Find the range of quadratic functions; examples and matched problems with their answers are located at the bottom of this page.

Graphical Analysis of Range of Quadratic Functions

The range of a function y = f(x) is the set of values y takes for all values of x within the domain of f.
The graph of any quadratic function, of the form f(x) = a x^{2} + b x + c, which can be written in vertex form as follows

f(x) = a(x - h)^{2} + k , where h = - b / 2a and k = f(h)

is either a parabola opening up, when a > 0, or a parabola opening down, when a < 0 (see graphs of several quadratic function below).
Therefore if a > 0, the graph of f has a minimum point and if a < 0, the graph of f has a maximum point. Both minima or maxima are the vertices of the parabolas with coordinates (k , k) where h = - b / 2a and k = f(h).

Examples with Solutions

Example 1

Find the range of function f
defined by

f(x) = - 2 x^{2} + 4 x + 2

Solution to Example 1

The vertex of the graph of f is at the point ( h , k ) where
h = - b / 2 a = - 4 / 2(-2) = 1 and k = f(1) = 4

The leading coefficient a = - 2 is negative and therefore the graph of f has a maximum at the point (1 , 4). The maximum value of f is 4. Hence the range of f is therefore given by the interval: (-∞ , 4 ] (see graph below to better understand)

Matched Problem 1:

Find the range of
function f defined by

f(x) = - 3 x^{2} - 6 x

Example 2

Find the range of function f
defined by

f(x) = 2 x^{2} + 12 x + 16

Solution to Example 2

The coordinates h and k of the vertex of the graph of f are given by
h = - b / 2a = - 12 / 2(2) = - 3 and k = f(-3) = - 2

The leading coefficient a = 2 is positive and therefore the graph of f has a minimum point at (h , k) = (-3 , -2). The range of f is given by the interval [-2 , +∞ ) (see graph of f below)