# Find Range of Quadratic Functions

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 $f(x) = 2x^2 - 10x + 19$

#### Answers for Matched Problems

1) $$(-\infty, 3]$$
2) $$[6.5, +\infty )$$

### More References and links

Find domain and range of functions,
Find the range of functions,
find the domain of a function and mathematics tutorials and problems.