Key Features of Quadratic Graphs

A quadratic function is a polynomial function of degree 2, typically in the form of $$f(x) = ax^2 + bx + c$$ , where a, b, and c are constants and a ≠ 0.

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

The vertex of a quadratic graph represents the highest or lowest point of the parabola, depending on whether it opens upwards or downwards.

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, given by the equation $$x = h$$ , where (h, k) is the vertex.

A quadratic function opens upwards if the coefficient 'a' is positive (a > 0) and downwards if 'a' is negative (a < 0).

The vertex (h, k) can be found using the formulas: $$h = -\frac{b}{2a}$$ and $$k = f(h)$$ .

The x-intercepts (or roots) are the points where the graph intersects the x-axis, found by solving the equation $$ax^2 + bx + c = 0$$ .

The quadratic formula is used to find the roots of a quadratic equation and is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ .

The discriminant, given by $$D = b^2 - 4ac$$ , determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.

For a quadratic function that opens upwards, the increasing interval is from the vertex to positive infinity, expressed as $$(h, \infty)$$ .