Chapter 3 – Solving Equations



Section 4.1: Quadratic Functions

Definition: A polynomial function has the form [pic] (page 326)

The coefficients ai are real numbers, n is a whole number. The domain of any polynomial is [pic].

Reminder:

1. [pic] is a constant function[pic].

2. [pic] is a linear function [pic]

3. Definition: The quadratic function [pic] can be written in the general form [pic].

Graphing Quadratic Functions

Example: Compare[pic], [pic], [pic], [pic].

a) Graphs with positive leading coefficients open up, and have a lowest point.

b) Graphs with negative leading coefficients open down and have a highest point.

c) The highest or lowest point on a parabola is the vertex (in these examples [pic]).

d) The vertical line passing through the vertex of a

parabola is the axis of symmetry, in the examples [pic].

Section 4.1: Quadratic Functions

Example: Compare: [pic], [pic], [pic]

Example: Compare: [pic],[pic],[pic]

Example: Compare [pic] and[pic]:

The parabola [pic] opens up, has vertex [pic] and axis of symmetry[pic]; the parabola[pic] opens down, has vertex [pic] and axis of symmetry[pic].

A quadratic function [pic]can be written in graphing form (page 328) [pic]. The graph is a parabola that opens upward if[pic], downward if[pic], has vertex [pic] and line of symmetry [pic].

Follow your author's instructions for graphing on pages 332 - 333.

Page 339 24, 38

Section 4.1: Quadratic Functions

If we use completing he square on the general parabola[pic],

[pic]

[pic]

[pic]

[pic]

Easy method of finding the vertex of the parabola:

The vertex of the parabola [pic] is given by [pic] = [pic].

Page 226 61

Section 4.1 Homework in book 1 – 8, 9, 13, 17, 23, 26, 28, 35, 38, 42, 46, 48, 49, 54, 62, 63

Use Completing the Square on 23, 26, 28, 35, 38

Section 4.2: Polynomial Functions of Higher Degree

Factoring easy quadratic polynomials reminder: Factor [pic]

Definition: A polynomial function has the form [pic] (page 326)

The coefficients ai are real numbers, n is a whole number. The domain of any polynomial is [pic].

Which of these are polynomial functions?

[pic] [pic]

[pic] [pic] [pic]

Use some simple observations to graph polynomials:

I. Polynomial Graph Qualities (See page 344.)

The graph of every polynomial function is

(a) smooth, contains no sharp corners or cusps;

(b) continuous, has no gaps or holes and can be drawn without lifting the pencil from the paper.

Section 4.2: Polynomial Functions of Higher Degree

II. The Leading Term Test (End Behavior)

Power functions, degree n, [pic]:

Example: Graph [pic]

[pic] is “u-shaped” if n is even, and opens upward if [pic]. The domain is[pic];

the range is [pic].

Example: Graph [pic]and [pic].

[pic] is “s-shaped” for n odd, rising on the right, falling on the left when [pic]. Both the domain and range are [pic].

The end behavior of a polynomial refers to “what the graph does beyond the right- and leftmost x-intercepts”.

End Behavior – Leading Coefficient Test (page 352)

As x moves without bound to the left or right, the polynomial function [pic]

has the behavior as the power function [pic]:

If n is odd and [pic] (positive), the graph of [pic] rises right and falls left:

As [pic] and as [pic].

If n is odd and [pic] is (negative), the graph of [pic] falls right and rises left:

As [pic] and as[pic].

Section 4.2: Polynomial Functions of Higher Degree End Behavior–Leading Coefficient Test (continued)

If n is even and [pic] positive,[pic] rises both right

and left: As [pic] and as [pic]

If n is even and [pic] negative,[pic] falls both right and left: As [pic] and as [pic]

III. Zeros (x-intercepts)

At x-intercepts of a graph, the graph either crosses or is tangent to (touches) the x-axis. Between consecutive x-intercepts, the graph is entirely above or entirely below the x-axis. A factored polynomial [pic] gives the x-intercepts [pic].

Example: The polynomial [pic] can be factored as[pic]. Setting each factor to 0, the intercepts are[pic], [pic], and[pic], having multiplicity 1, 1, and 2, respectively.

Definition: If [pic] is a polynomial function and c is a number for which[pic], then c is called a zero of P.

Section 4.2: Polynomial Functions of Higher Degree

Real Zeros of Polynomial Functions, page 259:

The following are equivalent:

1) a is a zero of the function [pic];

2) [pic] is a solution(root) of equation[pic];

3) [pic]is a factor of [pic];

4) [pic] is an x-intercept of [pic].

Even and Odd Multiplicity

A factor of [pic] yields a repeated zero of multiplicity k.

