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:
- The values of x that cause a quadratic expression to equal zero.
- The places where a quadratic function’s graph (the parabola) crosses the x axis.
- The roots or solutions of a quadratic equation.
- The factors of a quadratic expression.
Example: By studying figure (a) above we see that the following are equivalent:
- The quadratic function’s graph (i.e. the parabola)
y = x 2 − 3 x + 2
crosses the x axis at x = 1 and x = 2.
- The quadratic equation
x 2 − 3 x + 2 = 0
has solutions x = 1 and x = 2.
- The quadratic expression
x 2 − 3 x + 2
can be factored into (x − 1)(x − 2).
- Letting x = 1 makes the first factor equal zero and letting
x = 2 makes the second factor equal zero.
Either factor equalling zero makes the quadratic expression equal zero.
Example: By studying figure (d) above we see that the following are also equivalent:
- The quadratic function’s graph (parabola)
y = x 2 + 4
does not cross the x axis.
- The quadratic equation x 2 + 4 = 0 does not have real solutions.
(It does however have complex solutions.)
- The quadratic expression x 2 + 4 cannot be factored
over the real numbers.
(It can however be factored over the complex numbers.)