℤ 𝑫: ℝ 𝑹: 𝟐ℤ

𝑫: ℝ 𝑹: ℤ 𝑫: ℝ 𝑹: 𝟐ℤ + 𝟏

𝑫: ℝ 𝑹: ℤ 𝑫: ℝ 𝑹:𝟏

𝟐ℤ


Greatest Integer Functions – Homework


115. 𝑦 = [𝑥 + 2] 116. 𝑦 = [3𝑥]

117. 𝑦 = [3𝑥 + 2] 118. 𝑦 =2

3[𝑥]

119. 𝑦 = [𝑥] − 1 120. 𝑦 =2

3[𝑥] − 1

121. 𝑦 = −[𝑥 + 4] 122. 𝑦 = −1

4[𝑥] + 4

𝑫: ℝ 𝑹: ℤ 𝑫: ℝ 𝑹: ℤ

𝑫: ℝ 𝑹: ℤ 𝑫: ℝ 𝑹:𝟐

𝟑ℤ

𝑫: ℝ 𝑹: ℤ 𝑫: ℝ 𝑹:𝟐

𝟑ℤ

𝑫: ℝ 𝑹: ℤ 𝑫: ℝ 𝑹:𝟏

𝟒ℤ


Identifying Exponential Growth and Decay – Class Work

State whether the given function exponential growth or decay.

Identify the horizontal asymptote and the y-intercept.

123. A B

124. 𝑦 = 3(4)𝑥 125. 𝑦 =1

2(3)𝑥

126. 𝑦 = (1

2)

𝑥+ 4 127. 𝑦 = 2 (

1

4)

𝑥− 7

128. 𝑦 = 100 (1

3)

𝑥+ 50 129. 𝑦 = 17(4)−𝑥

130. 𝑦 = 12 (3

4)

−𝑥+ 6 131. 𝑦 =

1

2(5)−𝑥 − 2

𝑮𝒓𝒐𝒘𝒕𝒉

𝑯𝑨: 𝒚 = 𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟏)

𝑫𝒆𝒄𝒂𝒚

𝑯𝑨: 𝒚 = 𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟏)


𝑯𝑨: 𝒚 = 𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟑)


𝑯𝑨: 𝒚 = 𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎,𝟏

𝟐)


𝑯𝑨: 𝒚 = 𝟒

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟓)


𝑯𝑨: 𝒚 = −𝟕

𝒚 − 𝒊𝒏𝒕: (𝟎, −𝟓)


𝑯𝑨: 𝒚 = 𝟓𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟏𝟓𝟎)


𝑯𝑨: 𝒚 = 𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟏𝟕)


𝑯𝑨: 𝒚 = 𝟔

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟏𝟖)


𝑯𝑨: 𝒚 = 𝟐

𝒚 − 𝒊𝒏𝒕: (𝟎, −𝟏𝟏

𝟐)


Identifying Exponential Growth and Decay – Homework

State whether the given function exponential growth or decay.

Identify the horizontal asymptote and the y-intercept.

132. A B

133. 𝑦 = 3 (2

5)

𝑥 134. 𝑦 = 5(3)−𝑥

135. 𝑦 = 6 (1

2)

𝑥+ 4 136. 𝑦 = 4(25)𝑥 − 7

137. 𝑦 = 100 (1

3)

−𝑥+ 50 138. 𝑦 = 17(4)𝑥

139. 𝑦 = 12 (3

4)

𝑥+ 6 140. 𝑦 =

2

5(7)𝑥 − 1


𝑯𝑨: 𝒚 = 𝟐

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟑)


𝑯𝑨: 𝒚 = 𝟐

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟑)


𝑯𝑨: 𝒚 = 𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟑)


𝑯𝑨: 𝒚 = 𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟓)


𝑯𝑨: 𝒚 = 𝟒

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟏𝟎)


𝑯𝑨: 𝒚 = −𝟕

𝒚 − 𝒊𝒏𝒕: (𝟎, −𝟑)


𝑯𝑨: 𝒚 = 𝟓𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟏𝟓𝟎)


𝑯𝑨: 𝒚 = 𝟎

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟏𝟕)


𝑯𝑨: 𝒚 = 𝟔

𝒚 − 𝒊𝒏𝒕: (𝟎, 𝟏𝟖)


𝑯𝑨: 𝒚 = −𝟏

𝒚 − 𝒊𝒏𝒕: (𝟎, −𝟑

𝟓)


Graphing Exponentials – Class Work

Graph each equation.

141. 𝑦 = 3(4)𝑥 142. 𝑦 =1

2(3)𝑥

143. 𝑦 = (1

2)

𝑥+ 4 144. 𝑦 = 2 (

1

4)

𝑥− 7

145. 𝑦 = 100 (1

3)

𝑥+ 50 146. 𝑦 = 17(4)−𝑥

147. 𝑦 = 12 (3

4)

−𝑥+ 6


Graphing Exponentials – Homework

Graph each equation

148. 𝑦 = 3 (2

5)

𝑥 149. 𝑦 = 5(3)−𝑥

150. 𝑦 = 6 (1

2)

𝑥+ 4 151. 𝑦 = 4(25)𝑥 − 7

152. 𝑦 = 100 (1

3)

−𝑥+ 50 153. 𝑦 = 17(4)𝑥

154. 𝑦 = 12 (3

4)

𝑥+ 6


Intro to Logs – Class Work

Write each of the following in log form.

155. 102 = 100 156. 24 = 16 157. 27 = 33

Write each of the following in exponential form.

