Vertical Stretches and Compressions

Pre-Calculus · Axiom Academy

LESSON Vertical Stretches and Compressions Multiply a function by a constant and every point slides straight up or down — the graph stretches taller or squashes shorter. 1. Multiplying by a Scales the Height Take any function — here f(x) = x^2 — and build a new one, . The input x is untouched, so each point keeps its horizontal position. What changes is the height : every y -value is multiplied by a . Watch a ramp from 1 up to 2 and each point pull straight up, away from the x -axis. Every height is multiplied by the same factor a 2. A Fraction Squashes It Down What if a sits between 0 and 1 ? Then multiplying shrinks every height, so each point slides toward the x -axis. Watch a ease from 1 down to : the parabola flattens to half its height. The grid below tracks the exact heights — every f(x) value is cut in half. Heights grow; points move away from the x -axis. The graph looks taller and narrower. Heights shrink; points move toward the x -axis. The graph looks shorter and wider. When a = 1 nothing moves — . So |a| = 1 is the dividing line: cross above it to stretch, drop below it to compress. A negative factor does two jobs at once. The size of a still scales the height, but the minus sign sends each point through the x -axis to the other side. Watch a travel from 1 down through 0 to -1 : every point crosses the axis, landing as a mirror image. The one point that never moves is the spot already on the axis.

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