Polynomial and Rational Functions

Chapter 3

Quadratic Functions and Models

Section 3.1

Quadratic Functions

Quadratic function: Function of the form

f(x) = ax2 + bx + c(a, b and c real numbers, a ≠ 0)

Quadratic Functions

Example. Plot the graphs of

f(x) = x2, g(x) = 3x2 and

-10 -8 -6 -4 -2 2 4 6 8 10

-30

-20

-10

10

20

30

Quadratic Functions

Example. Plot the graphs of

f(x) = {x2, g(x) = {3x2 and

-10 -8 -6 -4 -2 2 4 6 8 10

-30

-20

-10

10

20

30

Parabolas

Parabola: The graph of a quadratic functionIf a > 0, the parabola opens upIf a < 0, the parabola opens

downVertex: highest / lowest point

of a parabola

Parabolas

Axis of symmetry: Vertical line passing through the vertex

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Parabolas

Example. For the function f(x) = {3x2 +12x { 11

(a) Problem: Graph the function Answer:

Parabolas

Example. (cont.) (b) Problem: Find the vertex and

axis of symmetry. Answer:

Parabolas

Locations of vertex and axis of

symmetry:Set

Set

Vertex is at:

Axis of symmetry runs through

vertex

Parabolas

Example. For the parabola defined by

f(x) = 2x2 { 3x + 2

(a) Problem: Without graphing,

locate the vertex.

Answer:

(b) Problem: Does the parabola

open up or down?

Answer:

x-intercepts of a Parabola

For a quadratic function f(x) = ax2 + bx + c: Discriminant is b2 { 4ac.Number of x-intercepts depends on

the discriminant.Positive discriminant: Two x-interceptsNegative discriminant: Zero x-

interceptsZero discriminant: One x-intercept

(Vertex lies on x-axis)

x-intercepts of a Parabola

Graphing Quadratic Functions

Example. For the function f(x) = 2x2 + 8x + 4

(a) Problem: Find the vertex Answer:

(b) Problem: Find the intercepts. Answer:

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Graphing Quadratic Functions

Example. (cont.)(c) Problem: Graph the function

Answer:

Graphing Quadratic Functions

Example. (cont.)(d) Problem: Determine the

domain and range of f. Answer:

(e) Problem: Determine where f is increasing and decreasing.

Answer:

Graphing Quadratic Functions

Example. Problem: Determine the

quadratic function whose vertex is (2, 3) and whose y-intercept is 11.

Answer:

-14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

Graphing Quadratic Functions

Method 1 for Graphing Complete the square in x to

write the quadratic function in the form y = a(x { h)2 + k

Graph the function using transformations

Graphing Quadratic Functions

Method 2 for Graphing Determine the vertex Determine the axis of

symmetry

Determine the y-intercept f(0) Find the discriminant b2 { 4ac.

If b2 { 4ac > 0, two x-intercepts If b2 { 4ac = 0, one x-intercept

(at the vertex) If b2 { 4ac < 0, no x-intercepts.

Graphing Quadratic Functions

Method 2 for Graphing Find an additional point

Use the y-intercept and axis of symmetry.

Plot the points and draw the graph

Graphing Quadratic Functions

Example. For the quadratic function

f(x) = 3x2 { 12x + 7 (a) Problem: Determine whether

f has a maximum or minimum value, then find it.

Answer:

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Graphing Quadratic Functions

Example. (cont.)(b) Problem: Graph f

Answer:

Quadratic Relations

Quadratic Relations

Example. An engineer collects the following data showing the speed s of a Ford Taurus and its average miles per gallon, M.

Quadratic Relations

Speed, s Miles per Gallon, M

30 18

35 20

40 23

40 25

45 25

50 28

55 30

60 29

65 26

65 25

70 25

Quadratic Relations

Example. (cont.)(a) Problem: Draw a scatter

diagram of the data Answer:

Quadratic Relations

Example. (cont.)(b) Problem: Find the quadratic

function of best fit to these data.

Answer:

Quadratic Relations

Example. (cont.)(c) Problem: Use the function to

determine the speed that maximizes miles per gallon.

