Let $f(x)=a{x}^2+bx+c$ and consider the curve

1. By completing the square on $f(x),$ find the coordinates of the vertex.

2. Find the coordinates of the vertex using differentiation (isn't that easier!)

3. Use the second derivative test to prove that the parabola is $\cup$ shaped when $a>0$ and $\cap$ shaped when $a<0.$

For (a), you need to remove the $a$ from the first two terms:

Now you're ready to complete the square.

For (b), note that $c$ is a constant so differentiates to $0.$