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.