Why graph of +ve ax² open upwards and graph of -ve ax² open downwards?

Question

While studying about quadratic equations i always wondered that why graph of +ve ax² open upwards and graph of -ve ax² open downwards. I even asked my teacher but she just gave reason that because it does... And that makes no sense If there's a quadratic equation ax²+bx+c then changing the value of a,b & c should affect the graph but I dont know how it affects the graph actually? I mean what is the significance of value of c in graph similarly of a and b?

Answer 1

The $ax^2$ term dominates as $x \to \infty$. If $a$ is positive the graph will go to $+\infty$ and open upwards. If $a$ is negative the graph will go to $-\infty$ and open downwards. $c$ shifts the graph upwards or downwards. $\frac b {2a}$ is the $x$ position of the vertex, so shifts the graph left or right.