2.4 Polynomial and Rational Functions

Polynomial Functions

Given a linear function f(x) = mx + b, we can add a square term, and get a quadratic function g(x) = ax2 + f(x) = ax2 + mx + b. We can continue adding terms of higher degrees, e.g. we can add a cube term and get h(x) = cx3 + g(x) = cx3 + ax2 + mx + b, and so on. f(x), g(x), and h(x) are all special cases of a polynomial function.

Deﬁnition (Polynomial Function) A polynomial function is a function that can be written in the form n n−1 f(x) = anx + an−1x + ... + a1x + a0 for n a nonnegative integer, called the degree of the polynomial. The coeﬃcients an, an−1, . . . , a1, a0 are real numbers with an 6= 0.

Note that although an 6= 0, the remaining coeﬃcients an−1, an−2, . . . , a1, a0 can very well be 0.

Domain of Polynomial Function The domain of a polynomial function is R, the set of all real numbers.

The domain of f(x) = xn is R regardless the value of n (any nonnegative integer), and so is the domain of g(x) = axn, where a is some real number. Clearly, if you add, say k, such functions with diﬀerent degrees (n) the domain of the resulting function will still be R. Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Consider a function f(x) = (x − 1)(x − 2)(x − 3). It could be rewritten as f(x) = (x − 1)(x − 2)(x − 3) = = (x − 1)(x2 − 2x − 3x + 6) = = (x − 1)(x2 − 5x + 6) = = x3 − 5x2 + 6x − x2 + 5x − 6= = x3 − 6x2 + 11x − 6. So, f(x) is a polynomial function of degree 3.

Question: How many y intercepts does f(x) have?

Answer: Only one, y = f(0) = −6. Any function can have at most one y intercept, otherwise it will not pass the vertical line test.

y Intercept of a Polynomial Function n n−1 If f(x) = anx + an−1x + ... + a1x + a0 is a polynomial function, it has exactly one y intercept y = a0.

Question: How many x intercepts does f(x) have?

Answer: f(x) has 3 intercepts. 0 = (x − 1)(x − 2)(x − 3) =⇒ x = 1 or x = 2 or x = 3.

x Intercept of a Polynomial Function A polynomial of degree n can have, at most, n linear factors. Therefore, the graph of a polynomial function of positive degree n can intersect the x axis at most n times. The x intercepts of n n−1 f(x) = anx + an−1x + ... + a1x + a0 could be found by solving n n−1 anx + an−1x + ... + a1x + a0 = 0.

2 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

y 7 6 5 4 f(x) 3 2 1 0 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 x −1 −2 −3 −4 −5 −6 −7

Consider a function h(x) = (x2 + 1)(x − 2)(x − 3).

h(x) = (x2 + 1)(x − 2)(x − 3) = = (x2 + 1)(x2 − 2x − 3x + 6) = = (x2 + 1)(x2 − 5x + 6) = = x4 − 5x3 + 6x2 + x2 − 5x + 6= = x4 − 5x3 + 7x2 − 5x + 6. h(x) is a polynomial function of degree 4, but has just 2 x intercepts, because the equation 0 = (x2 +1)(x−2)(x−3) has just 2 roots (zeros), which are x = 2 and x = 3.

3 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

20y h(x)

−1 0 1 2 3 4 x −4

Note that f(x) = x3 − 6x2 + 11x − 6 has degree 3, which is an odd number. It starts negative, ends positive, and crosses the x axis odd number of times. h(x) = x4 − 5x3 + 7x2 − 5x + 6 has degree 4, which is an even number. It starts positive, ends positive, and cross the x axis even number of times.

Consider m(x) = −f(x) = −(x3 −6x2 +11x−6) = −x3 +6x2 −11x+6, and n(x) = −g(x) = −(x4−5x3+7x2−5x+6) = −x4+5x3−7x2+5x−6.

y y 7 8 5 m(x) 4 n(x) 3 −1 0 1 2 3 4 x 1 −4 x −7 −6 −5 −4 −3 −2 −−11 0 1 2 3 4 5 6 7 −8 −3 −12

−5 −16

−7 −20

4 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Definition (Leading Coefficient) n n−1 Given a polynomial function f(x) = anx +an−1x +...+a1x+a0, the coefficient an of the highest-degree term is called the leading coefficient of a polynomial function f(x).

