Transforming Root Functions

Algebra 1 · Axiom Academy

LESSON Transforming Root Functions Watch the parent curve slide, stretch, and flip into — and see exactly how its start point, domain, and range move with it. 1. The Parent Curve and Its Start Point Everything starts with . A point rides along the curve from its start point at the origin and climbs to the right — never to the left, because we cannot take the square root of a negative number. That single fact fixes the domain. The parent curve we will transform Subtract h inside the root and the whole curve slides right by h . The reason: to start the climb we still need the inside to be zero, and x-h=0 now happens at x=h . So the start point walks from (0,0) over to (h,0) , and the domain shifts with it. Add k outside the root and every height rises by k , so the whole curve lifts up by k (or drops if k is negative). The start point rises from (h,0) to (h,k) , and because that is the curve's lowest point, the range floor rises with it. Multiply by a in front and every height above the start point scales by a . With a>1 the curve climbs steeper; with 0<a<1 it climbs flatter. The start point itself does not move — at the start the root is 0 , and — so it stays anchored at (h,k) while the rest of the curve fans out. Heights multiply, so the curve climbs steeper. is twice as tall at every x . Heights shrink, so the curve climbs flatter. is half as tall. At the start the root is 0 , and , so (h,k) never moves under a stretch. A positive stretch leaves both alone: domain , range .

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