in quadratic polynomial if leading Coefficient is - V then its graph is opened
Answers
Recall that a quadratic function has the form
f
(
x
)
=
a
x
2
+
b
x
+
c
.
where
a
,
b
, and
c
are constants, and
a
≠
0
.
The graph of a quadratic function is a U-shaped curve called a parabola. This shape is shown below.
A parabola has a maximum or a minimum, called the vertex, an axis of symmetry through the middle of the parabola, a y-intercept where it crosses the y-axis, and can have as many as two x-intercepts where it crosses the x-axis.
Parabola : The graph of a quadratic function is a parabola.
In graphs of quadratic functions, the sign on the coefficient
a
affects whether the graph opens up or down. If
a
<
0
, the graph makes a frown (opens down) and if
a
>
0
then the graph makes a smile (opens up). This is shown below.
image
Direction of Parabolas: The sign on the coefficient
a
determines the direction of the parabola.
Features of Parabolas
Parabolas have several recognizable features that characterize their shape and placement on the Cartesian plane.
Vertex
One important feature of the parabola is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph.
Axis of Symmetry
Parabolas also have an axis of symmetry, which is parallel to the y-axis. The axis of symmetry is a vertical line drawn through the vertex.
y
-intercept
The y-intercept is the point at which the parabola crosses the y-axis. There cannot be more than one such point, for the graph of a quadratic function. If there were, the curve would not be a function, as there would be two
y
values for one
x
value, at zero.
x
-intercepts
The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of
x
at which
y
=
0
. There may be zero, one, or two
x
-intercepts. The number of
x
-intercepts varies depending upon the location of the graph (see the diagram below).