Absolute Value Inequality Graphs in Two Variables

Videos, worksheets, games and activities to help Algebra 1 students learn how to graph absolute value inequality in two variables on a coordinate plane.

Absolute Value Inequality Graphs in Two Variables
With absolute value graphing, if the inequality is similar to the equation of a line, (for example y > m|x| + b), then we get a V shape, and we shade above or below the V. This is very similar to graphing inequalities with two variables. The difference is that we are graphing inequalities with absolute values which makes the V-shape.

How to make a table of values for absolute value functions so you can graph them?
Graphs of Absolute Value Equations: An Application (Algebra I)
This video provides an explanation of the concept of the graphs of absolute value equations.
Examples:
Graph

1. y = |x - 1|
   
2. y = |x + 5|

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How to solve and graph absolute value inequalities with x and y using a table?
Graphs of Absolute Value Equations: An Explanation
This video demonstrates a sample use of the graphs of absolute value equations.
Example:
x = |y - 2|

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Graphing Linear & Absolute Value Inequalities
The following video examines how to graph linear and absolute value inequalities.
Examples:
Graph each function.
4. y ≥ x - 5
5. y > |x| - 2
6. y ≥ -|3x|

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How to use vertical and horizontal translations to graph absolute value functions?
A family of functions is a group of functions with common characteristics. A parent function is the simplest function with these characteristics.
A translation is a transformation that slides a graph or figure.
Examples:

1. Graph the 3 functions on the same graph, then describe the differences and similarities.
   a) y = |x|
   b) y = |x| + 3
   c) y = |x| - 3
2. Graph the 3 functions on the same graph, then describe the differences and similarities.
   a) y = |x|
   b) y = |x + 2|
   c) y = |x - 2|
3. Graph the equation: use a translation of the parent function.
   y = |x - 3| + 2
4. Write the equation to translate the function 4 units down.
   y = |3x|
5. Write the equation to translate the function 2 units up.
   y = |x + 1|
6. Write the equation to translate the function 3 units to the left.
   y = |-3x|
7. Write the equation to translate the function 2 units to the right.
   y = |2x + 3|

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How to use stretch and shrink transformations to graph absolute value functions?
A vertical stretch of a function f by the factor a, a > 1, is a transformation of f that multiplies all y-values by a.
A vertical shrink of a function f by the factor a, 0 < a < 1, is a transformation of f that multiplies all y-values by a.
A reflection or flip, is a transformation that maps a point in the plane to its mirror image, using a specific line as a mirror.
Examples:

1. Graph the 3 functions on the same graph, then describe the differences and similarities.
   a) y = |x|
   b) y = 3|x|
   c) y = 1/3|x|
   
2. Describe and then graph the function y = 2|x|
   
3. Describe and then graph the function y = -|x + 2|
   
4. Describe and then graph the function y = -|x - 4|

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How to graph absolute value inequalities in a plane using the general formula?
y = a|x - h| + k where the vertex is (h, k)
Examples:

1. Graph each absolute inequality
   a) y ≥ |x + 2| - 3
   b) y ≤ |2x| - 1
   c) y + |x - 3| < 3
   
2. Write an inequality for each boundary.

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Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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