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Comparing smooth and continuous graphs

The degree of a polynomial function helps us to determine the number of x -intercepts and the number of turning points. A polynomial function of n th degree is the product of n factors, so it will have at most n roots or zeros, or x -intercepts. The graph of the polynomial function of degree n must have at most n 1 turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

A continuous function    has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve    is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

Intercepts and turning points of polynomials

A polynomial of degree n will have, at most, n x -intercepts and n 1 turning points.

Determining the number of intercepts and turning points of a polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of x -intercepts and turning points for f ( x ) = 3 x 10 + 4 x 7 x 4 + 2 x 3 .

The polynomial has a degree of 10 , so there are at most 10 x -intercepts and at most 9 turning points.

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Without graphing the function, determine the maximum number of x -intercepts and turning points for f ( x ) = 108 13 x 9 8 x 4 + 14 x 12 + 2 x 3 .

There are at most 12 x - intercepts and at most 11 turning points.

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Drawing conclusions about a polynomial function from the graph

What can we conclude about the polynomial represented by the graph shown in [link] based on its intercepts and turning points?

Graph of an even-degree polynomial.

The end behavior of the graph tells us this is the graph of an even-degree polynomial. See [link] .

Graph of an even-degree polynomial that denotes the turning points and intercepts.

The graph has 2 x -intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.

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What can we conclude about the polynomial represented by the graph shown in [link] based on its intercepts and turning points?

Graph of an odd-degree polynomial.

The end behavior indicates an odd-degree polynomial function; there are 3 x - intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.

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Drawing conclusions about a polynomial function from the factors

Given the function f ( x ) = 4 x ( x + 3 ) ( x 4 ) , determine the local behavior.

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

f ( 0 ) = 4 ( 0 ) ( 0 + 3 ) ( 0 4 = 0

The y -intercept is ( 0 , 0 ) .

The x -intercepts are found by determining the zeros of the function.

0 = −4 x ( x + 3 ) ( x 4 )
x = 0 or x + 3 = 0 or x 4 = 0 x = 0 or x = 3 or x = 4

The x -intercepts are ( 0 , 0 ) , ( –3 , 0 ) , and ( 4 , 0 ) .

The degree is 3 so the graph has at most 2 turning points.

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Given the function f ( x ) = 0.2 ( x 2 ) ( x + 1 ) ( x 5 ) , determine the local behavior.

The x - intercepts are ( 2 , 0 ) , ( 1 , 0 ) , and ( 5 , 0 ) , the y- intercept is ( 0 , 2 ) , and the graph has at most 2 turning points.

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Questions & Answers

bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
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Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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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
hello
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
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
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
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
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Miranda
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Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
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
which equation
Miranda
I dont know. lol
Jeffrey
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Miranda
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Jeffrey

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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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