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Polynomials

A linear function is a special type of a more general class of functions: polynomials. A polynomial function    is any function that can be written in the form

f ( x ) = a n x n + a n 1 x n 1 + + a 1 x + a 0

for some integer n 0 and constants a n , a n 1 ,…, a 0 , where a n 0 . In the case when n = 0 , we allow for a 0 = 0 ; if a 0 = 0 , the function f ( x ) = 0 is called the zero function . The value n is called the degree    of the polynomial; the constant a n is called the leading coefficient . A linear function of the form f ( x ) = m x + b is a polynomial of degree 1 if m 0 and degree 0 if m = 0 . A polynomial of degree 0 is also called a constant function . A polynomial function of degree 2 is called a quadratic function    . In particular, a quadratic function has the form f ( x ) = a x 2 + b x + c , where a 0 . A polynomial function of degree 3 is called a cubic function    .

Power functions

Some polynomial functions are power functions. A power function    is any function of the form f ( x ) = a x b , where a and b are any real numbers. The exponent in a power function can be any real number, but here we consider the case when the exponent is a positive integer. (We consider other cases later.) If the exponent is a positive integer, then f ( x ) = a x n is a polynomial. If n is even, then f ( x ) = a x n is an even function because f ( x ) = a ( x ) n = a x n if n is even. If n is odd, then f ( x ) = a x n is an odd function because f ( x ) = a ( x ) n = a x n if n is odd ( [link] ).

An image of two graphs. Both graphs have an x axis that runs from -4 to 4 and a y axis that runs from -6 to 7. The first graph is labeled “a” and is of two functions. The first function is “f(x) = x to the 4th”, which is a parabola that decreases until the origin and then increases again after the origin. The second function is “f(x) = x squared”, which is a parabola that decreases until the origin and then increases again after the origin, but increases and decreases at a slower rate than the first function. The second graph is labeled “b” and is of two functions. The first function is “f(x) = x to the 5th”, which is a curved function that increases until the origin, becomes even at the origin, and then increases again after the origin. The second function is “f(x) = x cubed”, which is a curved function that increases until the origin, becomes even at the origin, and then increases again after the origin, but increases at a slower rate than the first function.
(a) For any even integer n , f ( x ) = a x n is an even function. (b) For any odd integer n , f ( x ) = a x n is an odd function.

Behavior at infinity

To determine the behavior of a function f as the inputs approach infinity, we look at the values f ( x ) as the inputs, x , become larger. For some functions, the values of f ( x ) approach a finite number. For example, for the function f ( x ) = 2 + 1 / x , the values 1 / x become closer and closer to zero for all values of x as they get larger and larger. For this function, we say f ( x ) approaches two as x goes to infinity,” and we write f ( x ) 2 as x . The line y = 2 is a horizontal asymptote for the function f ( x ) = 2 + 1 / x because the graph of the function gets closer to the line as x gets larger.

For other functions, the values f ( x ) may not approach a finite number but instead may become larger for all values of x as they get larger. In that case, we say f ( x ) approaches infinity as x approaches infinity,” and we write f ( x ) as x . For example, for the function f ( x ) = 3 x 2 , the outputs f ( x ) become larger as the inputs x get larger. We can conclude that the function f ( x ) = 3 x 2 approaches infinity as x approaches infinity, and we write 3 x 2 as x . The behavior as x and the meaning of f ( x ) as x or x can be defined similarly. We can describe what happens to the values of f ( x ) as x and as x as the end behavior of the function.

To understand the end behavior for polynomial functions, we can focus on quadratic and cubic functions. The behavior for higher-degree polynomials can be analyzed similarly. Consider a quadratic function f ( x ) = a x 2 + b x + c . If a > 0 , the values f ( x ) as x ± . If a < 0 , the values f ( x ) −∞ as x ± . Since the graph of a quadratic function is a parabola, the parabola opens upward if a > 0 ; the parabola opens downward if a < 0 . (See [link] (a).)

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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