Polynomial and Rational Functions … · quadratic function in the form y = a(x — h)2 + k zGraph...

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 function

If a > 0, the parabola opens up

If a < 0, the parabola opens down

Vertex: 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-intercepts

Negative discriminant: Zero x-intercepts

Zero discriminant: One x-intercept

(Vertex lies on x-axis)

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


Example. (cont.)

(c) Problem: Graph the function

Answer:


Example. (cont.)

(d) Problem: Determine the domain and range of f.

Answer:

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

Answer:


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


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


Method 2 for GraphingDetermine 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.


Method 2 for Graphing

Find an additional point

Use the y-intercept and axis of symmetry.

Plot the points and draw the graph


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


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 Functions

Parabolas



Quadratic Relations

Polynomial Functions and Models

Section 3.2

Polynomial Functions

Polynomial function: Function of the form

f(x) = anxn + an —1x

n —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


Degrees: Zero function: undefined degree

Constant functions: degree 0.

(Non-constant) linear functions: degree 1.

Quadratic functions: degree 2.


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:


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 number

n > 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


Example.

Problem: Graph f(x) = x5 + 2

Answer:

Zeros of a Polynomial

Zero or root of a polynomial f:

r a real number for which f(r) = 0

r is an x-intercept of the graph of f.

(x — r) is a factor of f.


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


Repeated or multiple zero or root of f:

Same factor (x — r) appears more than once

Zero of multiplicity m:

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


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


Example. For the polynomial

f(x) = —x3(x — 3)2(x + 2)

(a) Problem: Graph the polynomial

Answer:


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 r

Graph 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 r

Graph crosses x-axis at r

-4 -2 2 4

-40

-20

20

40

Turning Points

Turning 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—1x

n—1 + L + a1x + a0

resembles the graph of the power function

y = anxn

End Behavior

End behavior of:


n—1 + L + 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:


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


Example. (cont.)

(f) Problem: Graph f using a graphing utility

Answer:


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:


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


Power Functions



Multiplicity

Turning Points

End Behavior


Cubic 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 dividend

q(x) is the quotient

g(x) is the divisor

r(x) is the remainder

Remainder Theorem

First-degree divisor

Has form g(x) = x — c

Remainder r(x)

Either the zero polynomial or a polynomial of degree 0,

Either way a number R.

Becomes f(x) = (x — c)q(x) + R

Substitute x = c

Becomes 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 off(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


n—1 + L + 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.


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.


Test values

Test a potential rational zero

Each time a zero is found, repeat on the depressed equation.


Example.

Problem: Find the rational zeros of the polynomial in the last example.

f(x) = 3x3 + 8x2 — 7x — 12

Answer:


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


Example.

Problem: Factor

f(x)=2x5 — 9x4 + 20x3 — 40x2 + 48x —16

Answer:

Bounds on Zeros

Bound on the zeros of a polynomial

Positive 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 + L + a1x + a0

A bound M on the zeros of f is the smaller of the two numbers

Max{1, |a0| + |a1| + L + |an-1|}, 1 + Max{|a0| ,|a1| , … , |an-1|}

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.


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 Algorithm

Remainder Theorem

Factor Theorem

Number of Real Zeros




Bounds on Zeros


Complex Zeros; Fundamental Theorem of Algebra

Section 3.7

Complex Polynomial Functions

Complex polynomial function: Function of the form

f(x) = anxn + an —1x

n —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 + bi

Number a — bi

Written

Properties:

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.


Theorem. Every complex polynomial function f(x) of degree n ≥ 1 can be factored into n linear factors (not necessarily distinct) of the form

f(x) = an(x — r1)(x — r2) L (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.


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:


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:


Example.

Problem: Find the complex zeros of the polynomial function

f(x) = x4 + 2x3 + x2 — 8x — 20

Answer:

Key Points

Complex Polynomial Functions

Complex Arithmetic



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 factors

x-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


As x approaches 0, is unbounded in the positivedirection.

Write f(x) → ∞

Read “f(x) approaches infinity”Also:

May write f(x) → ∞ as x → 0

May read: “f(x) approaches infinity as xapproaches 0”

-6 -4 -2 2 4

-4

-2

2

4


Example. For

Problem: Use transformations to graph f.

Answer:

Asymptotes

Horizontal asymptotes:

Let R denote a function.

Let x → —∞ or as x → ∞,

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 → c

If the values |R(x)| → ∞, 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 asymptote

0 horizontal and 1 oblique asymptotes

0 horizontal and 0 oblique asymptotes

There 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.


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

May be written

where is proper. (Long division!)


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 Rapproaches the graph of f, and there are no horizontal or oblique asymptotes.

This is all higher-degree polynomials


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

(a) Problem:

Answer:

(b) Problem:

Answer:


Example. (cont.)

(c) Problem:

Answer:

(d) Problem:

Answer:

Key Points

Rational Functions


Vertical 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.


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


Example.

Problem: Analyze the graph of the rational function

Answer:

Domain:

R in lowest terms:

x-intercepts:

y-intercept:

Symmetry:


Example. (cont.)

Answer: (cont.)


Horizontal asymptote:

Oblique asymptote:

-4 -2 2 4

-4

-2

2

4


Example. (cont.)

Answer: (cont.)


Example.

Problem: Analyze the graph of the rational function

Answer:

Domain:

R in lowest terms:

x-intercepts:

y-intercept:

Symmetry:


Example. (cont.)

Answer: (cont.)


Horizontal asymptote:

Oblique asymptote:

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6


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


Variation

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: Zero

Should have one of following forms

f(x) > 0

f(x) ≥ 0

f(x) < 0

f(x) ≤ 0


Determine where left side is 0 or undefined.

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


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.


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.


Example.

Problem: Solve the inequality x5 ≥ 16x

Answer:

Key Points


Polynomial and Rational Functions … · quadratic function in the form y = a(x — h)2 + k zGraph...

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