Answer:

Key Points

Quadratic FunctionsParabolasx-intercepts of a ParabolaGraphing Quadratic

FunctionsQuadratic Relations

Polynomial Functions and Models

Section 3.2

Polynomial Functions

Polynomial function: Function of the form

f(x) = anxn + an {1xn {1 + + a1x + a0

an, an {1, …, a1, a0 real numbers

n is a nonnegative integer (an 0)

Domain is the set of all real numbers Terminology

Leading coefficient: an

Degree: n (largest power) Constant term: a0

Polynomial FunctionsDegrees:

Zero function: undefined degreeConstant functions: degree 0.(Non-constant) linear functions:

degree 1.Quadratic functions: degree 2.

Polynomial Functions Example. Determine which of the

following are polynomial functions? For those that are, find the degree.

(a) Problem: f(x) = 3x + 6x2

Answer: (b) Problem: g(x) = 13x3 + 5 + 9x4

Answer: (c) Problem: h(x) = 14

Answer: (d) Problem:

Answer:

Polynomial Functions

Graph of a polynomial function will be smooth and continuous. Smooth: no sharp corners or cusps. Continuous: no gaps or holes.

Power Functions

Power function of degree n:Function of the form

f(x) = axn

a 0 a real numbern > 0 is an integer.

Power Functions

The graph depends on whether n is even or odd.

Power Functions

Properties of f(x) = axn

Symmetry:If n is even, f is even.

If n is odd, f is odd.

Domain: All real numbers.

Range:If n is even, All nonnegative real

numbers

If n is odd, All real numbers.

Power Functions

Properties of f(x) = axn

Points on graph:If n is even: (0, 0), (1, 1) and ({1, 1)

If n is odd: (0, 0), (1, 1) and ({1, {1)

Shape: As n increases Graph becomes more vertical if |x| > 1

More horizontal near origin

-4 -2 2 4

-4

-2

2

4

Graphing Using Transformations

Example. Problem: Graph f(x) = (x { 1)4 Answer:

-4 -2 2 4

-4

-2

2

4

Graphing Using Transformations

Example. Problem: Graph f(x) = x5 + 2Answer:

Zeros of a Polynomial

Zero or root of a polynomial f:r a real number for which f(r) =

0r is an x-intercept of the graph

of f.(x { r) is a factor of f.

Zeros of a Polynomial

Zeros of a Polynomial

Example.Problem: Find a polynomial of

degree 3 whose zeros are {4, {2 and 3.

Answer:

-10 -5 5 10

-40

-30

-20

-10

10

20

30

40

Zeros of a Polynomial

Repeated or multiple zero or root of f: Same factor (x { r) appears

more than onceZero of multiplicity m:

(x { r)m is a factor of f and (x { r)m+1 isn’t.

Zeros of a Polynomial

Example.Problem: For the polynomial, list

all zeros and their multiplicities.f(x) = {2(x { 2)(x + 1)3(x { 3)4

Answer:

-4 -2 2 4

-40

-20

20

40

Zeros of a Polynomial

Example. For the polynomialf(x) = {x3(x { 3)2(x + 2)

(a) Problem: Graph the polynomial Answer:

Zeros of a Polynomial

Example. (cont.)(b) Problem: Find the zeros and

their multiplicities Answer:

Multiplicity

Role of multiplicity:r a zero of even multiplicity:

f(x) does not change sign at rGraph touches the x-axis at r, but

does not cross

-4 -2 2 4

-40

-20

20

40

Multiplicity

Role of multiplicity:r a zero of odd multiplicity:

f(x) changes sign at rGraph crosses x-axis at r

-4 -2 2 4

-40

-20

20

40

Turning PointsTurning points:

Points where graph changes from increasing to decreasing function or vice versa

Turning points correspond to local extrema.

Theorem. If f is a polynomial function of degree n, then f has at most n { 1 turning points.

