A parabola with equation $y=x^2+bx+c$ passes through the points $(-1,-11)$ and $(3,17)$. What is $c$?
I know that a parabola is the graph of a quadratic function, so I can use the given points to form a system of two equations with two unknowns: $y=x^2+bx+c$. Plugging in $(-1,-11)$, I get $-11=(-1)^2+b(-1)+c$, or $-11=1-b+c$. Plugging in $(3,17)$, I get $17=(3)^2+b(3)+c$, or $17=9+3b+c$. Subtracting the first equation from the second, I get $28=8+4b$, or $b=5$. Substituting $b=5$ into either equation, I can solve for $c$. I'll use the first one: $-11=1-5+c$, or $c=-7$. Therefore, the value of $c$ is -7. 