Vertex Form Quadratics

The vertex form of a quadratic function is f(x) = a(x-h)² + k, where (h, k) is the vertex of the parabola.

The 'a' value determines the direction the parabola opens: if 'a' is positive, it opens upward; if 'a' is negative, it opens downward.

Changing the 'h' value shifts the graph horizontally. If 'h' increases, the graph shifts to the right; if 'h' decreases, it shifts to the left.

Changing the 'k' value shifts the graph vertically. If 'k' increases, the graph shifts up; if 'k' decreases, it shifts down.

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, given by the equation x = h in vertex form.

The vertex is the highest or lowest point of the parabola, depending on whether it opens upward or downward.

To find the vertex from standard form f(x) = ax² + bx + c, use the formula h = -b/(2a) and then calculate k by substituting h back into the function.

The x-intercepts are the points where the graph crosses the x-axis, found by solving the equation f(x) = 0.

The vertex lies on the axis of symmetry, which is the line that passes through the vertex and divides the parabola into two equal halves.

If the 'a' value in the vertex form f(x) = a(x-h)² + k is positive, the parabola opens upward; if 'a' is negative, it opens downward.