How do you determine the concavity of a quadratic function?

Question

Charles Anctil · Accepted Answer

For a quadratic function #f(x)=ax^2+bx+c#,
if #a>0#, then #f# is concave upward everywhere,
if #a<0#, then #f# is concave downward everywhere.

Emilia Aguilar · Answer

For a quadratic function #ax^2+bx+c#, we can determine the concavity by finding the second derivative. #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 #2a#, the sign of #a# directly correlates with the concavity of the function, in that if #a# is positive, #2a# is positive so the function is concave up, and the same can be said for a negative #a# value making #2a# negative resulting in the function being concave down. This can be shown graphically: The function #f(x)=6x^2+3x-5#, where #a>0#, should be concave up. graph{6x^2+3x-5 [-18.5, 17.54, -10.35, 7.68]} The function #f(x)= -1/2x^2-7x+1#, where #a<0#, should be concave down. graph{-1/2x^2-7x+1 [-64.2, 52.83, -24.88, 33.7]}

Paisley Johnson · Answer

undefined To determine the concavity of a quadratic function, you need to analyze its second derivative.

1. Compute the first derivative of the quadratic function.
2. Then, compute the second derivative of the function.
3. If the second derivative is positive for all x-values in the domain, the function is concave up.
4. If the second derivative is negative for all x-values in the domain, the function is concave down.
5. If the second derivative changes sign at a point within the domain, the function changes concavity at that point.