How do you determine the concavity of a quadratic function?

Sep 21, 2014

For a quadratic function $f \left(x\right) = a {x}^{2} + b x + c$,
if $a > 0$, then $f$ is concave upward everywhere,
if $a < 0$, then $f$ is concave downward everywhere.

Jan 22, 2016

For a quadratic function $a {x}^{2} + b x + c$, we can determine the concavity by finding the second derivative.

$f \left(x\right) = a {x}^{2} + b x + c$
$f ' \left(x\right) = 2 a x + b$
$f ' ' \left(x\right) = 2 a$

In any function, if the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.

Since the second derivative of any quadratic function is just $2 a$, the sign of $a$ directly correlates with the concavity of the function, in that if $a$ is positive, $2 a$ is positive so the function is concave up, and the same can be said for a negative $a$ value making $2 a$ negative resulting in the function being concave down.

This can be shown graphically:

The function $f \left(x\right) = 6 {x}^{2} + 3 x - 5$, where $a > 0$, should be concave up.

graph{6x^2+3x-5 [-18.5, 17.54, -10.35, 7.68]}

The function $f \left(x\right) = - \frac{1}{2} {x}^{2} - 7 x + 1$, where $a < 0$, should be concave down.

graph{-1/2x^2-7x+1 [-64.2, 52.83, -24.88, 33.7]}