158. log5 125 = 3 159. log6 36 = 2 160. log7 343 = 3

Solve the following equations

161. log4 64 = 𝑥 162. log2 64 = 𝑥

163. log3 𝑦 = 5 164. log6 𝑦 = 3

165. log𝑏 81 = 4 166. log𝑏 10 = 1

167. log5(𝑥 − 2) = log5(2𝑥 − 8) 168. log4(2𝑥 + 7) = log4(4𝑥 − 9)

𝐥𝐨𝐠 𝟏𝟎𝟎 = 𝟐 𝐥𝐨𝐠𝟐 𝟏𝟔 = 𝟒 𝐥𝐨𝐠𝟑 𝟐𝟕 = 𝟑

𝟓𝟑 = 𝟏𝟐𝟓 𝟔𝟐 = 𝟑𝟔 𝟕𝟑 = 𝟑𝟒𝟑

𝒙 = 𝟑 𝒙 = 𝟔

𝒚 = 𝟐𝟒𝟑 𝒚 = 𝟐𝟏𝟔

𝒃 = 𝟑 𝒃 = 𝟏𝟎

𝒙 = 𝟔 𝒙 = 𝟖


Intro to Logs – Homework

Write each of the following in log form.

169. 92 = 81 170. 25 = 32 171. 81 = 34

Write each of the following in exponential form.

172. log8 64 = 2 173. log4 256 = 4 174. log3 81 = 4


175. log4 1024 = 𝑥 176. log2 128 = 𝑥

177. log5 𝑦 = 4 178. log7 𝑦 = 4

179. log𝑏 1000 = 3 180. log𝑏 1024 = 10

181. log5(𝑥 + 2) = log5(3𝑥 − 8) 182. log4(3𝑥 − 6) = log4(𝑥 + 10)

𝐥𝐨𝐠𝟗 𝟖𝟏 = 𝟐 𝐥𝐨𝐠𝟐 𝟑𝟐 = 𝟓 𝐥𝐨𝐠𝟑 𝟖𝟏 = 𝟒

𝟖𝟐 = 𝟔𝟒 𝟒𝟒 = 𝟐𝟓𝟔 𝟑𝟒 = 𝟖𝟏

𝒙 = 𝟓 𝒙 = 𝟕

𝒚 = 𝟔𝟐𝟓 𝒚 = 𝟐𝟒𝟎𝟏

𝒃 = 𝟏𝟎 𝒃 = 𝟐

𝒙 = 𝟓 𝒙 = 𝟖


Properties of Logs – Class Work

Using Properties of Logs, fully expand each expression

183. log4 𝑥𝑦3𝑧4 184. log6𝑤

𝑥2𝑦 185. log7

7𝑚2

(𝑢𝑣)2

Using 𝑙𝑜𝑔25 ≈ 2.3219 and 𝑙𝑜𝑔210 ≈ 3.3219, evaluate the following

186. log2 50 187. log2 100 188. log2 20

Using Properties of Logs, rewrite the expression as a single log.

189. log 𝑥 + log 𝑦 − log 𝑧 190. 1 − 3 log5 𝑚 191. 5 log 𝑘 − 3(log 𝑟 + log 𝑡)


192. log3 𝑥 + log3 4 = 5 193. log2 𝑥 + log2(𝑥 + 3) = 2

194. log3 𝑥 +1

2log3 4 =

1

4log3 16 195. log3 𝑥 + log3(𝑥 − 2) = log3 35

196. 2 log3 𝑥 − log3 4 = log3 16 197. log3(𝑥 + 3) + log3(𝑥 − 2) = log3 66

𝐥𝐨𝐠𝟒 𝒙 + 𝟑 𝐥𝐨𝐠𝟒 𝒚 + 𝟒 𝐥𝐨𝐠𝟒 𝒛

𝐥𝐨𝐠𝟔 𝒘 − 𝟐 𝐥𝐨𝐠𝟔 𝒙 − 𝐥𝐨𝐠𝟔 𝒚

𝟏 + 𝟐 𝐥𝐨𝐠𝟕 𝒎 − 𝟐(𝐥𝐨𝐠𝟕 𝒖 + 𝐥𝐨𝐠𝟕 𝒗)

𝟓. 𝟔𝟒𝟑𝟖 𝟔. 𝟔𝟒𝟑𝟖 𝟒. 𝟑𝟐𝟏𝟗

𝐥𝐨𝐠𝒙𝒚

𝒛 𝐥𝐨𝐠𝟓

𝟓

𝒎𝟑 𝐥𝐨𝐠𝒌𝟓

(𝒓𝒕)𝟑

𝒙 = 𝟔𝟎. 𝟕𝟓 𝒙 = 𝟏

𝒙 = 𝟏 𝒙 = 𝟕

𝒙 = 𝟖 𝒙 = 𝟖


Properties of Logs – Home Work

Using Properties of Logs, fully expand each expression

198. log3 3𝑥𝑦2𝑧5 199. log44𝑤

𝑥2 200. log88𝑚4

(𝑢2𝑣)3

Using 𝑙𝑜𝑔23 ≈ 1.5850 and 𝑙𝑜𝑔218 ≈ 4.1699, evaluate the following

201. 𝑙𝑜𝑔21

6 202. 𝑙𝑜𝑔236 203. 𝑙𝑜𝑔281

Using Properties of Logs, rewrite the expression as a single log.

