2.8 Absolute Value Functions
- 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
- 14. Example: • 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.