Absolute Value as Piecewise

Algebra 2 · Axiom Academy

LESSON Absolute Value as Piecewise Discover how absolute value functions are really piecewise functions in disguise — and learn to write any absolute value expression as a piecewise definition. The absolute value of a number is its distance from zero on the number line. Distance is always non-negative! |5| = 5 (5 is 5 units from zero) |−3| = 3 (−3 is 3 units from zero) |0| = 0 (0 is 0 units from zero) Here's the key insight: absolute value behaves differently for positive and negative numbers. We can write this as a piecewise function: The number is already positive (or zero), so |x| = x The number is negative, so we flip its sign: |x| = −x When we graph y = |x| using its piecewise definition, we see why absolute value functions create that characteristic "V" shape: The two pieces meet at the vertex (the point of the V), which is at the origin (0, 0) for y = |x|. What happens with |x − 3|? The piecewise form reveals how the vertex moves! Key Question: When is x − 3 positive or negative? • If x ≥ 3, then x − 3 ≥ 0, so |x − 3| = x − 3 • If x < 3, then x − 3 < 0, so |x − 3| = −(x − 3) = −x + 3 |x + 2| = |x − (−2)| has vertex at x = −2 5. Complex Expressions: |2x − 3| + 1 Let's write a more complex absolute value expression as a piecewise function. The process is the same: find when the inside is positive or negative! Example: Write y = |2x − 3| + 1 as a piecewise function Step 1: When is 2x − 3 positive? When x = 2: Use top piece → y = (2x − 3) + 1 = 4 − 3 + 1 = 2

This is the written version of the interactive lesson above. See the full Algebra 2 course.