# How do you determine the concavity of a quadratic function?

##### 2 Answers

For a quadratic function

if

if

For a quadratic function

#f(x)=ax^2+bx+c#

#f'(x)=2ax+b#

#f''(x)=2a#

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

This can be shown graphically:

The function

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

The function

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