The Square Root Function

Algebra 1 · Axiom Academy

Meet y= : it starts at the origin, climbs but ever more gently, and is just the right half of y=x^2 seen in a mirror. A curve that starts at zero and never stops climbing — slowly Squaring is everywhere — areas, distances, energy. The square root runs squaring backwards : it asks "what number, squared, gives me this?" Watch that answer get drawn as a curve, and a whole new function appears. Watch the pen leave the origin and draw y= from left to right. Every time it reaches a perfect square — 1, 4, 9 — it lands on a point with a whole-number height: (1,1) , (4,2) , (9,3) . Notice it never goes left of 0 , and it rises but flattens as it climbs. Each whole-number height marks a perfect square underneath it: √1 = 1, √4 = 2, √9 = 3. Drive the point along the curve Drag the handle to slide a point along y= . Read off x and its square root y= as you go. Two things to feel: the curve has no left half (you can't go below x=0 ), and each step to the right lifts y by less than the last — the rise gets ever gentler. Domain: x 0 . Range: y 0 . Both start at the origin (0,0) . Where comes from: the mirror of x^2 Squaring and square-rooting undo each other, so their graphs are mirror images across the line y=x . Drag the slider to fold the right half of y=x^2 across that line — it lands exactly on y= . Watch the points swap: (2,4) becomes (4,2) , and (3,9) becomes (9,3) . Swap the coordinates of any point on y=x^2 and you land on a point of y= . One curve, four things to remember

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