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# Rational functions, polynomials

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Explain what makes a function a polynomial. Give an example of a function that is a polynomial and a function that is not a polynomial.

Given the polynomial f(x) = 2x3 + 5x2 - x - 5, answer the following questions:

Graph the polynomial. What is the degree of the polynomial?
Explain the constant in the polynomial and how it relates to the graph.
Define the x-intercepts of the polynomial.
Explain the relative maximum and relative minimum of the polynomial in the interval [-3,2]. Is there a different between the relative maximum and minimum and the absolute maximum and minimum?
Part II:

What is a rational function? Give an example. For your example, answer or perform the following:

Graph the rational function.
Give the domain and range of the rational function.
Find the vertical and horizontal asymptotes of the rational function.

https://brainmass.com/math/graphs-and-functions/rational-functions-polynomials-253851

## SOLUTION This solution is FREE courtesy of BrainMass!

A function f(x) is a polynomial if it has the form

a_0 + a_1 x + a_2 x^2 + ... + a_n x^n

where a_0, a_1, a_2, .., a_n are numbers.

Given the polynomial f(x) = 2x^3 + 5x^2 - x - 5

(a) visit

http://www.walterzorn.com/grapher/grapher_e.htm

and enter 2x^3 + 5x^2 - x - 5

in the box

(b) the polynomial has degree 3, because 3 is the largest exponent on x

(c) the number at the end (also referred to as the constant coefficient of the polynomial) is precisely the y-intercept
of the graph (namely, where the graph intersects the y-axis)

(d) the x-intercepts are the points at which the graph (function) crosses the x-axis

(e) the relative maximum (resp. minimum) points on some interval are points in this interval whose y-values
are greatest (resp. smallest)

(f) a function can have no absolute max/min points, but it may well have relative max/min points; as an example, consider the straight line f(x) = x + 1

A rational function is a quotient of two polynomial, that is R(x) is rational if R(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials, and Q(x) is nonzero for all x.

An example of a rational function would be 1/(x - 2); another example: (x - 2)/(x - 3)

Go to the website above and enter (x - 2)/(x - 3)
to graph this function.

(a) This function is defined for all x for which the denominator is not zero, i.e. Domain = {x : x =/= 3} (where =/= means not equal to).

(b) Range = {x : -oo < x < 1} U {x : 1 < x < oo}; that is, all real numbers except 1, i.e. R - {1}.

(c) It has a vertical asymptote at x = 3; it has a horizontal asymptote at y = 1.

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