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Graphing Absolute Value Equations

This comprehensive guide explores absolute value equations, focusing on their V-shaped graphs and how to translate these graphs vertically, horizontally, and diagonally. It provides step-by-step instructions for graphing absolute value equations using a calculator and examples of writing equations based on specified translations. Readers will learn how to identify vertices and describe translations in terms of direction and magnitude. This resource is ideal for students looking to enhance their understanding of absolute value functions and their graphical representations.

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Graphing Absolute Value Equations

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  1. Graphing Absolute Value Equations

  2. Absolute Value Equation A V-shaped graph that points upward or downward is the graph of an absolute value equation. Translation A translation is a shift of a graph horizontally, vertically or diagonally (combination of vertical and horizontal translation).

  3. Graphing Absolute Value Equations Vertical Translation The graph of y = │x │ + k is a translation of y = │x │ . if k is positive  up k units if k is negative  down k units Horizontal Translation The graph of y = │x – h│ is a translation of y = │x │ . if h is positive  right h units if h is negative  left h units

  4. Graphing Absolute Value Equations Diagonal Translation A combination of vertical and horizontal translation. The graph of y = │x – h│ + k is a translation of y = │x │.

  5. Graphing Absolute Value Equations Example 1: Graph the absolute value equation y = │x │- 5. Solution: Step 1. Press y = key on you calculator. Step 2. Use the NUM feature of MATH screen on your graphing calculator. Step 3. Choose 1: abs( feature on you calculator. Step 4. Enter the given equation. X,Τ,θ,n , ) , - , 5 Step 5. Press GRAPH.

  6. Cont(example 1)…

  7. Do these… Graph each function. Identify the vertex of each function. • y = │x │+ 2 • y = │x - 4 │ • y = │x - 6│-2 • y = │x - 2│+ 1.

  8. Answers: 1. 2. 3. 4.

  9. Writing an Absolute Value Equation Write an equation for each translation of y = │x│ Example 1: 9 units up 9 units up is vertical translation so use y = │x│ + k Since k is positive, the equation is y = │x│ + 9 Example 2: 2 units down 2 units down is vertical translation so use y = │x│ + k Since k is negative, the equation is y = │x│ - 2

  10. Writing an Absolute Value Equation Write an equation for each translation of y = -│x│. Example 3: 5 units right 5 units right is horizontal translation so use y = -│x - h│ Since k is positive, the equation is y = -│x - 5│ Example 4: 3 units left 3 units left is horizontal translation so use y = -│x -h│ Since k is negative, the equation is y = -│x – (-3)│  y = -│x +3│

  11. Writing an Absolute Value Equation Write an equation for each translation of y = │x│. Example 5: 2 units up and 1 unit left since this involves horizontal and vertical translations use y = │x - h│+ k Since k is positive and h is negative, the equation is y = │x – (-1)│+ 2  y = │x +1│+ 2 Example 6: 5 units down and 4 units left since this involves horizontal and vertical translations use y = │x - h│+ k Since k is negative and h is negative, the equation is y = │x – (-4)│+ (-5)  y = │x +4│ - 5

  12. Do these… Write an equation for each translation of the parent function y = x. • Left 9 units • Right 2 unit • Up 1 unit • Down 2/3 unit • Left 3 units and down 4 units • Right 5 units and up 1 unit Answers: 1. y = │x + 9│ 2. y = │x - 2│ 3. y = │x │ + 1 4. y = │x │ - 2/3 5. y = │x +3│- 4 6. y = │x - 5│ + 1

  13. Describing a Translation Graph each absolute value equation then describe the translation of the parent function. Example 1. y = x - 7 + 2 Answer: y = xis translated 7 units to the right and 2 units up.

  14. Describing a Translation Graph each absolute value equation then describe the translation of the parent function. Example 2. y = -x + 3 - 1 Answer: y = -xis translated 3 units to the left and 1 unit down.

  15. Do these… Graph each absolute value equation then describe the translation of the parent function. • y = x + 1- 3 • Y = x – 3 - 10 • Y = x + 2+ 1 • Y = -x - 1 - 6 • Y = -x - 5+ 7

  16. Write the equation of the given graph. Answer: Since the vertex is at the point (5, 3), then h= 5 and k = 3. Therefore, the equation is y = │x – 5│+ 3.

  17. Describe the translation from y = x+1- 2 to y = x - 3 + 4 Answer: 6 units up, 4 units right

  18. Describe the translation from y = x – 3+ 1 to y = x - 1 - 2 Answer: 3 units down, 2 units left

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