204. 2 log 𝑥 + 3 log 𝑦 − 4 log 𝑧 205. 1 − 2 log4 𝑚 206. 5 log 𝑓 − 2 log 𝑔 − 6 log ℎ


207. 𝑙𝑜𝑔34𝑥 + 𝑙𝑜𝑔32 = 3 208. 𝑙𝑜𝑔2𝑥 + 𝑙𝑜𝑔2(𝑥 − 3) = 2

209. 𝑙𝑜𝑔3𝑥 − 𝑙𝑜𝑔34 = 𝑙𝑜𝑔316 210. 𝑙𝑜𝑔3𝑥2 + 𝑙𝑜𝑔3𝑥 = 𝑙𝑜𝑔327

211. 2𝑙𝑜𝑔3𝑥 − 𝑙𝑜𝑔39 = 𝑙𝑜𝑔325 212. 𝑙𝑜𝑔3(2𝑥 + 3) + 𝑙𝑜𝑔3(𝑥 − 2) = 𝑙𝑜𝑔372

𝟏 + 𝐥𝐨𝐠𝟑 𝒙 + 𝟐 𝐥𝐨𝐠𝟑 𝒚 + 𝟓 𝐥𝐨𝐠𝟑 𝒛

𝟏 + 𝐥𝐨𝐠𝟒 𝒘 − 𝟐 𝐥𝐨𝐠𝟒 𝒙

𝟏 + 𝟒 𝐥𝐨𝐠𝟖 𝒎 − 𝟑(𝟐 𝐥𝐨𝐠𝟖 𝒖 + 𝐥𝐨𝐠𝟖 𝒗)

−𝟐. 𝟓𝟖𝟒𝟗 𝟓. 𝟏𝟔𝟗𝟗 𝟔. 𝟑𝟒

𝐥𝐨𝐠𝒙𝟐𝒚𝟑

𝒛𝟒 𝐥𝐨𝐠𝟒𝟒

𝒎𝟐 𝐥𝐨𝐠𝒇𝟓

𝒈𝟐𝒉𝟔

𝒙 = 𝟑. 𝟑𝟕𝟓 𝒙 = 𝟒

𝒙 = 𝟔𝟒 𝒙 = 𝟑

𝒙 = 𝟏𝟓 𝒙 = 𝟔. 𝟓


Common Logs – Class Work

Solve for the variable.

213. 7𝑥 = 18 214. 3𝑥+4 = 27𝑥 215. 4b−2 = 8

216. 52𝑑−3 = 29 217. 7𝑛−2 = 3𝑛 218. 4𝑡+2 = 5𝑡−2

Find the approximate value for each

219. 𝑙𝑜𝑔36 220. log517 221. log637

Common Logs – Home Work

Solve for the variable.

