What the Factored Form of a Quadratic can tell you about the graph
Like standard and vertex forms, the factored form of a quadratic function:
y = a Â· (x  c) Â· (x  d)
can also tell you whether the given quadratic has a graph that â€œsmilesâ€ or â€œfrowns.â€ As
with the standard and vertex forms the key again is the sign (+ or ) of the number a. If a
is positive the graph â€œsmilesâ€ and if a is negative the graph â€œfrowns.â€ Sometimes you
will see an example of a factored form that does not appear to have a value of a, such as:
y = (x 1) Â· (x + 2).
In this case, the value of a is equal to one (which is positive) and the graph of the
quadratic will smile.
The factored form of a quadratic equation also tells you where the xintercepts
(sometimes called the roots or zeros of the quadratic function are located). The xintercepts
of the equation are the xvalues that will make y = 0.
For example, the xintercepts of the quadratic:
y = (x 1) Â· (x + 2)
are x = 1 and x = 2 as plugging either of these two values into the quadratic equation
will make y equal to zero.
Example
Figure 1 shows the graph of a quadratic function. Find the equation of
this quadratic and express your answer in both factored and standard forms.
Figure 1: Find the formula of this quadratic function.
Solution
Figure 1 clearly shows both the of the xintercepts of the quadratic function so it will be
easiest to find the factored form of the quadratic first and then convert this to standard
form by FOILing.
The xintercepts of the quadratic shown in Figure 4 are located at x = 0 and x = 4. This
means that the factored form of the quadratic function must look something like this:
y = a Â· (x  0) Â· (x  4) = a Â· x Â· (x  4) .
The factored form must have a factor of (x  0), or more simply a factor of just x, to
ensure that when you plug in x = 0 the value of y will be equal to zero. The factored form
must also have a factor of (x  4) to ensure that when you plug in x = 4 the value of y will
be equal to zero.
To determine the numerical value of a you can plug in the x and ycoordinates of any
other point on the quadratic graph (i.e. any point other than one of the xintercepts) and
solve for a. Figure 1 shows that the point (2, 2) lies on the graph, so you can plug in x =
2 and y = 2 into the factored form. Doing this:
2 = a Â· 2 Â· (2  4)
2 = a Â· (4)
So, the equation of the quadratic function from Figure 1 (written in factored form) is:
To convert this equation to standard form, you can expand by FOILing and then simplify
(if necessary). Doing this:
(Expand by FOILing)
(Multiply through by
 beware of â€œâ€ signs)
The formula for the quadratic shown in Figure 1 (expressed in standard form) is:
