The a term:

Begin with a standard quadratic in which b = c = 0 and a = 1. (On all graphs below, you may click on the image to download the original file for your investigation.)

By graphing several quadratics in which you vary the value of a, the role a seems to play becomes quite evident. Observe the following graphs:

If a is positive, the graph is concave up. If a is negative, the graph is concave down. Furthermore, as a gets larger, the graph gets more narrow. Notice the impact the a coefficient has on the y-coordinate for varying x's. Herein lies a justification of these conclusions.

 

The c term:

Again, begin with a standard quadratic in which b = c = 0 and a = 1. By graphing several quadratics in which you vary the value of c, the role c seems to play becomes quite evident. Observe the following graphs:

If c is positive, the graph slides up. If c is negative, the graph slides down. In fact, it appears the entire graph (i.e. each associated value of y) slides exactly the amount and direction of c. These conclusions can be justified by analysis of the function. Additionally, one may be tempted to suggest that the value of c is the y-intercept of the function. One counterexample will disprove that notion, and since we haven't set b to be anything but zero, it is worth a look at a few more examples.

This property appears to hold true. See that the y-intercept occurs when the value of x is zero. In each function above, when x = 0, the quadratic and linear terms become zero and all that is left is become zero and all that is left is y = c. Thus, the y-intercept must always be exactly c.