222. 8𝑥 = 21 223. 64𝑥−1 = 42𝑥+5 224. 9b−6 = 42

225. 193𝑑−1 = 40 226. 25−𝑛 = 7𝑛 227. 18𝑡+1 = 32𝑡−1

Find the approximate value for each

228. 𝑙𝑜𝑔310 229. log520 230. log630

𝒙 =𝐥𝐨𝐠 𝟏𝟖

𝐥𝐨𝐠 𝟕≈ 𝟏. 𝟒𝟖𝟓 𝒙 = 𝟐 𝒃 =

𝐥𝐨𝐠 𝟖

𝐥𝐨𝐠 𝟒+ 𝟐 ≈ 𝟑. 𝟓

𝒅 =

𝐥𝐨𝐠 𝟐𝟗

𝐥𝐨𝐠 𝟓+𝟑

𝟐≈ 𝟐. 𝟓𝟒𝟔

𝒏 =𝟐 𝐥𝐨𝐠 𝟕

𝐥𝐨𝐠 𝟕−𝐥𝐨𝐠 𝟑≈ 𝟒. 𝟓𝟗𝟑 𝒕 =

𝟐 𝐥𝐨𝐠 𝟓+𝟐 𝐥𝐨𝐠 𝟒

𝐥𝐨𝐠 𝟓−𝐥𝐨𝐠 𝟒≈ 𝟐𝟔. 𝟖𝟓

𝟏. 𝟔𝟑𝟎𝟗 𝟏. 𝟕𝟔𝟎𝟒 𝟐. 𝟎𝟏𝟓𝟑

𝒙 =𝐥𝐨𝐠 𝟐𝟏

𝐥𝐨𝐠 𝟖≈ 𝟏. 𝟒𝟔𝟒 𝒙 = 𝟖 𝒃 =

𝐥𝐨𝐠 𝟒𝟐

𝐥𝐨𝐠 𝟗+ 𝟔 ≈ 𝟕. 𝟕𝟎𝟏

𝒅 =

𝐥𝐨𝐠 𝟒𝟎

𝐥𝐨𝐠 𝟏𝟗+𝟏

𝟑≈ 𝟎. 𝟕𝟓𝟏

𝒏 =𝟓 𝐥𝐨𝐠 𝟐

𝐥𝐨𝐠 𝟕+𝐥𝐨𝐠 𝟐≈ 𝟏. 𝟑𝟏𝟑 𝒕 =

𝐥𝐨𝐠 𝟑𝟐+𝐥𝐨𝐠 𝟏𝟖

𝐥𝐨𝐠 𝟑𝟐−𝐥𝐨𝐠 𝟏𝟖≈ 𝟏𝟏. 𝟎𝟒𝟕

𝟐. 𝟎𝟗𝟓𝟗 𝟏. 𝟖𝟔𝟏𝟒 𝟏. 𝟖𝟗𝟖𝟐


e and ln – Class Work


231. 𝑒ln 𝑥 = 6 232. 𝑒ln 𝑥 − 4 = 6 233. ln 𝑒𝑥+5 = 6𝑥

234. 3 ln 𝑒2𝑥 − 8 = 4 235. 𝑒2𝑥 = 7 236. 3𝑒(𝑥−1) + 9 = 10

237. ln(𝑥 + 1) = 7 238. ln(𝑥) + 1 = 7

e and ln – Homework


239. 𝑒ln 2𝑥 = 6 240. 5𝑒ln 𝑥 − 4 = 6 241. ln 𝑒2𝑥−5 = 6 + 𝑥

242. 4 ln 𝑒3𝑥 + 9 = 21 243. 𝑒3𝑥+1 = 6 244. 4𝑒(2𝑥+1) + 8 = 10

245. ln(𝑥 − 1) = 9 246. ln(𝑥) − 1 = 9

𝒙 = 𝟔 𝒙 = 𝟏𝟎 𝒙 = 𝟏

𝒙 = 𝟐 𝒙 =𝐥𝐧 𝟕

𝟐≈ 𝟎. 𝟗𝟕𝟑 𝒙 = 𝐥𝐧

𝟏

𝟑− 𝟏 ≈ −𝟐. 𝟎𝟗𝟗

𝒙 = 𝒆𝟕 − 𝟏 ≈ 𝟏𝟎𝟗𝟓. 𝟔𝟑𝟑 𝒙 = 𝒆𝟔 ≈ 𝟒𝟎𝟑. 𝟒𝟐𝟗

𝒙 = 𝟑 𝒙 = 𝟐 𝒙 = 𝟏𝟏

𝒙 = 𝟏 𝒙 =𝐥𝐧 𝟔−𝟏

𝟑≈ 𝟎. 𝟐𝟔𝟒 𝒙 =

𝐥𝐧𝟏

𝟐−𝟏

𝟐≈ −𝟎. 𝟖𝟒𝟕

𝒙 = 𝒆𝟗 + 𝟏 ≈ 𝟖𝟏𝟎𝟒. 𝟎𝟖𝟒 𝒙 = 𝒆𝟏𝟎 ≈ 𝟐𝟐𝟎𝟐𝟔. 𝟒𝟔𝟔


Growth and Decay – Class Work

Solve the following problems

247. $250 is deposited in an account earning 5% that compounds quarterly, what is the balance in the

account after 3 years?

248. A bacteria colony is growing at a continuous rate of 3% per day. If there were 5 grams to start, what is

the mass of the colony in 10 days?

249. A bacteria colony is growing at a continuous rate of 4% per day. How long till the colony doubles in

size?

250. If a car depreciates at an annual rate of 12% and you paid $30,000 for it, how much is it worth in 5

years?

251. An unknown isotope is measured to have 250 grams on day 1 and 175 grams on day 30. At what rate is

the isotope decaying? At what point will there be 100 grams left?

252. An antique watch made in 1752 was worth $180 in 1950; in 2000 it was worth $2200. If the watch’s

value is appreciating continuously, what would its value be in 2010?

253. A furniture store sells a $3000 living room and doesn’t require payment for 2 years. If interest is

charged at a 5% daily rate and no money is paid early, how much money is repaid at the end?

$𝟐𝟗𝟎. 𝟏𝟗

𝟔. 𝟕𝟓 𝒈𝒓𝒂𝒎𝒔

𝟏𝟕. 𝟑𝟑 𝒅𝒂𝒚𝒔

$𝟏𝟓, 𝟖𝟑𝟏. 𝟗𝟔

𝒓 = 𝟎. 𝟎𝟏𝟐 = 𝟏. 𝟐%

𝟕𝟕. 𝟎𝟕 𝒅𝒂𝒚𝒔

$𝟑, 𝟔𝟐𝟗. 𝟓𝟓

$𝟑, 𝟑𝟏𝟓. 𝟒𝟗


Growth and Decay – Homework

Solve the following problems

254. $50 is deposited in an account that earns 4% compounds monthly, what is the balance in the account

after 4 years?

255. A bacteria colony is growing at a continuous rate of 5% per day. If there were 7 grams to start, what is

the mass of the colony in 20 days?

256. A bacteria colony is growing at a continuous rate of 6% per day. How long till the colony doubles in

size?

257. If a car depreciates at an annual rate of 10% and you paid $20,000 for it, how much is it worth in 4

years?

258. An unknown isotope is measured to have 200 grams on day 1 and 150 grams on day 30. At what rate is

the isotope decaying? At what point will there be 50 grams left?

259. An antique watch made in 1752 was worth $280 in 1940; in 2000 it was worth $3200. If the watch’s

value is appreciating continuously, what would its value be in 2010?

260. A $9000 credit card bill isn’t paid one month, the credit company charges .5% continuously on unpaid

amounts. How much is owed 30 days later? (assume no other charges are made)

$𝟓𝟖. 𝟔𝟔

𝟏𝟗. 𝟎𝟐𝟖 𝒈𝒓𝒂𝒎𝒔

𝟏𝟏. 𝟓𝟓 𝒅𝒂𝒚𝒔

$𝟏𝟑, 𝟏𝟐𝟐

𝒓 = 𝟎. 𝟎𝟎𝟗𝟔 = 𝟎. 𝟗𝟔%

𝟏𝟒𝟒. 𝟓𝟕 𝒅𝒂𝒚𝒔

$𝟒, 𝟖𝟎𝟐. 𝟔𝟔

$𝟏𝟎, 𝟒𝟓𝟔. 𝟓𝟏


Logistic Growth – Class Work

Scientists measure a wolf population growing at a rate of 3% annually. They calculate the carrying capacity of

the region to be 100 members.

