Graphing Absolute Value Functions Using Transformations
Parent Function of Absolute Value
- The basic graph of ( |x| ) looks like a V pointing upward.
- Center: (0,0)
- Key Points:
- (|0| = 0)
- (|1| = 1)
- (|-1| = 1)
- (|-2| = 2)
Reflecting Over Axes
- Negative Outside: Reflects over the x-axis, opens downward.
- Negative Inside: Still opens upward, as (|-x| = |x|).
Domain and Range
- Domain: All real numbers, ((-\infty, \infty))
- Range:
- Upward opening: ([0, \infty))
- Downward opening: ((-\infty, 0])
Graphing Transformations
Horizontal and Vertical Shifts
- Leftward Shift: Function moves left.
- Rightward Shift: Function moves right.
- Upward Shift: Function moves up.
- Downward Shift: Function moves down.
Using Slope
- Positive Slope: Opens upwards.
- Negative Slope: Opens downwards.
- Slope affects steepness:
- (y = 2|x|) is steeper than (y = |x|).
Example Graphs
- Example 1: (|x+3|)
- Shifts 3 units to the left
- Opens upward
- Example 2: (|x-2|+1)
- Shifts 2 units right, 1 unit up
- Opens upward
Range Details
- Upward Graph: Lowest Y is minimum point’s Y value; highest is infinity.
- Downward Graph: Lowest is negative infinity; highest is maximum point’s Y value.
Multiple Transformations
- Example: (y = 4 - |x+1|)
- Shifts left 1 unit, up 4 units
- Opens downward due to negative sign
Sketching Using Points
- Method:
- Find vertex by setting inside of absolute value to zero.
- Calculate Y for points around the vertex.
- Plot and reflect across vertex for symmetry.
Additional Concepts
- Vertical Stretch/Compression: Multiplier in front of (|x|) changes steepness.
- Horizontal Shift: Given by value inside absolute value.
- Sign Impact: Positive opens upward, negative opens downward.
Practice Example
- (y = 5 - 3|x - 1|)
- Vertex: (1, 5)
- Slope of -3: steep and downward
- Plot points: Shift right 1, up 5, downward slope
These notes summarize the key points for graphing absolute value functions using transformations, focus on understanding shifts, reflections, and the impact of slopes.