SECTION B Q17. In the figure, graph can be related to representation of quadratic polynomial. 1. The shape of graph is shown as ii. The graph of parabol a open downwards, if i. How many zeroes are shown in the above graph? x iv.How many zeroes are there for the polynomial?
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.
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.
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).
Notice that, for parabolas with two
x
-intercepts, the vertex always falls between the roots. Due to the fact that parabolas are symmetric, the
x
-coordinate of the vertex is exactly in the middle of the
x
-coordinates of the two roots.
A Graphical Interpretation of Quadratic Solutions
The roots of a quadratic function can be found algebraically or graphically.
Answer:
5 zero are their in the figure