9.1 - Introduction to quadratics


Definition: A quadratic expression is an expression of the form:
a x 2 + b x + c,
where x is a variable and a, b, and c are constants, and a is not equal to zero.

The term a x 2 is called the quadratic term, b x is called the linear term and c is called the constant term.

The constant a is called the leading coefficient, b is called the linear coefficient, and c is called the additive constant.




Example: These are quadratic expressions:

We can create a quadratic function called, say f, whose input is x and whose output, f (x), is the quadratic expression evaluated at x:
a x 2 + b x + c


Example:  f (x) = x 2 − 3 x + 2 is a quadratic function. If we let the input be, say x = 4, then the output is f (4) = 4 2 − 3 · 4 + 2 = 6.



We can give the output of the quadratic function the name “y” and make a graph of y versus x. We will get a curve like the four curves shown below. These curves are called parabolas. They feature a point called the vertex where the parabola reaches its maximum height or depth and turns around. The parabola opens upward if a > 0 and opens downward if a < 0 but its exact location depends on the values of all three constants a, b and c. In the picture below, parabolas (a), (c) and (d) open upward and have their vertex at the bottom. Parabola (b) opens downward and has its vertex at the top.




If we set the quadratic expression equal to zero or if we set y = 0 or f (x) = 0 then we get the so-called quadratic equation:
a x 2 + b x + c = 0.
(Note that setting y = 0 in the graph means that we are looking at points where the parabola crosses the x axis, and setting f (x) = 0 in the quadratic function means that we are looking for values of x for which the output of the quadratic function is zero.)

There is a close connection between:

Example: By studying figure (a) above we see that the following are equivalent:



Example: By studying figure (d) above we see that the following are also equivalent: