-
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
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what are the types of wave
Maurice
fine, how about you?
Mohammed
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
Who can show me the full solution in this problem?
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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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