What is the Concavity of Quadratic Functions?

This tutorial explains how to determine the concavity of quadratic functions.

Concavity of Quadratic Functions

The concavity of a function is determined by the sign of its second derivative. For a quadratic function of the form

\[ f(x) = a x^{2} + b x + c, \quad a \neq 0 \]

The first and second derivatives are

\[ f'(x) = 2 a x + b \] \[ f''(x) = 2 a \]

Since \( f''(x) \) is constant and depends solely on \( a \), the concavity of the parabola depends on the sign of \( a \):

If \( a > 0 \), then \( f''(x) > 0 \) and the graph is concave up.
If \( a \lt 0 \), then \( f''(x) \lt 0 \) and the graph is concave down.

Below are examples illustrating these cases with detailed solutions.

Example 1

Determine the concavity of the quadratic function:

\[ f(x) = (2 - x)(x - 3) + 3 \]

Solution to Example 1

First, expand and rewrite \( f(x) \): \[ f(x) = -x^{2} + 5x - 3 \] The leading coefficient \( a = -1 \) is negative, so the graph is concave down. See the figure below. Graph of quadratic function Example 1

Example 2

Determine the concavity of the quadratic function:

\[ f(x) = -2(x - 1)(x - 2) + 3 x^{2} \]

Solution to Example 2

Expand \( f(x) \): \[ f(x) = x^{2} + 6x - 4 \] The leading coefficient \( a = 1 \) is positive, so the graph is concave up. See the figure below. Graph of quadratic function Example 2

Exercises With Answers

Determine the concavity of each quadratic function below:

\( f(x) = 2x^{3} + 6x - 13 \) (Note: This is cubic, not quadratic)
\( f(x) = (2 - x)(4 - x) \)
\( f(x) = -2(x - 3)^{2} - 5 \)
\( f(x) = x(x + 3) - 2(x - 3)^{2} \)

Answers to Exercises

Not quadratic — cubic function, concavity varies.
Concave up
Concave down
Concave down

Try another interactive tutorial on the concavity of quadratic functions available on this site.