A parabola with equation $y=x^2+bx+c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$?
I know that a parabola is symmetric about its vertex, so if it passes through two points with the same y-coordinate, the x-coordinate of the vertex must be halfway between them. That means the vertex has x-coordinate $3$, and I can use the formula $x=-\frac{b}{2a}$, where $a=1$ in this case, to find $b$. Solving for $b$, I get $b=-2a(3)=-6$. Now I have $y=x^2-6x+c$, and I can plug in either of the given points to find $c$. I'll use $(2,3)$, so I have $3=2^2-6(2)+c$.
[* so I have $3=4-12+c$, and simplifying, I get $c=11$. 