261. Write a logistic equation to model this situation.

262. Create a table that shows the pack population over the next 10 years if P1=30

263. Draw a graph of the equation

Logistic Growth – Homework

A calculus class determines that a rumor spreads around the school at a rate of 15% per hour. The school

population is 1600.

264. Write a logistic equation to model this situation.

265. Create a table that shows the number of people who know the rumor over the next 10 hours if the

class that starts it has 20 members

266. Draw a graph of the equation

𝑷𝒕+𝟏 = 𝑷𝒕 + 𝟎. 𝟎𝟑𝑷𝒕(𝟏 −𝑷𝒕

𝟏𝟎𝟎)

𝑷𝒕+𝟏 = 𝑷𝒕 + 𝟎. 𝟏𝟓𝑷𝒕(𝟏 −𝑷𝒕

𝟏𝟔𝟎𝟎)


Trig Functions – Class Work

Use the appropriate triangle to answer questions 267-274.

267. sin 𝜃 = 268. cos 𝜃 =

269. tan θ = 270. sin 𝛼 =

271. cos α = 272. tan α =

273. 𝜃 = 274. 𝛼 =

275. A right triangle has a hypotenuse of 7 and an angle of 40°, find the larger leg.

276. A right triangle has legs of 6 and 10 find the smaller acute angle.

277. A right triangle has an angle of 50° and a longer leg of 8, find the hypotenuse.

𝟏𝟖

𝟏𝟗.𝟓≈ 𝟎. 𝟗𝟐𝟑

𝟕.𝟓

𝟏𝟗.𝟓≈ 𝟎. 𝟑𝟖𝟓

𝟏𝟖

𝟕.𝟓≈ 𝟐. 𝟒

𝟏𝟎.𝟓

𝟏𝟕.𝟓≈ 𝟎. 𝟔

𝟏𝟒

𝟏𝟕.𝟓≈ 𝟎. 𝟖

𝟏𝟎.𝟓

𝟏𝟒≈ 𝟎. 𝟕𝟓

𝟔𝟕. 𝟑𝟖° 𝟑𝟔. 𝟖𝟕°

𝟓. 𝟑𝟔

𝟑𝟎. 𝟗𝟔𝟒°

𝟏𝟎. 𝟒𝟒


Trig Functions – Homework

Use the appropriate triangle to answer questions 278-285.

278. sin 𝜃 = 279. cos 𝜃 =

280. tan θ = 281. sin 𝛼 =

282. cos α = 283. tan α =

284. 𝜃 = 285. 𝛼 =

286. A right triangle has a hypotenuse of 9 and an angle of 60°, find the larger leg.

287. A right triangle has legs of 7 and 12 find the smaller acute angle.

288. A right triangle has an angle of 20° and a longer leg of 5, find the hypotenuse.

𝟖

𝟏𝟕≈ 𝟎. 𝟒𝟕𝟏

𝟏𝟓

𝟏𝟕≈ 𝟎. 𝟖𝟖𝟐

𝟖

𝟏𝟓≈ 𝟎. 𝟓𝟑𝟑

𝟏𝟔

𝟐𝟎≈ 𝟎. 𝟖

𝟏𝟐

𝟐𝟎≈ 𝟎. 𝟔

𝟏𝟔

𝟏𝟐≈ 𝟏. 𝟑𝟑𝟑

𝟐𝟖. 𝟎𝟕° 𝟓𝟑. 𝟏𝟑°

𝟕. 𝟕𝟗𝟒

𝟑𝟎. 𝟐𝟓𝟔°

𝟓. 𝟑𝟐𝟏


Converting Degrees and Radians – Class work

Convert the following degree measures to radians and radian measures to degrees.

289. 2𝜋

3 290. 35° 291. 225°

292. π

5 293. 150° 294.

14𝜋

9

295. 310° 296. 10π

7 297. 270°

Converting Degrees and Radians – Class work

Convert the following degree measures to radians and radian measures to degrees.

298. 5𝜋

3 299. 75° 300. 200°

301. 4π

5 302. 175° 303.

17𝜋

9

304. 350° 305. 9π

7 306.

