2-7 Absolute Value Functions and Graphs

Quick Review

The absolute value function y = | x | y equals vertical line x vertical line  is the parent function for the family of functions of the form y = a | y equals eh vertical line   x − h | + k . x minus h vertical line plus k .  The maximum or minimum point of the graph is the vertex of the graph.

y = 2 | x + 3 | + 1 a = 2 , h = − 3 , k = 1 y equals 2 vertical line x plus 3 vertical line plus 1 eh equals 2 comma h equals negative 3 comma k equals 1

• Vertex is at ( − 3 , 1 ) negative 3 comma 1 close
• Translated left 3 units
• Stretched by a factor of 2
• Translated up 1 unit

Example

Write an equation for the translation of the graph y = | x | y equals vertical line x vertical line  up 5 units.

Because the graph is translated up, k is positive, so the equation of the translated graph is y = | x | + 5 . y equals vertical line x vertical line plus 5 .

Exercises

Write an equation for each translation of the graph of y = | x | . y equals vertical line x vertical line .

40. up 4 units, right 2 units
41. vertex ( − 3 , 0 ) negative 3 comma 0 close
42. vertex (5, 2)
43. vertex (4, 1)

Graph each function.

44. f ( x ) = | x | − 8 f open x close equals vertical line x vertical line negative 8
45. f ( x ) = 2 | x − 5 | f open x close equals 2 vertical line x minus 5 vertical line
46. y = − 1 4 | x − 2 | + 3 y equals negative , 1 fourth , vertical line x minus 2 vertical line plus 3
47. y = − 2 | x + 1 | − 1 y equals negative 2 vertical line x plus 1 vertical line negative 1

Without graphing, identify the vertex and axis of symmetry of each function.

48. y = 2 | x − 4 | y equals 2 vertical line x minus 4 vertical line
49. y = − | x | + 2 y equals negative vertical line x vertical line plus 2

2-8 Two Variable Inequalities

Quick Review

An inequality describes a region of the coordinate plane that has a boundary. To graph an inequality involving two variables, first graph the boundary. Then determine which side of the boundary contains the solutions. Points on a dashed boundary are not solutions. Points on a solid boundary are solutions.

Example

Graph the inequality y ≥ 2 x + 3 . y greater than or equal to 2 x plus 3 .

Graph the solid boundary line y = 2 x + 3 . y equals 2 x plus 3 .

Since y is greater than 2 x + 3 , 2 x plus 3 comma  shade above the boundary.

Exercises

Graph each inequality.

50. y ≥ − 2 y greater than or equal to , minus 2
51. y < 3 x + 1 y less than , 3 x plus 1
52. y < − | x − 5 | y less than , minus vertical line x minus 5 vertical line
53. y > | 2 x + 1 | y greater than , vertical line 2 x plus 1 vertical line
54. Transportation An air cargo plane can transport as many as 15 regular shipping containers. One super-size container takes up the space of 3 regular containers.
    1. Write an inequality to model the number of regular and super size containers the plane can transport.
    2. Describe the domain and range.
    3. Graph the inequality you wrote in part (a).
55. Open-Ended Write an absolute value inequality with a solid boundary that only has solutions below the x-axis.

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