Definition: Vertex form writes a quadratic function as y = a(x – h)^2 + k, so the vertex (h, k) and the direction of opening can be read directly.
What does vertex form show?
The vertex is (h, k). The sign of a gives the direction, and |a| gives the width. The axis of symmetry is x = h.
How is it related to transformations?
Start with y = x2. Replacing x with x – h slides the graph right by h. Adding k lifts it by k. Multiplying by a stretches, shrinks, or flips it.
Example. Write y = x2 – 8x + 19 in vertex form. Complete the square: x2 – 8x + 16 + 3, so y = (x – 4)2 + 3. The vertex is (4, 3) and the parabola opens up. Another case: y = -2(x + 1)2 + 6 has vertex (-1, 6), opens downward, and its maximum value is 6.
What mistakes occur?
The sign inside the parentheses is opposite to the coordinate: (x + 1) means h = -1. Students also forget that a multiplies the squared part only after squaring. To convert back to standard form, expand the square and combine terms.