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Given the function f ( x ) = 0.2 ( x 2 ) ( x + 1 ) ( x 5 ) , express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

The leading term is 0.2 x 3 , so it is a degree 3 polynomial. As x approaches positive infinity, f ( x ) increases without bound; as x approaches negative infinity, f ( x ) decreases without bound.

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Identifying local behavior of polynomial functions

In addition to the end behavior of polynomial functions, we are also interested in what happens in the “middle” of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

We are also interested in the intercepts. As with all functions, the y- intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one y- intercept ( 0 , a 0 ) . The x- intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x- intercept. See [link] .

Intercepts and turning points of polynomial functions

A turning point    of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The y- intercept is the point at which the function has an input value of zero. The x -intercepts are the points at which the output value is zero.

Given a polynomial function, determine the intercepts.

  1. Determine the y- intercept by setting x = 0 and finding the corresponding output value.
  2. Determine the x -intercepts by solving for the input values that yield an output value of zero.

Determining the intercepts of a polynomial function

Given the polynomial function f ( x ) = ( x 2 ) ( x + 1 ) ( x 4 ) , written in factored form for your convenience, determine the y - and x -intercepts.

The y- intercept occurs when the input is zero so substitute 0 for x .

f ( 0 ) = ( 0 ) 4 4 ( 0 ) 2 45 = −45

The y- intercept is (0, 8).

The x -intercepts occur when the output is zero.

0 = ( x 2 ) ( x + 1 ) ( x 4 )
x 2 = 0 or x + 1 = 0 or x 4 = 0 x = 2 or x = −1 or x = 4

The x -intercepts are ( 2 , 0 ) , ( 1 , 0 ) , and ( 4 , 0 ) .

We can see these intercepts on the graph of the function shown in [link] .

Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.
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Determining the intercepts of a polynomial function with factoring

Given the polynomial function f ( x ) = x 4 4 x 2 45 , determine the y - and x -intercepts.

The y- intercept occurs when the input is zero.

f ( 0 ) = ( 0 ) 4 4 ( 0 ) 2 45 = −45

The y- intercept is ( 0 , −45 ) .

The x -intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.

f ( x ) = x 4 4 x 2 45 = ( x 2 9 ) ( x 2 + 5 ) = ( x 3 ) ( x + 3 ) ( x 2 + 5 )
0 = ( x 3 ) ( x + 3 ) ( x 2 + 5 )
x 3 = 0 or x + 3 = 0 or x 2 + 5 = 0 x = 3 or x = 3 or ( no real solution)

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

We can see these intercepts on the graph of the function shown in [link] . We can see that the function is even because f ( x ) = f ( x ) .

Graph of f(x)=x^4-4x^2-45, which labels all the intercepts at (-3, 0), (3, 0), and (0, -45).
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Given the polynomial function f ( x ) = 2 x 3 6 x 2 20 x , determine the y - and x -intercepts.

y -intercept ( 0 , 0 ) ; x -intercepts ( 0 , 0 ) , ( 2 , 0 ) , and ( 5 , 0 )

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Questions & Answers

0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
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tan20?×tan40?×tan80?
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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?
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-3 and -2
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tan(?cosA)=cot(?sinA) then prove cos(A-?/4)=1/2?2
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sin x(1+tan x)+cos x(1+cot x) = sec x +cosec
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The length is one inch more than the width, which is one inch more than the height. The volume is 268.125 cubic inches.
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Using Earth’s time of 1 year and mean distance of 93 million miles, find the equation relating ?T??T? and ?a.?
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cos(x-45)°=Sin x ;x=?
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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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