Graph of a Polynomial Function n n−1 Given a polynomial function f(x) = anx +an−1x +...+a1x+a0:

(a) if an > 0 and n is odd, then the graph of f(x) starts negative, ends positive, and crosses the x axis odd number of times but at least once;

(b) if an < 0 and n is odd, then the graph of f(x) starts positive, ends negative, and crosses the x axis odd number of times but at least once;

(c) if an > 0 and n is even, then the graph of f(x) starts positive, ends positive, and crosses the x axis even number of times or does not cross it at all;

(d) if an < 0 and n is even, then the graph of f(x) starts negative, ends negative, and crosses the x axis even number of times or does not cross it at all.

Note: (c) is a reﬂection in the x axis of (a), and (d) is a reﬂection in the x axis of (b).

Also note that a polynomial function always either increases or decreases without bound as x goes to either negative or positive in- ﬁnity.

5 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Continuity and ”Smoothness” of Polynomial Func- tion

2|x| Consider f(x) = x . f(x) has a discontinuous break at x = 0. y 4 f(x) 2

−5 −3 −1 1 3 5x −2

−4

6 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Consider g(x) = |x| − 2. g(x) is continuous, but not smooth due to a sharp corner at (0, −2). y 4 g(x) 2

−5 −3 −1 1 3 5x −2

−4

2 Consider h(x) = x−1. h(x) has a discontinuous break at x = 1. y 5

3 h(x)

−5 −3 −1 1 3 5x −1

−3

−5

Graph of a Polynomial Function The graph of a polynomial function is continuous, with no holes or breaks. That is, the graph can be drawn without removing a pen from the paper. Also, the graph of a polynomial is ”smooth”, i.e. has no sharp corners.

7 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Rational Functions

Just as rational numbers are deﬁned in terms of quotients of integers, rational functions are deﬁned in terms of quotients of polynomials.

Deﬁnition (Rational Function)

A rational function is any function that can be written in the form n(x) f(x) = , d(x) 6= 0 d(x) where n(x) and n(x) are polynomials.

For example,

1 x − 2 x13 − 8 f(x) = , g(x) = , h(x) = , x x2 − x − 6 x5 p(x) = x4 − 5x3 + 7x2, q(x) = 123, r(x) = 0 are all rational functions.

If n(x) and d(x) are polynomials, then they both have domain R. However,

Domain of a Rational Function

n(x) If f(x) = d(x) is a rational function, then its domain is the set of all real numbers such that d(x) 6= 0.

8 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Example 1 x2+1 Find the domain of f(x) = x2−7x+10

9 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Vertical and Horizontal Asymptotes

Recall that a polynomial function is always continuous and ”smooth”. It is also true that if x increases or or decreases without bound, then function also increases or decreases without bound. However, this may not be true for a rational function. Also, a rational function may not have a y intercept.

x−3 Consider a rational function f(x) = x−2. Its domain (−∞, 2] ∪ [2, ∞), or all real numbers except for x = 2,

x f(x) 1.5−3 −1.5 1.5 1.5−2 = −0.5 = 3 1.75−3 −1.25 1.75 1.75−2 = −0.25 = 5 1.9−3 −1.1 1.9 1.9−2 = −0.1 = 11 1.95−3 −1.05 1.95 1.95−2 = −0.05 = 21 1.99−3 −1.01 1.99 1.99−2 = −0.01 = 101 1.999−3 −1.001 1.999 1.999−2 = −0.001 = 1001 1.9999−3 −1.0001 1.9999 1.9999−2 = −0.0001 = 10001 1.99999−3 −1.00001 1.99999 1.99999−2 = −0.00001 = 100001 2 undeﬁned 2.00001−3 −0.99999 2.00001 2.00001−2 = 0.00001 = −99999 2.0001−3 −0.9999 2.0001 2.0001−2 = 0.0001 = −9999 2.001−3 −0.999 2.001 2.001−2 = 0.001 = −999 2.01−3 −0.99 2.01 2.01−2 = 0.01 = −99 2.05−3 −0.95 2.05 2.05−2 = 0.05 = −19 2.1−3 −0.9 2.1 2.1−2 = 0.1 = −9 2.25−3 −0.75 2.25 2.25−2 = 0.25 = −3 2.5−3 −0.5 2.5 2.5−2 = 0.5 = −1

