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 Â· x^{ 2} + b
Â· 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.
Sign of the number a in the quadratic
equation 
Appearance of the graph of the quadratic
equation 
Negative 
Frowning 
Positive 
Smiling 
Table 1: Effect of sign of a on appearance of quadratic graph.