If a is a zero of odd multiplicity, the graph of the function crosses the x-axis at[pic].

If a is a zero of even multiplicity, the graph of the function is tangent to the x-axis at[pic].

Analyzing the Graph of a Polynomial (example 7 page 353)

Wiley Examples Section 4.2: 65, 71, 73,

Section 4.3: Polynomial Division

Example:

How do you check a division problem? [pic]

To check the problem: [pic] or

dividend =divisor ·quotient + remainder

[pic]

Division Algorithm for Polynomials (page 360)

[pic], [pic] is the dividend, [pic]is the divisor, [pic]is the quotient, [pic] is the remainder.

Synthetic Division—shortcut to divide a polynomial by a polynomial of degree one, [pic].

Do the following example twice, using long division and synthetic division. When writing the coefficients, use a zero for a missing term.

Example: [pic]

Example: [pic]

Section 4.4: Real Zeros of a Polynomial

Factoring easy quadratic polynomials reminder: Factor:

[pic]

[pic]

Remainder Theorem (page 367): If [pic] is a polynomial divided by[pic], then the remainder is[pic], [pic].

Example: Show the remainder theorem and synthetic division yield the same remainder when [pic] is divided by[pic].

Example: Show the remainder theorem and synthetic division yield the same remainder when [pic] is divided by[pic].

Factor Theorem (page 367): A polynomial [pic] has a factor [pic] if and only if [pic].

Example: Does[pic]or[pic]divide[pic]evenly?

On calculator, use TABLE.

Section 4.4: Real Zeros of a Polynomial

The Remainder in Synthetic Division Summary:

1. The remainder r, gives the value of f at [pic], that is [pic].

2. If [pic][pic] is a factor of[pic].

3. If [pic] [pic] is an x-intercept of the graph of f.

Page 380 11

Example: Factor [pic]:

[pic]

[pic]

What is the leading coefficient?

What is the constant term?

What are the zeros?

The real zeros of a polynomial are related to the leading coefficient and the constant term. The relationship is indicated in the Rational Zero Test.

Rational Zeros Test (RZT) (page 369)

If [pic] has integer coefficients, every rational zero of P has the form p/q where p divides [pic]evenly and q divides [pic]evenly.

[pic]

Section 4.4: Real Zeros of a Polynomial

Example: List the possible rational zeros of [pic]

Page 380 56, 57

Section 4.5: Complex Zeros, Fundamental Theorem of Algebra

The Fundamental Theorem of Arithmetic: Any number can be factored into primes, not necessarily distinct.

Example: Factor 24 into primes.

The Fundamental Theorem of Algebra (page 382)

Every [pic] polynomial of degree n has at least one zero in the complex number system.

n Zeros Theorem (page 383)

Every [pic] polynomial of degree n can be expressed as a product of n linear factors so has n linear factors, not necessarily distinct.

Look at examples a. through e. on page 383.

Example: Solve [pic]

Complex Zeros Occur in Conjugate Pairs

Let [pic] be a polynomial function that has real coefficients. If [pic] is a zero of f, the conjugate [pic] is also a zero of f (page 383).

Page 388 8, 14, 24, 34, 50

Section 4.6: Rational Functions

Reminders: An equation of a vertical line has the form ________________; an equation of a horizontal line has the form ________________.

Definition: A rational function has the form [pic], [pic]and [pic] are polynomials and [pic].

Example: Graph the rational function [pic] using transformations.

The domain of [pic] is _________________.

Function [pic] is a transformation (shift right 2, up 1) of the graph of[pic].

Note: As [pic]

As [pic]

The line [pic] is a vertical asymptote.

Also note:

As [pic] and

as [pic]

The line [pic] is a horizontal asymptote.

Asymptotes are lines that a function graph approaches but never touches (sometimes horizontal asymptotes are crossed).

Section 4.6: Rational Functions

Diagrams of Vertical Asymptotes (page 392):

[pic] [pic]

[pic] [pic]

Locating Vertical Asymptotes (page 393)

Vertical Asymptotes occur at zeros of the

denominator. See “Note”: If a rational function is not in lowest terms, the common factor x – a results in a hole in the graph.

Section 4.6: Rational Functions

Diagrams of Horizontal Asymptotes (page 392):

[pic] [pic]

[pic] [pic]

Locating Horizontal Asymptotes (page 395) Determine the degrees of the numerator and denominator, n and m respectively.

(a) If [pic], the x-axis, [pic], is a horizontal asymptote.

(b) If [pic], horizontal asymptote [pic].

(c) If [pic], there is no horizontal asymptote (there is another type of asymptote).

Graphing Rational Functions, page 398

Page 405 23, 46, 56

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