Overview
The lecture reviews absolute value functions, covering their properties, graphing, solving equations, and key features for pre-calculus students.
Prerequisites & Properties of Absolute Value
- Absolute value, denoted |x|, represents the distance from zero; it is always non-negative.
- The absolute value of negative numbers equals the absolute value of their positive counterparts (e.g., |−2x| = |2x|).
- To solve |A| = B (with B ≥ 0), set A = B or A = −B.
- |A| = negative value has no solution; |A| = 0 only if A = 0.
General Form & Graphing
- The general form: f(x) = a|x − h| + k, where (h, k) is the vertex.
- The basic graph V-shape opens upwards for a > 0, downwards for a < 0.
- Only translations and vertical stretches/compressions apply; no horizontal flips or compressions are needed.
Graph Features & Intercepts
- The vertex is at (h, k); determine by reading from the formula.
- Number of x-intercepts depends on values of a and k: two, one, or none.
- The y-intercept is found by evaluating f(0).
Domain and Range
- Domain is all real numbers (−∞, ∞).
- Range: For a > 0, y ≥ k; for a < 0, y ≤ k.
Additional Properties
- No asymptotes; the function is continuous everywhere.
- Increasing/decreasing: For a > 0, decreases left of vertex, increases right; for a < 0, vice versa.
- Not one-to-one but each linear branch is one-to-one and invertible.
Symmetry & Piecewise Representation
- Axis of symmetry: vertical line x = h.
- When h = 0, the function is even (symmetric about y-axis).
- Absolute value can be written as a piecewise function:
- f(x) = x for x ≥ 0; f(x) = −x for x < 0.
Applications
- Absolute value functions represent distances (e.g., |x − 3| is the distance from x to 3).
- Useful for practicing linear transformations due to simple graph.
Key Terms & Definitions
- Absolute Value (|x|) — the distance from x to 0 on the number line, always non-negative.
- Vertex — the point (h, k) where the graph changes direction.
- Axis of Symmetry — vertical line x = h dividing the graph into two symmetric halves.
- Piecewise Function — a function defined by different expressions on different intervals.
- Even Function — a function symmetric about the y-axis (f(−x) = f(x)).
Action Items / Next Steps
- Practice solving absolute value equations.
- Review graphing absolute value functions using transformations.
- Study how to write and interpret absolute value functions as piecewise functions.