What is the Concavity of Quadratic Functions?
Tutorial on how to determine the concavity of quadratic functions.
f(x) = a x 2 + b x + c , with a not equal to 0
The concavity of functions may be determined using the sign of the second derivative. For a quadratic function f of the form
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:
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Updated: 2 April 2013
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