I need some help figuring out how to solve this problem.
Which of the following could be a graph of the equation $ y= ax^2 + bx + c$ where $b^2 - 4ac = 0$
The picture below was the correct answer. So the first equation told me that the the shape of the graph would be a parabola. Then the second equation is what I assume tells you that this parabola is shifted to the right and is facing upwards although I don't see what about $b^2 - 4ac = 0$ tells me this.
I'm assuming that it's $ax^2+bx+c$, not $ax^2+bx+x$.
Actually, nothing you've included tells you that it's shifted to the right or facing upward. It faces upward if and only if $a > 0$. It's shifted to the right if and only if $b < 0$ (given that $a > 0$). What $b^2-4ac = 0$ tells you is that it's tangent to the $x$-axis. (Equivalently, it has a double root.)
I'm assuming this was a multiple-choice problem. The critical part would have been recognizing what $b^2-4ac = 0$ tells you, and to identify the one parabola that was tangent to the $x$-axis.