Parent Functions Catalog

Algebra 2 · Axiom Academy

LESSON Parent Functions Catalog Master the fundamental function families and their transformations — the building blocks of every algebraic function you will meet. The linear parent is the simplest function there is: the output is the input. Its defining behaviour is a constant rate of change — take one step across and you always climb the same amount, no matter where you stand. Every line is this parent, transformed Constant rate of change; the graph is a straight line through the origin with slope 1 . Squaring destroys the sign of the input, so x and -x produce the same output. That single fact is the whole shape: the parabola is a mirror image across the y -axis , and its lowest point sits where the two arms meet. Vertex form — every parabola is this parent, transformed U-shaped parabola; vertex at the origin; axis of symmetry at x = 0 ; minimum value of 0 . Absolute value answers one question: how far is x from zero? A distance is never negative, so the graph can never dip below the axis — it folds up into a V whose corner sits exactly where the distance is 0 . Every V is this parent, transformed V-shaped graph; vertex at the origin; symmetric about the y -axis; slope of 1 for and -1 for . The square root undoes squaring — but only on the half of the parabola where squaring is reversible. Reflecting y = x^2 (for ) across the line y = x swaps inputs and outputs , and because a square is never negative, the new inputs can never be negative either.

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