As we’ve already seen, f(x) = x2 graphs
into a PARABOLA.
This is the simplest quadratic function
we can think of. We will use this one as
a model by which to compare all other
quadratic functions we will examine.
All parabolas have a VERTEX,
VERTEX,
the lowest or highest point on
the graph (depending upon
whether it opens up or down.
All parabolas have an AXIS OF
SYMMETRY,
SYMMETRY, an imaginary line which
goes through the vertex and about
which the parabola is symmetric.
Some parabolas open up and some
open down.
Parabolas will all have a different vertex
and a different axis of symmetry.
Some parabolas will be wide and some
will be narrow.
The standard form of a quadratic
function is:
f(x) = ax2 + bx + c
The position, width, and orientation of a
particular parabola will depend upon
the values of a, b, and c.
Now compare f(x) = x2 to the following:
f(x) = x 2 + 3
f(x) = x 2 - 2
Vertical shift up
Vertical shift down
Now compare f(x) = x2 to the following:
f(x) = (x + 2)2
f(x) = (x – 3)2
Horizontal shift to
the left
Horizontal shift to
the right
When the standard form of a quadratic
function f(x) = ax2 + bx + c is written in the
form:
f(x) = a(x - h) 2 + k
We can tell by horizontal and vertical shifting
of the parabola where the vertex will be.
The parabola will be shifted h units
horizontally and k units vertically.
Thus, a quadratic function written in the form
y = a(x - h) 2 + k
will have a vertex at the point (h,k)
h,k).
The value of “a” will determine whether the
parabola opens up or down (positive or
negative) and whether the parabola is narrow
or wide.