11𝜋

18

𝟏𝟐𝟎°

𝟑𝟔°

𝟑𝟏𝝅

𝟏𝟖

𝟕𝝅

𝟑𝟔

𝟓𝝅

𝟔

𝟐𝟓𝟕. 𝟏𝟒°

𝟓𝝅

𝟒

𝟐𝟖𝟎°

𝟑𝝅

𝟐

𝟑𝟎𝟎° 𝟓𝝅

𝟏𝟐

𝟏𝟒𝟒° 𝟑𝟒𝟎°

𝟐𝟑𝟏. 𝟒𝟑° 𝟏𝟏𝟎°

𝟏𝟎𝝅

𝟗

𝟑𝟓𝝅

𝟑𝟔

𝟑𝟓𝝅

𝟏𝟖


Graphing Sin, Cos, and Tan – Class Work

305. What is the relationship between when a sine max/min value occurs and when a cosine zero occurs?

306. How do the zeros of the cosine and sine graphs relate to a tangent graph?

Consider the domain of (– 𝜋, 𝜋) for sin 𝜃 , cos 𝜃, 𝑎𝑛𝑑 tan 𝜃

307. Which function(s) is increasing on the entire interval?

308. Which function(s) has a relative max?

309. Which function(s) has a concave down interval followed by a concave up interval?

310. Which function(s) has an undefined value of x on the interval?

Graphing Sin, Cos, and Tan – Homework

311. What is the relationship between when a cosine max/min value occurs and when a sine zero occurs?

312. How does the tangent function’s concavity change between asymptotes?

Consider the domain of (– 𝜋, 𝜋) for sin 𝜃 , cos 𝜃, 𝑎𝑛𝑑 tan 𝜃

313. Which function(s) is decreasing on the entire interval?

314. Which function(s) has a zero?

315. Which function(s) has a concave up interval followed by a concave down interval?

316. Which function(s) has a relative minimum?

A sine max/min occurs at the exact same x-value as cosine zeros.

The sine zeros are where tangent also has zeros. The cosine zeros are where

tangent has vertical asymptotes.

tan 𝑥

sin 𝑥 , cos 𝑥

cos 𝑥 , tan 𝑥

tan 𝑥

A cosine max/min occurs at the exact same x-value as sine zeros.

Tangent is concave down, then changes to concave up between the vertical

asymptotes.

𝑛𝑜𝑛𝑒

sin 𝑥 , cos 𝑥 , tan 𝑥

sin 𝑥 , tan 𝑥

sin 𝑥 , cos 𝑥


Positive and Negative Integer Power Functions – Class Work

For each equation find lim𝑥→−∞

, lim𝑥→0−

𝑓(𝑥) , lim𝑥→0+

𝑓(𝑥) , 𝑎𝑛𝑑 lim𝑥→∞

𝑓(𝑥).

317. 𝑓(𝑥) = 4𝑥3 318. 𝑓(𝑥) = 5𝑥6

319. 𝑓(𝑥) = −1

3𝑥5 320. 𝑓(𝑥) = −𝑥2

321. 𝑓(𝑥) = 4𝑥−3 322. 𝑓(𝑥) = 5𝑥−6

323. 𝑓(𝑥) = −1

3𝑥−5 324. 𝑓(𝑥) = −𝑥−2

lim𝑥→−∞

𝑓(𝑥) = −∞

lim𝑥→0−

𝑓(𝑥) = 0

lim𝑥→0+

𝑓(𝑥) = 0

lim𝑥→∞

𝑓(𝑥) = ∞

lim𝑥→−∞

𝑓(𝑥) = ∞

lim𝑥→0−

𝑓(𝑥) = 0

lim𝑥→0+

𝑓(𝑥) = 0

lim𝑥→∞

𝑓(𝑥) = ∞

lim𝑥→−∞

𝑓(𝑥) = ∞

lim𝑥→0−

𝑓(𝑥) = 0

lim𝑥→0+

𝑓(𝑥) = 0

lim𝑥→∞

𝑓(𝑥) = −∞

lim𝑥→−∞

𝑓(𝑥) = −∞

lim𝑥→0−

𝑓(𝑥) = 0

lim𝑥→0+

𝑓(𝑥) = 0

lim𝑥→∞

𝑓(𝑥) = −∞

lim𝑥→−∞

𝑓(𝑥) = 0

lim𝑥→0−

𝑓(𝑥) = −∞

lim𝑥→0+

𝑓(𝑥) = ∞

lim𝑥→∞

𝑓(𝑥) = 0

lim𝑥→−∞

𝑓(𝑥) = 0

lim𝑥→0−

𝑓(𝑥) = ∞

lim𝑥→0+

𝑓(𝑥) = ∞

lim𝑥→∞

𝑓(𝑥) = 0

lim𝑥→−∞

𝑓(𝑥) = 0

lim𝑥→0−

𝑓(𝑥) = ∞

lim𝑥→0+

𝑓(𝑥) = −∞

lim𝑥→∞

𝑓(𝑥) = 0

lim𝑥→−∞

𝑓(𝑥) = 0

lim𝑥→0−

𝑓(𝑥) = −∞

lim𝑥→0+

𝑓(𝑥) = −∞

lim𝑥→∞

𝑓(𝑥) = 0


Positive and Negative Integer Power Functions – Homework

For each equation find lim𝑥→−∞

𝑓(𝑥) , lim𝑥→0−

𝑓(𝑥) , lim𝑥→0+

𝑓(𝑥) , 𝑎𝑛𝑑 lim𝑥→∞

𝑓(𝑥).

