# What is the Concavity of Quadratic Functions?

Tutorial on how to determine the concavity of quadratic functions.

## Concavity of Quadratic FunctionsThe concavity of functions may be determined using the sign of the second derivative. For a quadratic function f is of the formf(x) = a x^{ 2} + b x + c , with a not equal to 0The 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 1What is the concavity of the following quadratic function?f(x) = (2 - x)(x - 3) + 3## Solution to Example 1Expand f(x) and rewrite it as followsf(x) = -x ^{ 2} + 5x -3
The leading coefficient a is negative and therefore the graph of is concave down. see figure below.
## Example 2What is the concavity of the following quadratic function?f(x) = -2(x - 1)(x - 2) + 3 x^{ 2}## Solution to Example 2Expand f(x) as followsf(x) = x ^{ 2} + 6 x - 4
The leading coefficient a is positive and therefore the graph of is concave up. see figure below.
## Exercises With AnswersDetermine 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 Exercisesa) concave upb) 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.
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