Quadratic Functions & Completing the Square

Pre-Calculus · Axiom Academy

LESSON Quadratic Functions & Completing the Square Turn any f(x)=ax^2+bx+c into vertex form, derive the quadratic formula from it, and read the discriminant b^2-4ac . Every quadratic is a parabola. Written three ways, each form reveals something different — and vertex form a(x-h)^2+k is just the base parabola y=x^2 slid so its turning point lands on the vertex (h,k) . Watch y=x^2 shift onto f(x)=x^2+6x+5 . Standard form — reads off a , b , c Vertex form — reads off the vertex (h,k) Factored form — reads off the roots (when they are real) The parabola has a lowest point. The vertex (h,k) is the minimum , and k is the smallest output. The parabola has a highest point. The vertex (h,k) is the maximum , and k is the largest output. To reach vertex form by hand, look at x^2+bx as a picture: a square of side x with a strip of area bx beside it. Split that strip in half, wrap the halves around two sides, and one corner is missing — a square of area . Add that corner to complete the square ; subtract it right back so nothing changes. If the leading coefficient is not 1 , factor it out of the x -terms before completing the square. Convert 2x^2+8x+5 : Half of 4 is 2 , and 2^2=4 , so add and subtract 4 inside: Distributing the 2 carries it onto the correction — the inside -4 becomes an outside -8 : -8+5=-3 , so the vertex is (-2,-3) . Key move: because the 2 multiplies the whole bracket, the compensation term is scaled by a , not just subtracted as is.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.