End Behavior Theorem. [End Behavior]

For large values of x, either positive or negative, that is, for large |x|, the graph of the polynomial

f(x) = anxn + an{1xn{1 + + a1x + a0

resembles the graph of the power function

y = anxn

End Behavior

End behavior of:f(x) = anxn + an{1xn{1 + + a1x

+ a0

Analyzing Polynomial Graphs

Example. For the polynomial: f(x) =12x3 { 2x4 { 2x5

(a) Problem: Find the degree. Answer:

(b) Problem: Determine the end behavior. (Find the power function that the graph of f resembles for large values of |x|.)

Answer:

Analyzing Polynomial Graphs

Example. (cont.)(c) Problem: Find the x-

intercept(s), if any Answer:

(d) Problem: Find the y-intercept. Answer:

(e) Problem: Does the graph cross or touch the x-axis at each x-intercept:

Answer:

-4 -2 2 4

-80

-60

-40

-20

20

40

60

80

Analyzing Polynomial Graphs

Example. (cont.)(f) Problem: Graph f using a

graphing utilityAnswer:

Analyzing Polynomial Graphs

Example. (cont.)(g) Problem: Determine the

number of turning points on the graph of f. Approximate the turning points to 2 decimal places.

Answer:

(h) Problem: Find the domain Answer:

Analyzing Polynomial Graphs

Example. (cont.)(i) Problem: Find the range

Answer:(j) Problem: Find where f is

increasing Answer:

(k) Problem: Find where f is decreasing

Answer:

Cubic Relations

Cubic Relations

Example. The following data represent the average number of miles driven (in thousands) annually by vans, pickups, and sports utility vehicles for the years 1993-2001, where x = 1 represents 1993, x = 2 represents 1994, and so on.

Cubic Relations

Year, x Average Miles Driven, M

1993, 1 12.4

1994, 2 12.2

1995, 3 12.0

1996, 4 11.8

1997, 5 12.1

1998, 6 12.2

1999, 7 12.0

2000, 8 11.7

2001, 9 11.1

Cubic Relations

Example. (cont.)(a) Problem: Draw a scatter

diagram of the data using x as the independent variable and M as the dependent variable.

Answer:

Cubic Relations

Example. (cont.)(b) Problem: Find the cubic

function of best fit and graph it

Answer:

Key Points

Polynomial FunctionsPower FunctionsGraphing Using TransformationsZeros of a PolynomialMultiplicityTurning PointsEnd BehaviorAnalyzing Polynomial GraphsCubic Relations

The Real Zeros of a Polynomial Function

Section 3.6

Division Algorithm Theorem. [Division Algorithm]

If f(x) and g(x) denote polynomial functions and if g(x) is a polynomial whose degree is greater than zero, then there are unique polynomial functions q(x) and r(x) such that

where r(x) is either the zero polynomial or a polynomial of degree less than that of g(x).

Division Algorithm

Division algorithm

f(x) is the dividendq(x) is the quotientg(x) is the divisorr(x) is the remainder

Remainder Theorem

First-degree divisorHas form g(x) = x { cRemainder r(x)

Either the zero polynomial or a polynomial of degree 0,

Either way a number R.Becomes f(x) = (x { c)q(x) + RSubstitute x = cBecomes f(c) = R

Remainder Theorem

Theorem. [Remainder Theorem] Let f be a polynomial function. If f(x) is divided by x { c, the remainder is f(c).

Remainder Theorem

Example. Find the remainder if

f(x) = x3 + 3x2 + 2x { 6 is divided by:(a) Problem: x + 2

Answer:(b) Problem: x { 1

Answer:

Factor Theorem

Theorem. [Factor Theorem] Let f be a polynomial function. Then x { c is a factor of f(x) if and only if f(c) = 0.

If f(c) = 0, then x { c is a factor of f(x).

If x { c is a factor of f(x), then f(c) = 0.

Factor Theorem

Example. Determine whether the function

f(x) = {2x3 { x2 + 4x + 3 has the given factor:(a) Problem: x + 1

Answer:(b) Problem: x { 1

Answer:

Number of Real Zeros

Theorem. [Number of Real Zeros]A polynomial function of degree n, n ¸ 1, has at most n real zeros.

