TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:
Polynomial and Rational Functions Chapter 3 TexPoint fonts used in EMF. Read the TexPoint manual...
-
Upload
elisabeth-cummings -
Category
Documents
-
view
232 -
download
3
Transcript of Polynomial and Rational Functions Chapter 3 TexPoint fonts used in EMF. Read the TexPoint manual...
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