10 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

x f(x) −100000−3 −100003 -100000 −100000−2 = −100002 = 1.00001 −10000−3 −10003 -10000 −10000−2 = −10002 = 1.0001 −1000−3 −1003 -1000 −1000−2 = −1002 = 1.001 −1000−3 −1003 -1000 −1000−2 = −1002 = 1.001 −10−3 −13 -10 −10−2 = −12 = 1.08333 −5−3 −8 -5 −5−2 = −7 = 1.14286 0−3 −3 0 0−2 = −2 = 1.5 1.5−3 −1.5 1.5 1.5−2 = −0.5 = 3 2.5−3 −0.5 2.5 2.5−2 = 0.5 = −1 3−3 0 3 3−2 = 1 = 0 5−3 2 5 5−2 = 3 = 0.666667 10−3 7 10 10−2 = 8 = 0.875 100−3 97 100 100−2 = 98 = 0.989796 1000−3 997 1000 1000−2 = 998 = 0.998998 10000−3 9997 10000 10000−2 = 9998 = 0.9999 100000−3 99997 100000 100000−2 = 99998 = 0.99999 y x = 2 8 The graph of f(x) gets closer to 6 the line x = 2 as x gets closer to 2. 4 Line x = 2 is a vertical asymp-

2 tote for f(x). y = 1 −8 −6 −4 −2 0 2 4 6 8x The graph of f(x) gets closer to −2 the line y = 1 as x increases or decreases without bound. The line −4 y = 1 is a horizontal asymptote −6 for f(x). −8

11 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Deﬁnition (Vertical Asymptote) A vertical line x = a is called a vertical asymptote for a function f(x) if the graph of f(x) gets closer to the line x = a as x gets closer to a.

n(x) Note: the number of vertical asymptotes of a rational function f(x) = d(x) is at most equal to the degree of d(x).

Deﬁnition (Horizontal Asymptote) A horizontal line y = b is called a horizontal asymptote for a function f(x) if the graph of f(x) gets closer to the line y = b as x gets x increases or decreases without bound.

Note: a rational function has at most one horizontal asymptote. Moreover, the graph of a rational function approaches the horizontal asymptote (when one exists) both as x increases and decreases without bound.

x = −2 y x = 2 y x = 0 5 5 4 4 3 3 2 2 1 1

−5 −3 −1 1 3 5x −5 −3 −1 1 3 5x −1 −1 −2 −2 −3 −3 −4 −4 −5 −5

8 8 1 x2 + 1 f(x) = = f(x) = x + = x2 − 4 (x − 2)(x + 2) x x 12 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Example 2 3x+3 Given the rational function f(x) = x2−9 , (a) Find the domain. (b) Find the x and y intercepts. (c) Find the equations of all vertical asymptotes. (d) If there is a horizontal asymptote, ﬁnd its equation. (e) Using the information from parts (a)-(d) and additional points as necessary, sketch a graph of f for −10 ≤ x ≤ 10.

13 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

y 10 9 8 7 6 5 4 3 2 1

−10−9−8−7−6−5−4−3−2−−11 0 1 2 3 4 5 6 7 8 9 10x −2 −3 −4 −5 −6 −7 −8 −9 −10

14 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

3x2−3x−36 Consider the rational function g(x) = x3−4x2−9x+36.

x = −3 y x = 3 10 9 8 7 6 5 4 3 2 1

−10−9−8−7−6−5−4−3−2−−11 0 1 2 3 4 5 6 7 8 9 10x −2 −3 −4 −5 −6 −7 −8 −9 −10

15 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Example 3 Find the vertical and horizontal asymptotes of the rational function x3−4x f(x) = x2+5x.

16 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

17 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

Applications

Example 4 (Employee Training) A company that manufactures computers has established that, on the average, a new employee can assemble N(t) components per day after t days of on-the-job training, as given by 25t + 5 N(t) = , t ≥ 0 t + 5 Sketch a graph of N, 0 ≤ t ≤ 100, including any vertical or horizontal asymptotes. What does N(t) approach as t increases without bound?

18 Ch 2. Functions and Graphs 2.4 Polynomial and Rational Functions

N(t)