Rational Zeros Theorem

Theorem. [Rational Zeros Theorem]Let f be a polynomial function of degree 1 or higher of the form

f(x) = anxn + an{1xn{1 + + a1x + a0

an 0, a0 0, where each coefficient is an integer. If p/q, in lowest terms, is a rational zero of f, then p must be a factor of a0 and q must be a factor of an.

Rational Zeros Theorem

Example. Problem: List the potential

rational zeros of

f(x) = 3x3 + 8x2 { 7x { 12

Answer:

Finding Zeros of a Polynomial

Determine the maximum number of zeros.Degree of the polynomial

If the polynomial has integer coefficients:Use the Rational Zeros Theorem

to find potential rational zeros Using a graphing utility,

graph the function.

Finding Zeros of a Polynomial

Test values Test a potential rational zeroEach time a zero is found,

repeat on the depressed equation.

Finding Zeros of a Polynomial

Example. Problem: Find the rational zeros

of the polynomial in the last example.

f(x) = 3x3 + 8x2 { 7x { 12

Answer:

Finding Zeros of a Polynomial

Example.

Problem: Find the real zeros of

f(x) = 2x4 + 13x3 + 29x2 + 27x +

9

and write f in factored form

Answer:

Factoring Polynomials

Irreducible quadratic: Cannot be factored over the real numbers

Theorem. Every polynomial function (with real coefficients) can be uniquely factored into a product of linear factors and irreducible quadratic factors

Corollary. A polynomial function (with real coefficients) of odd degree has at least one real zero

Factoring Polynomials

Example.

Problem: Factor

f(x)=2x5 { 9x4 + 20x3 { 40x2 +

48x {16

Answer:

Bounds on Zeros

Bound on the zeros of a polynomialPositive number M Every zero lies between {M

and M.

Bounds on Zeros

Theorem. [Bounds on Zeros]Let f denote a polynomial whose leading coefficient is 1.

f(x) = xn + an{1xn{1 + + a1x + a0

A bound M on the zeros of f is the smaller of the two numbersMax{1, ja0j + ja1j + + jan-1j},

1 + Max{ja0j ,ja1j , … , jan-1j}

Bounds on Zeros

Example. Find a bound to the zeros of each polynomial.(a) Problem:

f(x) = x5 + 6x3 { 7x2 + 8x { 10 Answer:

(b) Problem: g(x) = 3x5 { 4x4 + 2x3 + x2 +5

Answer:

Intermediate Value Theorem

Theorem. [Intermediate Value Theorem]Let f denote a continuous function. If a < b and if f(a) and f(b) are of opposite sign, then f has at least one zero between a and b.

Intermediate Value Theorem

Example. Problem: Show that f(x) = x5 { x4 + 7x3 { 7x2 { 18x +

18 has a zero between 1.4 and 1.5. Approximate it to two decimal places.

Answer:

Key Points

Division AlgorithmRemainder TheoremFactor TheoremNumber of Real ZerosRational Zeros TheoremFinding Zeros of a PolynomialFactoring PolynomialsBounds on Zeros Intermediate Value Theorem

Complex Zeros; Fundamental Theorem of Algebra

Section 3.7

Complex Polynomial Functions

Complex polynomial function: Function of the formf(x) = anxn + an {1xn {1 + + a1x + a0

an, an {1, …, a1, a0 are all complex numbers,

an 0, n is a nonnegative integer x is a complex variable. Leading coefficient of f: an

Complex zero: A complex number r with f(r) = 0.

Complex Arithmetic

See Appendix A.6. Imaginary unit: Number i with

i2 = {1. Complex number: Number of the

form z = a + bi a and b real numbers. a is the real part of z b is the imaginary part of z

Can add, subtract, multiply Can also divide (we won’t)

Complex Arithmetic

Conjugate of the complex number a + biNumber a { biWrittenProperties:

Complex Arithmetic

Example. Suppose z = 5 + 2i and w = 2 { 3i. (a) Problem: Find z + w

Answer:(b) Problem: Find z { w

Answer:(c) Problem: Find zw

Answer:(d) Problem: Find

Answer:

Fundamental Theorem of Algebra

Theorem. [Fundamental Theorem of Algebra]Every complex polynomial function f(x) of degree n ¸ 1 has at least one complex zero.

