<< 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

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
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron 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