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5.3 Graphs of polynomial functions  (Page 4/13)

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Use the graph of the function of degree 5 in [link] to identify the zeros of the function and their multiplicities.

Graph of an even-degree polynomial with degree 6.

The graph has a zero of –5 with multiplicity 1, a zero of –1 with multiplicity 2, and a zero of 3 with even multiplicity.

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Determining end behavior

As we have already learned, the behavior of a graph of a polynomial function    of the form

f ( x ) = a n x n + a n − 1 x n − 1 + ... + a 1 x + a 0

will either ultimately rise or fall as   x   increases without bound and will either rise or fall as   x   decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say –100 or –1,000.

Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function,   a n x n ,   is an even power function, as   x   increases or decreases without bound,   f ( x )   increases without bound. When the leading term is an odd power function, as   x   decreases without bound,   f ( x )   also decreases without bound; as   x   increases without bound,   f ( x )   also increases without bound. If the leading term is negative, it will change the direction of the end behavior. [link] summarizes all four cases.

Graph of a polynomial function with degree 5.

Understanding the relationship between degree and turning points

In addition to the end behavior, recall that we can analyze a polynomial function’s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function   f ( x ) = x 4 − x 3 − 4 x 2 + 4 x   in [link] . The graph has three turning points.

Graph of an odd-degree polynomial with a negative leading coefficient. Note that as x goes to positive infinity, f(x) goes to negative infinity, and as x goes to negative infinity, f(x) goes to positive infinity.

This function   f   is a 4 th degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.

Interpreting turning points

A turning point    is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).

A polynomial of degree   n   will have at most   n − 1   turning points.

Finding the maximum number of turning points using the degree of a polynomial function

Find the maximum number of turning points of each polynomial function.

  1.   f ( x ) = − x 3 + 4 x 5 − 3 x 2 + 1
  2.   f ( x ) = − ( x − 1 ) 2 ( 1 + 2 x 2 )

  1. First, rewrite the polynomial function in descending order:   f ( x ) = 4 x 5 − x 3 − 3 x 2 + 1

    Identify the degree of the polynomial function. This polynomial function is of degree 5.

    The maximum number of turning points is   5 − 1 = 4.

  2. First, identify the leading term of the polynomial function if the function were expanded.

    Graph of f(x)=x^4-x^3-4x^2+4x which denotes where the function increases and decreases and its turning points.

    Then, identify the degree of the polynomial function. This polynomial function is of degree 4.

    The maximum number of turning points is   4 − 1 = 3.

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Graphing polynomial functions

We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.

Given a polynomial function, sketch the graph.

  1. Find the intercepts.
  2. Check for symmetry. If the function is an even function, its graph is symmetrical about the   y - axis, that is,   f ( − x ) = f ( x ) .   If a function is an odd function, its graph is symmetrical about the origin, that is,   f ( − x ) = − f ( x ) .
  3. Use the multiplicities of the zeros to determine the behavior of the polynomial at the   x - intercepts.
  4. Determine the end behavior by examining the leading term.
  5. Use the end behavior and the behavior at the intercepts to sketch a graph.
  6. Ensure that the number of turning points does not exceed one less than the degree of the polynomial.
  7. Optionally, use technology to check the graph.

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
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Haryormhidey Reply
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Felix Reply
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ALIYU
field is a region of space under the influence of some physical properties
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WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
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Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
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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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