Let's see what happens to graph if we set \(b = c = 0\), and \(a\) to different positive values. In the picture, the red graph shows \(y = x^2\), the blue graph is \(y = 2x^2\), the green graph is \(y = 3x^2\), and the orange graph is \(y = 4x^2\). I went all out for the purple graph: it is \(y = 11x^2\). Do you notice how, as we increase the value of \(a\), the graph gets steeper and narrower?

As advertised, we get a straight line. In this picture, the green graph is \(y = 2x + 3\), and the black graph is \(y = -2x + 4\).

Hold onto your hats! We're turning everything upside down. The red graph is \(y = -2x^2 + 3x + 4\), the black graph is \(y = -x^2 + 3x\), and the green graph is \(y = -x^2\).

We're going to fix \(a = 2\) for now, and set \(c = 0\), so that we can isolate the effect of changing the value of \(b\) in our equation. The red graph is \(y = 2x^2\), the green graph is \(y = 2x^2 + x\), the orange graph is \(y = 2x^2 + 3x\), the purple graph is \(y = 2x^2 - x\), and the black graph is \(y = 2x^2 - 3x\). Increasing the value of \(b\) shifts the graph to the right and down. The graph shifts to the left and down for larger negative values of \(b\).

Changing the value of \(c\), while keeping \(a\) and \(b\) fixed shifts the graph up (for positive values of \(c\)) and down (for negative values of \(c\)) the \(xy\)-plane. Here we have the graphs of \(y = 3x^2 - 2x\) in red, \(y = 3x^2 - 2x - 1\) in blue, \(y = 3x^2 - 2x - 4\) in green and \(y = 3x^2 - 2x + 1\) in orange.

The green graph is the graph of the equation \(y = x^2 - 4x + 4\). It has equal real roots at \(x = 2\). The black graph is the graph of the equation \(y = x^2 + 4x + 7\). It goes nowhere near the \(x\)-axis. The corresponding quadratic equation has no real roots. Finally, the blue graph is the graph of \(y = x^2 - 2x - 1\). As it cuts the \(x\)-axis twice, the corresponding quadratic equation has \(2\) distinct real roots.