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Graphing Absolute Value Equations

Learn how to graph and translate absolute value equations, including vertical, horizontal, and diagonal translations. Practice examples provided.

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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 h 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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