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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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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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