Answer :
Answer:
Step-by-step explanation:
To determine which function could represent the given graph based on the polynomial characteristics, let's analyze the graph first:
Given graph:
markdown
Copy code
123
4
567
Analysis of the Graph:
Intersections with the x-axis: The graph has three x-intercepts (assuming the numbers 1, 2, 3, 4, 5, 6, 7 represent positions on the x-axis).
Behavior at the x-intercepts: We need to determine how the graph behaves at each x-intercept (whether it crosses or touches the x-axis).
General Form of Polynomial Functions:
Polynomials are expressed in the form:
(
)
=
(
−
1
)
1
(
−
2
)
2
⋯
(
−
)
f(x)=k(x−r
1
)
m
1
(x−r
2
)
m
2
⋯(x−r
n
)
m
n
where
r
i
are the roots (x-intercepts), and
m
i
are the multiplicities of those roots.
Multiplicity and Behavior at the Roots:
Odd multiplicity: The graph crosses the x-axis.
Even multiplicity: The graph touches the x-axis and turns around.
Given Options:
Let's analyze each option:
(
)
=
(
−
)
3
(
−
)
3
f(x)=x(x−a)
3
(x−b)
3
:
Roots at
=
0
x=0,
=
x=a,
=
x=b.
Multiplicity:
1
1 at
=
0
x=0 (crosses),
3
3 at
=
x=a (crosses),
3
3 at
=
x=b (crosses).
(
)
=
(
−
)
2
(
−
)
4
f(x)=(x−a)
2
(x−b)
4
:
Roots at
=
x=a,
=
x=b.
Multiplicity:
2
2 at
=
x=a (touches),
4
4 at
=
x=b (touches).
(
)
=
(
−
)
6
(
−
)
2
f(x)=x(x−a)
6
(x−b)
2
:
Roots at
=
0
x=0,
=
x=a,
=
x=b.
Multiplicity:
1
1 at
=
0
x=0 (crosses),
6
6 at
=
x=a (touches),
2
2 at
=
x=b (touches).
(
)
=
(
−
)
5
(
−
)
f(x)=(x−a)
5
(x−b):
Roots at
=
x=a,
=
x=b.
Multiplicity:
5
5 at
=
x=a (crosses),
1
1 at
=
x=b (crosses).
Matching the Options to the Graph:
Given that the graph shows three distinct x-intercepts and no points where the graph only touches the x-axis (implying no even multiplicities), we can deduce the following:
The correct option should have three distinct roots with odd multiplicities.
Conclusion:
Among the given options,
(
)
=
(
−
)
3
(
−
)
3
f(x)=x(x−a)
3
(x−b)
3
fits the criteria as it has:
Three distinct x-intercepts.
Odd multiplicities (crosses the x-axis at all roots).
Thus, the function that could represent the given graph is:
(
)
=
(
−
)
3
(
−
)
3
f(x)=x(x−a)
3
(x−b)
3
To determine which function could represent the given graph based on the polynomial characteristics, let's analyze the graph first:
Given graph:
markdown
Copy code
123
4
567
Analysis of the Graph:
Intersections with the x-axis: The graph has three x-intercepts (assuming the numbers 1, 2, 3, 4, 5, 6, 7 represent positions on the x-axis).
Behavior at the x-intercepts: We need to determine how the graph behaves at each x-intercept (whether it crosses or touches the x-axis).
General Form of Polynomial Functions:
Polynomials are expressed in the form:
(
)
=
(
−
1
)
1
(
−
2
)
2
⋯
(
−
)
f(x)=k(x−r
1
)
m
1
(x−r
2
)
m
2
⋯(x−r
n
)
m
n
where
r
i
are the roots (x-intercepts), and
m
i
are the multiplicities of those roots.
Multiplicity and Behavior at the Roots:
Odd multiplicity: The graph crosses the x-axis.
Even multiplicity: The graph touches the x-axis and turns around.
Given Options:
Let's analyze each option:
(
)
=
(
−
)
3
(
−
)
3
f(x)=x(x−a)
3
(x−b)
3
:
Roots at
=
0
x=0,
=
x=a,
=
x=b.
Multiplicity:
1
1 at
=
0
x=0 (crosses),
3
3 at
=
x=a (crosses),
3
3 at
=
x=b (crosses).
(
)
=
(
−
)
2
(
−
)
4
f(x)=(x−a)
2
(x−b)
4
:
Roots at
=
x=a,
=
x=b.
Multiplicity:
2
2 at
=
x=a (touches),
4
4 at
=
x=b (touches).
(
)
=
(
−
)
6
(
−
)
2
f(x)=x(x−a)
6
(x−b)
2
:
Roots at
=
0
x=0,
=
x=a,
=
x=b.
Multiplicity:
1
1 at
=
0
x=0 (crosses),
6
6 at
=
x=a (touches),
2
2 at
=
x=b (touches).
(
)
=
(
−
)
5
(
−
)
f(x)=(x−a)
5
(x−b):
Roots at
=
x=a,
=
x=b.
Multiplicity:
5
5 at
=
x=a (crosses),
1
1 at
=
x=b (crosses).
Matching the Options to the Graph:
Given that the graph shows three distinct x-intercepts and no points where the graph only touches the x-axis (implying no even multiplicities), we can deduce the following:
The correct option should have three distinct roots with odd multiplicities.
Conclusion:
Among the given options,
(
)
=
(
−
)
3
(
−
)
3
f(x)=x(x−a)
3
(x−b)
3
fits the criteria as it has:
Three distinct x-intercepts.
Odd multiplicities (crosses the x-axis at all roots).
Thus, the function that could represent the given graph is:
(
)
=
(
−
)
3
(
−
)
3
f(x)=x(x−a)
3
(x−b)
3