325. 𝑓(𝑥) = 3𝑥9 326. 𝑓(𝑥) = −2𝑥8

327. 𝑓(𝑥) = −1

4𝑥11 328. 𝑓(𝑥) = 6𝑥20

329. 𝑓(𝑥) = 3𝑥−9 330. 𝑓(𝑥) = −2𝑥−8

331. 𝑓(𝑥) = −1

4𝑥−11 332. 𝑓(𝑥) = 6𝑥−20

lim𝑥→−∞

𝑓(𝑥) = −∞

lim𝑥→0−

𝑓(𝑥) = 0

lim𝑥→0+

𝑓(𝑥) = 0

lim𝑥→∞

𝑓(𝑥) = ∞

lim𝑥→−∞

𝑓(𝑥) = −∞

lim𝑥→0−

𝑓(𝑥) = 0

lim𝑥→0+

𝑓(𝑥) = 0

lim𝑥→∞

𝑓(𝑥) = −∞

lim𝑥→−∞

𝑓(𝑥) = ∞

lim𝑥→0−

𝑓(𝑥) = 0

lim𝑥→0+

𝑓(𝑥) = 0

lim𝑥→∞

𝑓(𝑥) = −∞

lim𝑥→−∞

𝑓(𝑥) = ∞

lim𝑥→0−

𝑓(𝑥) = 0

lim𝑥→0+

𝑓(𝑥) = 0

lim𝑥→∞

𝑓(𝑥) = ∞

lim𝑥→−∞

𝑓(𝑥) = 0

lim𝑥→0−

𝑓(𝑥) = −∞

lim𝑥→0+

𝑓(𝑥) = ∞

lim𝑥→∞

𝑓(𝑥) = 0

lim𝑥→−∞

𝑓(𝑥) = 0

lim𝑥→0−

𝑓(𝑥) = −∞

lim𝑥→0+

𝑓(𝑥) = −∞

lim𝑥→∞

𝑓(𝑥) = 0

lim𝑥→−∞

𝑓(𝑥) = 0

lim𝑥→0−

𝑓(𝑥) = ∞

lim𝑥→0+

𝑓(𝑥) = −∞

lim𝑥→∞

𝑓(𝑥) = 0

lim𝑥→−∞

𝑓(𝑥) = 0

lim𝑥→0−

𝑓(𝑥) = ∞

lim𝑥→0+

𝑓(𝑥) = ∞

lim𝑥→∞

𝑓(𝑥) = 0


Rational Power Functions – Class Work

Choose which function(s) fits the description provided.

(A) 𝑓(𝑥) = 𝑥1

3⁄ (B) 𝑓(𝑥) = √𝑥4 (C) 𝑓(𝑥) = 𝑥1

6⁄ (D) 𝑓(𝑥) = √𝑥7

333. undefined for negative real values 334. lim𝑥→−∞

𝑓(𝑥) = −∞

335. closest to (10,1) 336. lim𝑥→0+

𝑓(𝑥) = 0

337. lim𝑥→0−

𝑓(𝑥) = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 338. closest to (10 , 3)

Rational Power Functions – Homework

Choose which function(s) fits the description provided.

(A) 𝑓(𝑥) = −𝑥1

3⁄ (B) 𝑓(𝑥) = −√𝑥4 (C) 𝑓(𝑥) = −𝑥1

6⁄ (D) 𝑓(𝑥) = −√𝑥7

339. Domain is (−∞, ∞) 340. lim𝑥→∞

𝑓(𝑥) = ∞

341. closest to (5,-1.5) 342. lim𝑥→0

𝑓(𝑥) = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

343. Range is (−∞, 0] 344. closest to (10 ,- 1.5)

B and C A and D

D A, B, C, and D

B and C A

A and D None

B B and C


Rational Functions – Class Work

For each of the following functions, name any discontinuities and tell whether they are non-removable or

removable. Graph each function.

345. ℎ(𝑥) =3𝑥2−4𝑥−4

𝑥2−4 346. 𝑔(𝑥) =

𝑥−5

𝑥2−25

347. 𝑗(𝑥) = x2−12x+36

x−6 348. 𝑚(𝑥) =

𝑥2−9

𝑥2−6𝑥+9

clipboard(67).galleryitem

349. lim𝑥→3−

𝑓(𝑥) = lim𝑥→3+

𝑓(𝑥) = 2; 𝑓(3) 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

B and C C

𝑥 = 2 removable

𝑥 = −2 non-removable

𝑥 = 5 removable


𝑥 = 6 removable

𝑥 = 3 non-removable

𝑥 = 3 removable

Graphs will vary.

Any graph that would go through

the point (3,2), but instead has a

hole at that point.


Rational Functions – Homework

For each of the following functions, name any discontinuities and tell whether they are non-removable or

removable. Graph each function.

350. ℎ(𝑥) =𝑥2−4𝑥+4

𝑥2−4 351. 𝑔(𝑥) =

𝑥−4

𝑥2−16

352. 𝑗(𝑥) = 2x2−8x−24

x−6 353. 𝑚(𝑥) =

𝑥2−49

𝑥2+14𝑥+49

354. lim𝑥→−5−

𝑓(𝑥) = lim𝑥→−5+

𝑓(𝑥) = 1; 𝑓(−5) 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

𝑥 = 2 removable


𝑥 = 4 removable


𝑥 = 6 removable


𝑥 = −5 removable

Graphs will vary.

Any graph that would go through

the point (−5,1), but instead has a

hole at that point.


Unit Review - Multiple Choice

1. Which equation has an x-intercept of (5,0) and a y-intercept of (0, −5

2)

a. 𝑦 +5

2= 5(𝑥 − 0)

b. 𝑦 −5

2= 5(𝑥 − 0)

c. 𝑦 =1

2(𝑥 − 5)

d. 𝑦 =1

2(𝑥 + 5)

2. The equation of a line perpendicular to 2𝑥 + 3𝑦 = 7 and containing (5,6) is

a. 3𝑥 − 2𝑦 = 3

b. 𝑦 − 6 = −2

3(𝑥 − 5)

c. 3𝑥 − 2𝑦 = 4

d. 𝑦 =2

3(𝑥 − 6)

3. A line with no slope and containing (3,8) has equation

a. 𝑦 = 3

b. 𝑦 = 8

c. 𝑥 = 3

d. 𝑥 = 8

4. The vertex of 𝑦 = −2|𝑥 − 1| + 7 is

a. (2,7)

b. (−2,7)

c. (−1,7)

d. (1,7)

