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 \( f(x) = ax^2 + bx + c \), which can be written in vertex form as follows
\[ f(x) = a(x - h)^2 + k \]
where \( h = -\dfrac{b}{2a} \) and \( k = f(h) \)
is either a parabola opening up, when \( a > 0 \), or a parabola opening down, when \( a \lt 0 \) (see graphs of several quadratic functions below).
Therefore if \( a > 0 \), the graph of \( f \) has a minimum point and if \( a \lt 0 \), the graph of \( f \) has a maximum point.
Both minima or maxima are the vertices of the parabolas with coordinates \( (k, k) \) where \( h = -\dfrac{b}{2a} \) and \( k = f(h) \).

Examples with Solutions

Example 1

Find the range of function \( f \)
defined by
\[ f(x) = -2x^2 + 4x + 2 \]

Solution to Example 1

The vertex of the graph of \( f \) is at the point \( (h, k) \) where
\( h = -\dfrac{b}{2a} = -\dfrac{4}{2(-2)} = 1 \) 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: \( (-\infty, 4] \) (see graph below to better understand)

Matched Problem 1:

Find the range of
function \( f \) defined by
\[ f(x) = -3x^2 - 6x \]

Example 2

Find the range of function \( f \)
defined by
\[ f(x) = 2x^2 + 12x + 16 \]

Solution to Example 2

The coordinates \( h \) and \( k \) of the vertex of the graph of \( f \) are given by
\( h = -\dfrac{b}{2a} = -\dfrac{12}{2(2)} = -3 \) 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 \([-2, +\infty) \) (see graph of \( f \) below)

Matched Problem 2:

Find the range of
function \( f \) defined by
\[ f(x) = 2x^2 - 10x + 19 \]