As a graph of a function, a parabola is a smooth curve which opens upward or opens downward as shown in the figure below. The turning point on the parabola is called the vertex of the parabola. The vertical line through the vertex is called axis of the parabola. Axis of a parabola is its line of symmetry. That is, the portions of the curve that lies ‘to the left side’ and ‘to the right side’ of the axis are mirror images of one another through the axis. The parabola intersects its axis at its vertex.

The following features of parabolas are very helpful in graphing a quadratic functions and to determine their ranges.

1. Suppose the graph of f (x) is the parabola shown on the adjacent figure whose vertex is (h, k) and opens upward. Then, from the graph you can see the followings.

(a) y = f (x) ≥ k for all x . This implies that rang(f ) = { y∈ℝ | y ≥ k }

That is, k is the minimum value of f which occurs when x = h.

(b) The vertical line x = h is the axis of the parabola.

2. Suppose the graph of f(x) is the parabola shown on the adjacent graph whose vertex is (h, k) and opens downward. Then, from the graph you can see the followings.

(a) y = f (x) ≤ k for all x. This implies that
rang(f ) = { y ∈ℝ | y ≤ k }

That is, k is the maximum value of f which occurs when x = h.

(b) The vertical line x = h is the axis of the parabola.
Note also that if a vertical line x = h is the line of symmetry for the graph (parabola) of f then the values of f at r' units to the left from h’ and ‘r units to the right from h’ are equal. i.e.,

A vertical line x = h is the line of symmetry for the parabola (graph) of f if and only if f (h − r) = f (h + r) for every r∈ℝ.