<< Chapter < Page Chapter >> Page >

Sketching the graph of a polynomial function

Sketch a graph of f ( x ) = −2 ( x + 3 ) 2 ( x 5 ) .

This graph has two x -intercepts. At x = −3 , the factor is squared, indicating a multiplicity of 2. The graph will bounce at this x -intercept. At x = 5 , the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.

The y -intercept is found by evaluating f ( 0 ) .

f ( 0 ) = −2 ( 0 + 3 ) 2 ( 0 5 ) = −2 9 ( −5 ) = 90

The y -intercept is ( 0 , 90 ) .

Additionally, we can see the leading term, if this polynomial were multiplied out, would be 2 x 3 , so the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity. See [link] .

Showing the distribution for the leading term.

To sketch this, we consider that:

  • As x the function f ( x ) , so we know the graph starts in the second quadrant and is decreasing toward the x - axis.
  • Since f ( x ) = −2 ( x + 3 ) 2 ( x 5 ) is not equal to f ( x ) , the graph does not display symmetry.
  • At ( 3 , 0 ) , the graph bounces off of the x -axis, so the function must start increasing.

    At ( 0 , 90 ) , the graph crosses the y -axis at the y -intercept. See [link] .

Graph of the end behavior and intercepts, (-3, 0) and (0, 90), for the function f(x)=-2(x+3)^2(x-5).

Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at ( 5 , 0 ) . See [link] .

Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).

As x the function f ( x ) −∞ , so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.

Using technology, we can create the graph for the polynomial function, shown in [link] , and verify that the resulting graph looks like our sketch in [link] .

Graph of f(x)=-2(x+3)^2(x-5).
The complete graph of the polynomial function f ( x ) = 2 ( x + 3 ) 2 ( x 5 )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Sketch a graph of f ( x ) = 1 4 x ( x 1 ) 4 ( x + 3 ) 3 .

Graph of f(x)=(1/4)x(x-1)^4(x+3)^3.

Got questions? Get instant answers now!

Using the intermediate value theorem

In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x -axis, we can confirm that there is a zero between them. Consider a polynomial function f whose graph is smooth and continuous. The Intermediate Value Theorem    states that for two numbers a and b in the domain of f , if a < b and f ( a ) f ( b ) , then the function f takes on every value between f ( a ) and f ( b ) . (While the theorem is intuitive, the proof is actually quite complicated and requires higher mathematics.) We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function f at x = a lies above the x - axis and another point at x = b lies below the x - axis, there must exist a third point between x = a and x = b where the graph crosses the x - axis. Call this point ( c ,   f ( c ) ) . This means that we are assured there is a solution c where f ( c ) = 0.

In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the x - axis. [link] shows that there is a zero between a and b .

Graph of an odd-degree polynomial function that shows a point f(a) that’s negative, f(b) that’s positive, and f(c) that’s 0.
Using the Intermediate Value Theorem to show there exists a zero.

Intermediate value theorem

Let f be a polynomial function. The Intermediate Value Theorem    states that if f ( a ) and f ( b ) have opposite signs, then there exists at least one value c between a and b for which f ( c ) = 0.

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





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