Absolute Value Functions• The absolute value of x is defined by:• The graph of y = |x| looks like a v-shape. vertex
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.
Transformations• There are four ways the absolute value graph can be changed:1. Open Up or Open Down2. Change in Width – sides can be steeper or less steep3. Horizontal Shift – vertex moves left or right4. Vertical Shift- vertex moves up or down
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|.
ExamplesNotice:a is the slope of the right side of the graph!
Effect of h y = a |x – h| + k• h shifts the vertex left or right• The direction is opposite the sign of h• Examples:
Effects of k y = a |x – h| + k• k shifts the vertex up or down• Positive k shifts up• Negative k shifts down• Examples:
The Vertex• The vertex will be at (h, k).• Example:• Vertex: (-3, -4)• Axis of symmetry: Vertical line through the vertex (-3, -4)