Horizontal Stretches and Compressions

Pre-Calculus · Axiom Academy

LESSON Horizontal Stretches and Compressions Why multiplying the input by b scales every x -coordinate by 1/b — the backwards rule that makes f(2x) a squeeze, not a stretch. Take any point on the original graph. On y = f(bx) , the input only equals a — and so the output only equals f(a) — when bx = a , that is at x = a/b . Every output keeps its height but slides to 1/b of its old x . Same height, now reached at x = a/b 2. When b > 1 : a Squeeze Toward the y -Axis If b > 1 , then a/b < a , so every point moves closer to the y -axis — a horizontal compression by a factor of 1/b . Watch become : each crest and zero slides to half its old x , and the period drops from to . A feature of at x = a — say the crest at . The same feature of now at x = a/2 — the crest at . Halved: . The wave oscillates twice as fast. Unchanged: the wave still rises to +1 and falls to -1 . The bigger number squeezes the graph. To hit a target output you now need a smaller input, so the whole picture pulls inward — exactly opposite to a vertical stretch, where a bigger factor pulls the graph taller. 3. When 0 < b < 1 : a Stretch Away From the y -Axis If b is a fraction between 0 and 1 , then a/b > a , so every point moves farther from the y -axis — a horizontal stretch by a factor of 1/b . Watch y = x^2 become : the point at x = 2 slides out to x = 4 , and the parabola opens twice as wide. Every x -coordinate is divided by b , so the graph squeezes by 1/b .

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