Absolute value functions have a V-shape and model situations involving distance or edges. The graph can be transformed by changing the slope (a), shifting the vertex horizontally (h), or shifting the vertex vertically (k). To graph, identify the vertex and axis of symmetry, then use the slope to sketch the right side and symmetry to complete the left. Writing the equation involves identifying the vertex (h, k) and slope (a).

1. 2.8 Absolute Value Functions

2. Absolute Value Functions
• The absolute value of x is defined by:
• The graph of y = |x| looks like a v-shape.
vertex

3. Why are they important?
• Have you ever played pool or putt-putt
golf?
• The path of the ball when making a bank
shot is an example of an absolute value
function.

4. Transformations
• There are four ways the absolute value graph
can be changed:
1. Open Up or Open Down
2. Change in Width – sides can be steeper or
less steep
3. Horizontal Shift – vertex moves left or right
4. Vertical Shift- vertex moves up or down

5. General Form
• y = a |x – h| + k
• Effects of a:
• When a > 0 (positive), the V opens up.
• When a < 0 (negative), the V opens down.
• When |a| < 1, the sides are less steep than
y = |x|.
• When |a| > 1, the sides are steeper than
y = |x|.

6. Examples
Notice:
a is the slope of the right side of the graph!

7. Effect of h
y = a |x – h| + k
• h shifts the vertex left or right
• The direction is opposite the sign of h
• Examples:

8. Effects of k
y = a |x – h| + k
• k shifts the vertex up or down
• Positive k shifts up
• Negative k shifts down
• Examples:

9. The Vertex
• The vertex will be at (h, k).
• Example:
• Vertex: (-3, -4)
• Axis of symmetry:
Vertical line through
the vertex
(-3, -4)

10. Your Turn!
• Find the vertex:

11. Graphing Absolute Value Functions
• Plot the vertex.
• Sketch the axis of symmetry.
• Use a (the slope) to graph the right side.
• Use symmetry to draw in the left side.
• Example:
Graph

12. Example:
• Graph

13. Your Turn!
• Graph

14. Example:
• Graph

15. Your Turn!
• Graph

16. Writing Absolute Value Functions
• y = a| x – h | + k
• Find the vertex (h, k)
• Count the slope (of the right side) to find a.

17. Example:
• Write an equation of the graph shown.

18. Your Turn!
• Write an equation of the graph shown.