Vertical and Horizontal Asymptotes

The line x = a is a vertical asymptote of the graph of f if f(x) approaches ∞ or f(x) approaches -∞. It corresponds with the zeroes in the denominator of a rational function.

Example: f(x) = 2x^3
                       x^2-1

To find the vertical asymptote, you should set the denominator equal to zero. The denominator can't equal zero, so the vertical asymptote is the numbers that will make the denominator equal zero

x^2-1 = 0                    Factor the equation.
(x-1)(x+1) = 0            
x = 1, -1

There are two vertical asymptotes: x = 1, -1. 

The actual graph will never pass the vertical asymptotes because they restrict the graph's behavior and where it can and can't cross. Therefore, the graph will never touch or pass the vertical asymptote.

However, the graph is allowed to touch and even pass the horizontal asymptote. The horizontal asymptote gives a general idea of the graph's behavior horizontally, but do to the fact that it can sometimes be touched or passed, it is not as specific as the vertical asymptote is about the graph's behavior.

In order to find the horizontal asymptote you must take the ratio of the leading coefficients of the denominator and numerator. Only the leading coefficients matter when determining the horizontal asymptote.

There are three situations depending on the degree of the first term of the denominator and numerator:

Situation #1
If the degree of the numerator and denominator is the same, then the horizontal asymptote is the numerator divided by the denominator. (x --> ∞, f(x) --> 2)

2x^2
x^2 -1 = 2

So in this case the horizontal asymptote is y = 2. Note that the -1 doesn't matter because the numbers with the highest degree will be the largest in comparison to the numbers that follow, making them more dominate and more important when finding the horizontal asymptote.


Situation #2
If the degree of the numerator is lower than that of the denominator, then the horizontal asymptote is y = 0. The horizontal asymptote is zero because as the denominator gets bigger, the numbers in the numerator get closer to zero (x --> ∞, f(x) --> 0), logically making the asymptote zero.

2x^2
x^3-1 = 0

So the horizontal asymptote is y = 0.

Situation #3
If the degree of the numerator is bigger than the degree of the denominator, then the horizontal asymptote is nonexistent. This is because as the numerator approaches infinity, so does the denominator (x --> ∞, f(x) --> ∞). Therefore, there is no point in a horizontal asymptote because there are no horizontal limitations to where the graph can/can't pass or touch.

2x^3
x^2-1 = none

There is no horizontal asymptote.

Helpful site:
http://www.purplemath.com/modules/asymtote.htm

Questions/Muddy points for Chapter 2

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Write a question or concern you currently have regarding any of the material we've covered so far. 

Graphing Rational Functions

The basic formula for a rational function is:

 

Graph the function:

 

Find the x and y intercepts, vertical asymptotes, horizontal asymptotes if there are any.

 

To find the x-intercept you have to plug zero in for y and the only way the function can equal zero is if the numerator equals zero.

 

Your x-intercept equals -5.

 

To get the y-intercept you plug zero into x.





The y-intercept equals -7.5

 

To find the vertical asymptote you have to find out what makes the functions undefinded which is when the denominator equals zero.

The vertical asymptote is 

 

Finding the horizontal asymptote has to do with the highest power factor and there are three different ways of finding it:

 

1. If the numerator's highest power is higher than the denominator's then there is no horizontal asymptote.

2. If the numerator and denominator's highest powers are equal than it is the leading coefficients of the highest power over each other:

  This H.A is 2

 

3. If the denminator's highest power is higher then the numerator than the horizontal asymptote is y=0

 

This problems equation has equal powers so the horizontal asymptote is y=3.

The graph:

 

There is one exception about vertical asymptotes and it is if you plug that number in the top and both the top and bottom then there is a hole in the graph instead of a line.



 

Complex Numbers

Any number of the form a + bi, where a andb are real numbers and i is an imaginary number whose square equals -1.

If a and b are real numbers, the number a+bi is a complex number, and it is said to be written in standard form. If b=0, the number a + bi= a is a real number. If b doesn’t equal 0, the number a + bi is an imaginary number.

Examples:        3+5i

                        6+4i

Adding and Subtracting Complex Numbers

(3-i) + (2+3i)= 3-i+2+3i                Remove Parenthesis

                  = 3+2-i+3i               Group Like terms

                  = 5+2i                     Write in standard form

Multiplying Complex Numbers

(i)(3i)= 3i²                                             Multiply

            =(3)(-1)              i²=-1

            =-3                   Simplify

(2-i)(4+3i)= 8+6i-4i-3i²     Product of binomials

            = 8+6i-4i-3(-1)    i²=-1

            = 8+3+6i-4i        Group like terms

            =11+2i

Complex Conjugate

(a+bi)(a-bi)

Dividing Complex Numbers

2+3i/4-2i= 2-3i(4+2i)/4-2i(4+2i)     Multiply numerator and denominator by conjugate

            = 8+4i+12i+6i²/16-4i²       Expand

            = 8-6+16i/16+4              i²=-1

            =(2+16i)/20                    Simplify

            = 1/10 + 4/5i                  Write in standard form

Applications

Vertical axis of graph is imaginary axis

Horizontal axis of graph is real axis