Fundamental Theorem of Algebra

Theorem. Every complex polynomial function f(x) of degree n ¸ 1 can be factored into n linear factors (not necessarily distinct) of the formf(x) = an(x { r1)(x { r2) (x { rn)

where an, r1, r2, …, rn are complex numbers. That is, every complex polynomial function f(x) of degree n ¸ 1 has exactly n (not necessarily distinct) zeros.

Conjugate Pairs Theorem

Theorem. [Conjugate Pairs

Theorem]

Let f(x) be a polynomial

whose coefficients are real

numbers. If a + bi is a zero of

f, then the complex conjugate

a { bi is also a zero of f.

Conjugate Pairs Theorem

Example. A polynomial of degree 5 whose coefficients are real numbers has the zeros {2, {3i and 2 + 4i. Problem: Find the remaining

two zeros.Answer:

Conjugate Pairs Theorem

Example. Problem: Find a polynomial f of

degree 4 whose coefficients are real numbers and that has the zeros {2, 1 and 4 + i.

Answer:

Conjugate Pairs Theorem

Example. Problem: Find the complex zeros

of the polynomial function f(x) = x4 + 2x3 + x2 { 8x { 20

Answer:

Key Points

Complex Polynomial Functions

Complex ArithmeticFundamental Theorem of

AlgebraConjugate Pairs Theorem

Properties of Rational Functions

Section 3.3

Rational Functions

Rational function: Function of the form

p and q are polynomials,q is not the zero polynomial.

Domain: Set of all real numbers except where q(x) = 0

Rational Functions

is in lowest terms:The polynomials p and q have

no common factorsx-intercepts of R:

Zeros of the numerator p when R is in lowest terms

Rational Functions

Example. For the rational function

(a) Problem: Find the domain

Answer:

(b) Problem: Find the x-intercepts

Answer:

(c) Problem: Find the y-intercepts

Answer:

Graphing Rational Functions

Graph of

-10 -5 5 10

-10

-7.5

-5

-2.5

2.5

5

7.5

10

Graphing Rational Functions

As x approaches 0, is unbounded in the positive direction. Write f(x) ! 1Read “f(x) approaches infinity”Also:

May write f(x) ! 1 as x ! 0May read: “f(x) approaches

infinity as x approaches 0”

-6 -4 -2 2 4

-4

-2

2

4

Graphing Rational Functions

Example. ForProblem: Use transformations to

graph f.Answer:

Asymptotes

Horizontal asymptotes:Let R denote a function.Let x ! {1 or as x ! 1, If the values of R(x) approach

some fixed number L, then the line y = L is a horizontal asymptote of the graph of R.

Asymptotes

Vertical asymptotes:Let x ! cIf the values jR(x)j ! 1, then the

line x = c is a vertical asymptote of the graph of R.

Asymptotes

Asymptotes:Oblique asymptote: Neither

horizontal nor vertical Graphs and asymptotes:

Graph of R never intersects a vertical asymptote.

Graph of R can intersect a horizontal or oblique asymptote (but doesn’t have to)

Asymptotes

A rational function can have:Any number of vertical

asymptotes.1 horizontal and 0 oblique

asymptote0 horizontal and 1 oblique

asymptotes0 horizontal and 0 oblique

asymptotesThere are no other possibilities

Vertical Asymptotes

Theorem. [Locating Vertical Asymptotes]

A rational function

in lowest terms, will have a

vertical asymptote x = r if r

is a real zero of the

denominator q.

Vertical Asymptotes

Example. Find the vertical asymptotes, if any, of the graph of each rational function.

(a) Problem:

Answer:

(b) Problem:

Answer:

Vertical Asymptotes

Example. (cont.)

(c) Problem:

Answer:

(d) Problem:

Answer:

Horizontal and Oblique Asymptotes

Describe the end behavior of a rational function.

Proper rational function: Degree of the numerator is less

than the degree of the denominator.

Theorem. If a rational function R(x) is proper, then y = 0 is a horizontal asymptote of its graph.

Horizontal and Oblique Asymptotes

Improper rational function R(x): one that is not proper.May be written

where is proper. (Long division!)

