<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

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?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
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
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
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
Krampah Reply
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.
Sahid Reply
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
Samuel Reply
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?
Joseph Reply
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's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
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?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask