Quadratic Functions Vertex-Graphing Form

Quadratic Functions Quadratic Functions can be written in the form y = f(x) = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. y = ax2 + bx + c is called general form Quadratic functions have the unique curve known as a parabola

Quadratic Functions To graph quadratic functions, we need to write the equation in vertex-graphing form y = a(x – p)2 + q is called vertex-graphing form

Quadratic Functions The Role of “a”: Determines direction of opening for parabola If a is positive  opens up (concave up) If a is negative  opens down (concave down) If |a| = 1  parabola has a “normal width” As |a| ↑  parabola becomes narrower (skinny) As |a| ↓  parabola becomes wider (flat)

Quadratic Form The Role of “p” and “q”: Determine the vertex of the parabola The vertex is the highest or lowest point on a parabola. Also known as the “max” or “min” values

Quadratic Functions The axis of symmetry is the vertical line that passes through the vertex and divides the parabola into two equal halves. Equation of the axis of symmetry is given by x = p

Example Example 1: For the quadratic function f(x) = -3(x + 6)2 + 4, determine: The direction in which the parabola opens. The coordinates of the vertex. The equation of the axis of symmetry

f(x) = -3(x + 6)2 + 4 to y = a(x – p)2 + q Solution Begin by comparing: f(x) = -3(x + 6)2 + 4 to y = a(x – p)2 + q Solution to a): We can see that a is negative so the parabola opens downwards. Solution to b): Since x – p = x + 6, p must be -6 and q is 4, the vertex is at (-6, 4) Solution to c): The equation of the axis of symmetry is x = -6.