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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!
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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

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Mark Reply
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Miranda
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Miranda Drice
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
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Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
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What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
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something seems to wispher me to my ear that u are good at it. lol
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Jeffrey
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Miranda
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Miranda
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Jeffrey
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Jeffrey
I'm going to 11grade
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
Nelson Reply
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Miranda
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Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
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Nokwanda
C'est comment
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Hi
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SORIE
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
Practice Key Terms 4

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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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