What is the Concavity of Quadratic Functions?

Tutorial on how to determine the concavity of quadratic functions.

Concavity of Quadratic Functions

The concavity of functions may be determined using the sign of the second derivative. For a quadratic function f is of the form
f(x) = a x 2 + b x + c , with a not equal to 0

The first and second derivatives of are given by
f '(x) = 2 a x + b
f "(x) = 2 a
The sign of f " depends on the sign of coefficient a included in the definition of the quadratic function. Two cases are possible. If a is positive then f " is positive and the graph of f is concave up. If a is negative then the graph of f is concave down. Below are some examples with detailed solutions.

Example 1

What is the concavity of the following quadratic function?
f(x) = (2 - x)(x - 3) + 3

Solution to Example 1

Expand f(x) and rewrite it as follows
f(x) = -x 2 + 5x -3
The leading coefficient a is negative and therefore the graph of is concave down. see figure below.
graph of quadratic function , example 1

Example 2

What is the concavity of the following quadratic function?
f(x) = -2(x - 1)(x - 2) + 3 x 2

Solution to Example 2

Expand f(x) as follows
f(x) = x 2 + 6 x - 4
The leading coefficient a is positive and therefore the graph of is concave up. see figure below.
graph of quadratic function , example 1

Exercises With Answers

Determine the concavity of each quadratic function. a) f(x) = 2x 3 + 6 x - 13
b) f(x) = (2 - x)(4 - x)
c) f(x) = -2(x - 3) 2 - 5
d) f(x) = x(x + 3) - 2(x - 3) 2

Answers to Above Exercises

a) concave up
b) concave up
c) concave down
d) concave down

Another interactive tutorial, using an applet, on the concavity of graphs quadratic functions is included in this site.

More on Concavity and Differentiation

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