2.6 Rational Functions

A rational function can be written as






where N(x) and D(x) are polynomials and D(x) is not the zero polynomial.



Domain of a rational function of x is all real numbers except x-values that make the denominator zero.


Ex.




Domain:




Horizontal and Vertical Asymptotes

The line y = b is a horizontal asymptote of the graph of f if as or .

Ex 1

The graph of

has the line y = -1 as a horizontal asymptote. The degree of the of the numerator is equal to the degree of the denominator, and the horizontal asymptote is given by the ratio of the leading coefficents of the numerator and denominator.

Ex 2

The graph of

has no horizontal asymptote because the degree of the numerator is greater than the degree of the denominator.

Ex 3

The graph of

has the line y=0 (the x-axis) as a horizontal asymptote. Note that the degree of the numerator is less than the degree of the denominator.

The line x = a is a vertical asymptote of the graph of f if or


as , either from the left or right.


Ex.

The graph of

has a vertical asymptote of 3 because that is where D(x) = 0.



Intercepts


Y-intercept :





X-Intercept(s):

2.4: Complex Numbers

The Complex Number System:

i is an imaginary number

i = √-1, so i² = -1

standard form of a complex number: a + bi

Operations with Complex Numbers:

Addition: add like terms

(3 + 7i) + (9 + 6i)

3 + 9 + 7i + 6i

12 + 13i

Subtraction: subtract like terms

(4 + 5i) - (2-3i)

4-2 + 5i-(-3i)

2 + 8i

Multiplication: distribute

(3 + 2i)(1 + 2i)

3 + 6i + 2i + 2i²

3 + 8i + 2(-1)

1 + 8i

Division: multiply by the complex conjugate

Finding i to any power:

this pattern repeats itself forever and ever

So if you wanted to find i to a really high power, like:

you would need to divide the exponent, 327, by 4 to find what number in the pattern it stops at.

327/4 = 81 with a remainder of 3 (which is really the only part that matters to us)

Since the remainder is 3, the answer will be the same as i to the power of 3, which is -i

so:

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra

If F(x) is a polynomial of degree n, where n>0, f has at last one zero in the complex number system

Linear Factorization Theorem

If f(x) is a polynomial of degree n where n>0, f has precisely n linear factors

f(x)=an(x-c1)(x-c2). . . (x-cn)

where C1, C2......., Cn are complex numbers

Example: Real Zeros of Polynomial Functions

a.) The first degree polynomial F(x)=x-2 has exactly one zero: x=2

b.) Countring multiplicity, the second degree polynomial function

f(x)=x^2 - 6x + 9 = (x - 3)(x -3)

has exactly two zeros: x = 3 and x= 3.

A third-degree polynomial has THREE zeros

A fourth-degree polynomial has FOUR zeros. . . ect.

Example: Finding the Zeros of a Polynomial Function

f(x) = x^5 + x^3 + 2x^2 - 12x + 8

[ (+ -) (1,2,4,8) ] All possible Rational Zeros

Use synthetic division, or look to graph of equation for help.

After using synthetic division you can determine the zeros for the equation

F(x)= (x - 1)(x - 1)(x+ 2)(x^2 + 4)

How to factor x^2 + 4

X^2 - (-4) = (X- √-4) (X- √-4) = (X - 2i) (X - 2i)

All zeros = 1, 1, -2, 2i, and -2i

2.3 Real Zeros of Polynomial Functions-Synthetic and Long Division Blog

In section 2.3 we look at different ways to divide polynomials. The two major ways we do this is by long division of polynomials and synthetic division.

When doing long division of polynomials you treat the problem just as you would when doing a long division problem with simple numbers

Solve the polynomial just as you would this:

Another way to divide polynomials in Synthetic Division!

This is a nice shortcut for long division. It is a pattern. Vertically you must add the terms. And your diagonal patter is to multiply by the factor outside.

In synthetic division if the integers add up to 0, (X-1) is a factor.

The Rational Zero Test
The Rational Zero Test relates the possible rational zeros of a polynomial to the leading coefficient and to the constant term of the polynomial.
 Possible rational zeros=  (factors of constant term)/(factors of leading coefficient)

Chapter 2.2- Polynomial Functions of Higher Degree

Graphs of Polynomial Functions

The graph of a polynomial function is:

• continuous: does not contain breaks, holes, or gaps in the graph

• has only smooth, rounded turns

Note: the graphs cannot have sharp pointed turns







Example of polynomial function:

Note: this graph is continuous- has no breaks, holes or gaps- and has smooth,

rounded turns

Monomials of the form: (Note: n must be an integer greater than zero)

n is even n is odd

When n is even, the graph touches When n is odd, the graph crosses

the axis at the x-intercept. the axis at the x-intercept.

The Leading Coefficient Test

As x moves to the left or right,

the graph rises or falls based on these rules:

When n is odd

• leading coefficient = positive, the graph falls to the left and rises to the right

• leading coefficient = negative, the graph rises to the left and falls to the right

leading coefficient is positive leading coefficient is negative



When n is even

• leading coefficient = positive, the graph rises to the left and right

• leading coefficient = negative, the graph falls to the left and right

Example

Determine the left and right behavior of the graphs:

rises to the left and falls to the right

Why? The degree is odd and the leading coefficient is negative

rises to the left and right

Why? The degree is even and the leading coefficient is positive

Zeros of Polynomial Functions

• 
  
  
  the graph f has at most n real zeros. In other words, the highest degree of a polynomial function determines the number of zeros.
  

• 
  
  the function f has at most x-1 relative extrema.

Note: Extrema are the relative minimums and maximums combined

Zero Rules:

• 
  
  
  x=a is a zero of the function f



• 
  
  
  x=a is a solution of the polynomial equation f(x)=0



• 
  
  
  (x-a) is a factor of the polynomial f(x)



• 
  
  
  (a,0) is an x-intercept of the graph f

Section 2.1: Quadratic Functions

Polynomial functions are classified by degree.

• Constant Function- f(x) = a, a ≠0. This polynomial function has a degree of 0.
  

Example: f(x)=2

• Linear Function- f(x) = mx + b, m≠0. This polynomial function has a degree of 1. The slope of this function is m and y-intercept is (0,b).

• Example: f(x) = x
• Quadratic Function: f(x) = ax² + bx + c. Let a, b, and c be real numbers with a ≠0.

Example: f(x) = x²

• Standard form of a Quadratic Function: f(x) = a(x-h)² + k, a ≠0 where the vertex is ( h, k)

Example: Write a quadratic function in standard form by completing the square.

f(x) = 2x² + 8x + 7
f(x) = 2(x² + 4x) + 7
f(x) = 2(x² + 4x + 4) + 7
f(x) = 2(x² + 4x + 4) - 2(4) + 7
f(x) = 2(x + 2)² -1

• Maximum and Minimum- if a>0, f has a minimum that occurs at
  x = -b/2a.

If a<0, f has a maximum that occurs at x= -b/2a.