-
Home
- Algebra and trigonometry
- Polynomial and rational functions
- Graphs of polynomial functions
Key concepts
- Polynomial functions of degree 2 or more are smooth, continuous functions. See
[link] .
- To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See
[link]
,
[link]
, and
[link] .
- Another way to find the
intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the
axis. See
[link]
.
- The multiplicity of a zero determines how the graph behaves at the
intercepts. See
[link]
.
- The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
- The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
- The end behavior of a polynomial function depends on the leading term.
- The graph of a polynomial function changes direction at its turning points.
- A polynomial function of degree
has at most
turning points. See
[link]
.
- To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most
turning points. See
[link] and
[link]
.
- Graphing a polynomial function helps to estimate local and global extremas. See
[link]
.
- The Intermediate Value Theorem tells us that if
have opposite signs, then there exists at least one value
between
and
for which
See
[link]
.
Section exercises
Verbal
What is the difference between an
intercept and a zero of a polynomial function
The
intercept is where the graph of the function crosses the
axis, and the zero of the function is the input value for which
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Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.
If we evaluate the function at
and at
and the sign of the function value changes, then we know a zero exists between
and
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If the graph of a polynomial just touches the
x -axis and then changes direction, what can we conclude about the factored form of the polynomial?
There will be a factor raised to an even power.
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Algebraic
For the following exercises, find the
or
t -intercepts of the polynomial functions.
For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.
Questions & Answers
define biology infour way
Explain the following terms .
(1) Abiotic factors in an ecosystem
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc
Qasim
passive process of transport of low-molecular weight material according to its concentration gradient
AI-Robot
what is production?
Catherine
how did the oxygen help a human being
how did the nutrition help the plants
Biology is a branch of Natural science which deals/About living Organism.
evolutionary history and relationship of an organism or group of organisms
AI-Robot
cell is the smallest unit of the humanity biologically
Abraham
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Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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