Vertex form is y = a(x − h)² + k, where (h, k) is the vertex of the parabola and a determines the direction and width. This form makes graphing straightforward — plot the vertex and use the value of a to determine the shape. Converting from standard form (y = ax² + bx + c) to vertex form requires completing the square. Vertex form reveals the transformations applied to the parent function y = x²: the graph is shifted h units horizontally, k units vertically, and stretched or compressed by a factor of |a|.
Start with the parent function y = x² and apply transformations one at a time: vertical shift (y = x² + k), horizontal shift (y = (x − h)²), then both together. Show that a > 1 narrows the parabola and 0 < a < 1 widens it. Practice converting between vertex form and standard form by expanding and by completing the square. Graph directly from vertex form without converting.
From graphing quadratics, you know that every parabola has a vertex — the turning point where the function changes from decreasing to increasing (or vice versa). Vertex form, y = a(x − h)² + k, is designed to make that vertex visible at a glance. The vertex is (h, k), readable directly from the equation. This is the payoff: instead of finding the vertex by computing −b/2a from standard form, you simply read it off.
The form is built from transformations of the parent parabola y = x², which you can think of as the "default" parabola with vertex at the origin. Each parameter shifts or scales it. Adding k shifts the graph up or down by k units — this is a vertical translation. Replacing x with (x − h) shifts the graph right by h units (left if h is negative) — this is a horizontal translation. The sign asymmetry is the main trap: (x − 3)² pushes the vertex to x = 3, not x = −3, because you need x = 3 to make the expression equal zero. Finally, the coefficient a stretches or compresses the parabola. When |a| > 1, the parabola is narrower than y = x²; when 0 < |a| < 1, it is wider. A negative a flips the parabola upside down.
Converting from standard form y = ax² + bx + c to vertex form requires completing the square — a technique you'll master in the next topic. But even now, you can convert in reverse: expand y = a(x − h)² + k by multiplying out (x − h)² = x² − 2hx + h², then distribute a, and collect constants. This gives back y = ax² + (−2ah)x + (ah² + k). Matching with standard form: b = −2ah, so h = −b/2a, and c = ah² + k, so k = c − ah². These are the standard formulas for the vertex — but vertex form makes them unnecessary.
The deepest insight is that every quadratic function has a vertex, and that vertex is the geometric center of symmetry of the parabola. The axis of symmetry is the vertical line x = h. Any input h + d gives the same output as h − d (the parabola is mirror-symmetric about this axis), because a(h + d − h)² + k = a(h − d − h)² + k = ad² + k. Vertex form exposes this symmetry that is hidden in standard form. When you later study function transformations, vertex form will generalize: any function can be shifted and scaled in the same pattern, not just quadratics.