What are the important features of the graphs of quadratic functions?
Answers
A quadratic function has the general form:
y
=
a
x
2
+
b
x
+
c
(where
a
,
b
and
c
are real numbers) and is represented graphically by a curve called PARABOLA that has a shape of a downwards or upwards U.
The main features of this curve are:
1) Concavity: up or down. This depends upon the sign of the real number
a
:
enter image source here
2) Vertex. The vertex is the highes or lowest point of the parábola.
the coordinates of this point are:
x
=
−
b
2
a
and
y
=
−
Δ
4
a
Where
Δ
=
b
2
−
4
a
c
3) point of intercept with the y axis. This is the point where the parábola crosses the y axis and has coordinates:
(
0
,
c
)
enter image source here
4) Possible points of intercept with the x axis (there also can be none). These are the points where the parábola crosses the x axis.
They are obtained by putting y=0 and solving for x the 2nd degree equation:
a
x
2
+
b
x
+
c
=
0
, which will give the x coordinates of these points (2 solutions)
Depending on the discriminant
Δ
=
b
2
−
4
a
c
if it is <0 the parábola does not cross the x axis.
enter image source here