What does the coefficient of X^2 tell you about the shape of the graph?

The coefficient of x^2 tells you a lot about the shape of the quadratic. Firstly, a positive coefficient implies there is a minima (often known as a smiley graph) and a negative coefficient implies there is a maxima (often known as a sad graph). All quadratic graphs are symmetrical about their minima or maxima. The magnitude of the coefficient of x^2 tells you the steepness of the graph, with larger coefficients increasing the steepness (speed) of the increase or decrease of the function.