Absolute Value Transformations

Pre-Calculus · Axiom Academy

LESSON Absolute Value Transformations Every absolute-value graph is the same V — moved, stretched, and flipped by three numbers in . 1. The V Is Just Distance From Zero Pick any input x . Its absolute value is how far that point sits from the origin — always a non-negative number. Plot that distance as the height above each x , and the points sweep out a V whose corner (the vertex ) rests at (0, 0) . The output is the distance from 0 — never negative 2. Three Knobs: Slide, Stretch, Flip The general form hides three controls. The pair (h, k) is the vertex — move it anywhere. The number a sets how steep the arms are and which way the V opens. Watch the parent V become . The vertex slides to x = h . Note the minus sign: |x - 1| moves the corner right to x = 1 . The whole V lifts (or drops) so the vertex sits at y = k . Here k = 3 raises the corner to height 3 . Each arm has slope . makes a steeper, narrower V; makes a wider one. opens upward; reflects it to open downward , an inverted V. Here a = -2 . For : the vertex is (1, 3) , it opens downward because , and the arms fall with slope . Setting g(x) = 0 gives the x-intercepts and — the two places the inverted V crosses the axis. 3. Wrapping a Whole Function: y = |f(x)| You can take the absolute value of any function, not just x . The rule y = |f(x)| keeps every part of the graph that is already at or above the x-axis, and folds the part below the axis straight up — like reflecting it in a mirror lying along the x-axis.

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