IDNLearn.com is committed to providing high-quality answers to your questions. Get the information you need from our community of experts, who provide detailed and trustworthy answers.
Sagot :
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
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover the answers you need at IDNLearn.com. Thanks for visiting, and come back soon for more valuable insights.