Graphing Absolute Value Equations

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).

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

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 │.

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.

Cont (example 1)…

Do these… Graph each function. Identify the vertex of each function. 1. y = │x │ y = │x - 4 │ 3. y = │x - 6│-2 4. y = │x - 2│+ 1.

Answers:

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

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│

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

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 │ y = │x │ - 2/3 5. y = │x +3│ y = │x - 5│ + 1

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. 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.

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

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.

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

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