Horizontal and Oblique Asymptotes

If f(x) = b, (a constant)Line y = b is a horizontal

asymptote If f(x) = ax + b, a 0,

Line y = ax + b is an oblique asymptote

In all other cases, the graph of R approaches the graph of f, and there are no horizontal or oblique asymptotes.This is all higher-degree

polynomials

Horizontal and Oblique Asymptotes

Example. Find the hoizontal or oblique asymptotes, if any, of the graph of each rational function.

(a) Problem:

Answer:

(b) Problem:

Answer:

Horizontal and Oblique Asymptotes

Example. (cont.)

(c) Problem:

Answer:

(d) Problem:

Answer:

Key Points

Rational FunctionsGraphing Rational FunctionsVertical AsymptotesHorizontal and Oblique

Asymptotes

The Graph of a Rational Function; Inverse and Joint Variation

Section 3.4

Analyzing Rational Functions

Find the domain of the rational function.

Write R in lowest terms.Locate the intercepts of the

graph.x-intercepts: Zeros of numerator of

function in lowest terms.y-intercept: R(0), if 0 is in the

domain.Test for symmetry – Even, odd or

neither.

Analyzing Rational Functions

Locate the vertical asymptotes: Zeros of denominator of function

in lowest terms. Locate horizontal or oblique

asymptotes Graph R using a graphing

utility. Use the results obtained to

graph by hand

Analyzing Rational Functions

Example. Problem: Analyze the graph of

the rational functionAnswer:

Domain: R in lowest terms:x-intercepts:y-intercept: Symmetry:

Analyzing Rational Functions

Example. (cont.)Answer: (cont.)

Vertical asymptotes: Horizontal asymptote:Oblique asymptote:

-4 -2 2 4

-4

-2

2

4

Analyzing Rational Functions

Example. (cont.)Answer: (cont.)

Analyzing Rational Functions

Example. Problem: Analyze the graph of

the rational functionAnswer:

Domain:R in lowest terms:x-intercepts:y-intercept:Symmetry:

Analyzing Rational Functions

Example. (cont.)Answer: (cont.)

Vertical asymptotes:

Horizontal asymptote:Oblique asymptote:

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Analyzing Rational Functions

Example. (cont.)Answer: (cont.)

Variation

Inverse variation:Let x and y denote 2

quantities. y varies inversely with x

If there is a nonzero constant such that

Also say: y is inversely proportional to x

Variation

Joint or Combined Variation:Variable quantity Q

proportional to the product of two or more other variables

Say Q varies jointly with these quantities.

Combinations of direct and/or inverse variation are combined variation.

Variation

Example. Boyle’s law states that for a fixed amount of gas kept at a fixed temperature, the pressure P and volume V are inversely proportional (while one increases, the other decreases).  

Variation

Example. According to Newton, the gravitational force between two objects varies jointly with the masses m1 and m2 of each object and inversely with the square of the distance r between the objects, hence

Key Points

Analyzing Rational FunctionsVariation

Polynomial and Rational Inequalities

Section 3.5

Solving Inequalities Algebraically

Rewrite the inequality Left side: Polynomial or rational

expression f. (Write rational expression as a single quotient)

Right side: ZeroShould have one of following

formsf(x) > 0f(x) ¸ 0f(x) < 0f(x) · 0

Solving Inequalities Algebraically

Determine where left side is 0 or undefined.

Separate the real line into intervals based on answers to previous step.

Solving Inequalities Algebraically

Test Points:Select a number in each interval Evaluate f at that number.If the value of f is positive, then

f(x) > 0 for all numbers x in the interval.

If the value of f is negative, then f(x) < 0 for all numbers x in

the interval.

Solving Inequalities Algebraically

Test Points (cont.) If the inequality is strict (< or >)

Don’t include values where x = 0 Don’t include values where x is

undefined.If the inequality is not strict (·

or ¸)Include values where x = 0 Don’t include values where x is

undefined.

Solving Inequalities Algebraically

Example. Problem: Solve the inequality x5

¸ 16xAnswer:

Key Points

Solving Inequalities Algebraically