What the Standard Form of a Quadratic can tell you about the graph

The graphs of quadratic equations come in two varieties: smiling and frowning (see Figure 1, ). The standard form of a quadratic equation:

y = a · x ² + b · x + c,

can tell you whether a given graph will smile or frown. If the number (or coefficient), a, multiplying the x ² is positive then the quadratic is smiling. If the number a is negative then the quadratic is frowning.

Figure 1(a): The graph of a smiling quaratic equation opens upwards and outwards, like a smiling mouth.

Figure 1(b): The graph of a frowning quaratic equation opens downwards and outwards, like an unhappy mouth.

The appearance of the quadratic equation (smiling or frowning) is controlled by the sign of the number a.

Table 1: Effect of sign of a on appearance of quadratic graph.