5. 4[−3(2.7) + 0.5] =

a. −40

b. −36

c. −32

d. −28

6. The equation that models exponential decay passing through (0,5) and a lim𝑥→∞

𝑓(𝑥) = 4 is

a. 𝑓(𝑥) = 5𝑒𝑥 + 4

b. 𝑓(𝑥) = −1𝑒𝑥 + 4

c. 𝑓(𝑥) = 5𝑒−𝑥 + 4

d. 𝑓(𝑥) = −1𝑒−𝑥 + 4

7. A forest fire spreads continuously, burning 10% more acres every hour. How long will it take for 1000

acres to be on fire after 200 acres are burning?

a. 23.026 hours

b. 16.094 hours

c. 6.932 hours

C

A

B

D

C

D

B


d. not enough information

8. log6 5 =

a. .116

b. .898

c. 1.113

d. 1.308

9. Given 4x = 10, find 𝑥

a. 2.5

b. .602

c. .400

d. 1.661

10. log 𝑚 = .345 𝑎𝑛𝑑 log 𝑛 = 1.223, 𝑓𝑖𝑛𝑑 log10𝑚2

𝑛3

a. -1.979

b. .651

c. 6.507

d. 8.473

11. Which of the following would not influence the carrying capacity of a logistic growth model

a. the population of a town

b. the food supply in an ecological preserve

c. the rate of spread of the flu

d. the area inside a Petri dish

12. The larger leg of a right triangle is 6 and the smallest angle is 20°, what is the hypotenuse?

a. 6.385

b. 5.638

c. 17.543

d. 14.703

13. How many degrees is 4π

9?

a. 160°

b. 110°

c. 80°

d. 62°

14. An example of a function that is concave down and then concave up is

a. f(x) = sin 𝑥 𝑜𝑛 (0, 𝜋)

b. f(x) = tan 𝑥 𝑜𝑛 (0, 𝜋)

c. f(x) = cos 𝑥 𝑜𝑛 (0, 𝜋)

d. f(x) = sin 𝑥 𝑜𝑛 (−𝜋, 𝜋)

15. The function that has lim𝑥→0−

𝑓(𝑥) = ∞ 𝑎𝑛𝑑 lim𝑥→0+

𝑓(𝑥) = −∞ is

a. 𝑓(𝑥) = −3𝑥3

B

D

A

C

A

C

C


b. 𝑓(𝑥) = −4𝑥4

c. 𝑓(𝑥) = −3𝑥−3

d. 𝑓(𝑥) = −4𝑥−4

16. The function ℎ(𝑥) =4𝑥2−3𝑥−1

4𝑥2−1 has the following discontinuities

a. non-removable discontinuity at 𝑥 = ±1

2

b. removable discontinuity at 𝑥 = ±1

2

c. non-removable discontinuity at 𝑥 =1

2; removable discontinuity at 𝑥 = −

1

2

d. non-removable discontinuity at 𝑥 = −1

2; removable discontinuity at 𝑥 =

1

2

Extended Response

1. Consider the function ℎ(𝑥) =𝑓(𝑥)

𝑔(𝑥), if 𝑓(𝑥) = 𝑥2 − 2𝑥 − 8 and 𝑔(𝑥) = 𝑥2 + 8𝑥 + 15

a. Use limit notation to describe the end behavior of ℎ(𝑥)

b. Name any discontinuities and whether they are removable or non-removable.

c. 𝑗(𝑥) = ℎ(𝑥) ∙ 𝑚(𝑥) and has a removable discontinuity at 𝑥 = −3,what is 𝑚(𝑥)?

2. Entomologists introduce 20 of one variety of insect to a region and determine that the population

doubles every 6 hours.

a. Write an equation to model this situation.

b. What will the population be in 10 hours?

C

A

lim𝑥→−∞

ℎ(𝑥) = +1

lim𝑥→∞

ℎ(𝑥) = +1

𝑥 = −5, −3

𝑏𝑜𝑡ℎ 𝑛𝑜𝑛 − 𝑟𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒

𝑚(𝑥) is any polynomial with 𝑥 + 3 as a factor, or any rational

function with 𝑥 + 3 as a factor of the numerator

𝐴 = 20𝑒0.1155𝑡

63.48 ℎ𝑜𝑢𝑟𝑠


c. If those same scientists determine that the region can support a maximum of 100,000 of the

species, rewrite your equation from part a.

3. A compostable bag breaks down such that only 10% remains in 6 months.

a. If the decomposition is continual, at what rate is the bag decomposing?

b. How much of the bag remained after 4 months?

c. When will there be less than 1% of the bag remaining?

4. A 15’ ladder is rated to have no more than a 70° angle and no less than a 40° angle.

a. What is maximum rated distance the ladder can be placed from the wall?

b. How high up a wall can the ladder reach and be within the acceptable use limits?

c. At what base angle should the ladder be placed to reach 10’ up the wall?

𝑃𝑡+1 = 𝑃𝑡 + 0.1155𝑃𝑡(1 −𝑃𝑡

100000)

𝑟 = 38.4% 𝑑𝑒𝑐𝑎𝑦 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ

21.5% 𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔

𝑎𝑓𝑡𝑒𝑟 12 𝑚𝑜𝑛𝑡ℎ𝑠

11.5 𝑓𝑒𝑒𝑡

14.1 𝑓𝑒𝑒𝑡

41.8°

slope & y-intercept class work -...

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