Three Forms of Quadratics

Algebra 2 · Axiom Academy

LESSON Three Forms of Quadratics Standard, vertex, and factored form are three names for one parabola — each hands you a different feature for free. The same quadratic can be written three equivalent ways. Take one parabola, y = x^2 - 4x + 3 . Every form below is the same curve — they only differ in which feature they show you at a glance. with ( a sets the direction and width) Standard form hands you the y-intercept with no work: set x = 0 and every x -term vanishes, leaving y = c . So the curve crosses the y-axis at . is the vertex; x = h is the axis of symmetry Vertex form hands you the turning point (h, k) directly, and the parabola is a perfect mirror image across the vertical line x = h . Two points the same distance left and right of the axis always sit at the same height. r_1 and r_2 are the x-intercepts (roots / zeros) Factored form hands you the roots . A product equals zero exactly when one of its factors is zero, so y = 0 precisely at x = r_1 and x = r_2 . Here y = (x-1)(x-3) is zero at x = 1 and x = 3 . Each form is one algebra step from the others: Completing the square is literal : x^2 + bx is an x -by- x square plus a strip of area bx . Split the strip into two half-strips of width , wrap them on two sides, and the missing corner is exactly . Add that corner and you have completed a real square. The shortcut that falls out: the vertex sits at Mind the minus in — dropping it is the classic error. Convert y = x^2 - 6x + 8 . Then and k = f(3) = -1 :

This is the written version of the interactive lesson above. See the full Algebra 2 course.