<< Chapter < Page Chapter >> Page >

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.

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

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.

Got questions? Get instant answers now!

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.

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

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.

Got questions? Get instant answers now!

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.

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

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.

Got questions? Get instant answers now!

Questions & Answers

0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak Reply
tan20?×tan40?×tan80?
Santosh Reply
At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?
Kimberly Reply
-3 and -2
Julberte Reply
tan(?cosA)=cot(?sinA) then prove cos(A-?/4)=1/2?2
Chirag Reply
tan(pi.cosA)=cot(?sinA) then prove cos(A-?/4)=1/2?2
Chirag Reply
sin x(1+tan x)+cos x(1+cot x) = sec x +cosec
Ankit Reply
let p(x)xq
Sophie Reply
To the nearest whole number, what was the initial population in the culture?
Cheyenne Reply
do posible if one line is parallel
Fran Reply
The length is one inch more than the width, which is one inch more than the height. The volume is 268.125 cubic inches.
Vamprincess Reply
Using Earth’s time of 1 year and mean distance of 93 million miles, find the equation relating ?T??T? and ?a.?
James Reply
cos(x-45)°=Sin x ;x=?
Samaresh Reply
10-n ft
Nalin Reply

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