2.6: Rational Functions and Asymptotes

Section 2.6 is an introduction to rational functions.

The definition of a rational function is: "any function which can be written as the ratio of two polynomial functions."

According to our book, a rational function can be written in the form

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

We learned form Chapter 1 that the domain of a rational function of x includes all real numbers except x-values that make the denominator zero.

The other topic of 2.6 is asymptotes.

The definition of a vertical asymptote is "A vertical line which the graph of the line of a function approaches but never reaches."

The definition of a horizontal asymptote is "A horizontal line which the graph of a function approaches as variable tends to positive or negative infinity. It should be noted that the graph can cross the horizontal asymptote as many times as it likes (as with many oscillating functions). A horizontal asymptote occurs when the limit of a function as the variable approaches either positive or negative infinity is a constant."

Let "f" be the rational function f(x) where

where

N(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ and

D(x) = b_mx^m + b_m-1x^m-1 + ... + b₁x + b₀

Where N(x) and D(x) have no common factors.
1. The graph of "f" has vertical asymptotes at the zeros of D(x).
2. The graph of "f" has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x).
a. If n is less than m, the line y = 0 (x-axis) is a horizontal asymptote.
b. If n = m, the line is a horizontal asymptote.

c. If n is greater than m, the graph of "f" has no horizontal asymptote.

Example: Find the Domain and Asymptotes

1. Find the vertical asymptotes by taking D(x) and setting it equal to zero.

(x - 2)³ = 0
x - 2 = 0
x = 2

therefore a vertical asymptote is the line x = 2

2. Find the horizontal asymptotes
n = 0 and m = 3 so n is less than m, therefore the line y = 0 is the horizontal asymptote.

3. Graph it using a graphing utility.

4. Domain (- ∞, 2) U (2, ∞)

5. Range (- ∞, 0) U (0, ∞)

The x-intercepts are where the graph crosses the x-axis, and the y-intercepts are where the graph crosses the y-axis.

More specifically,

• an x-intercept is a point in the equation where the y-value is zero, and
• a y-intercept is a point in the equation where the x-value is zero.
  

• Find the x- and y-intercepts of 25x² + 4y² = 9

Using the definitions of the intercepts, I will proceed as follows:

x-intercept(s):

y = 0 for the x-intercept(s), so:

25x² + 4y² = 9
25x² + 4(0)² = 9
25x² + 0 = 9
x² =


x = ± ( )




Then the x-intercepts are the points (, 0) and ( , 0)

y-intercept(s):

x = 0 for the y-intercept(s), so:

25x² + 4y² = 9
25(0)² + 4y² = 9
0 + 4y² = 9
y² =


y = ± ( )

Then the y-intercepts are the points (0, ) and (0, )

Just remember: Whichever intercept you're looking for, the other variable gets set to zero.

December 6: Introduction to Functions

Section 1.1 is an introduction to functions. A function, according to our textbook is "... a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs).

Simplified, this means that a function is a relationship between two things in which for every x value (the input) there is only one y value (the output).

For example, the relationship between the side length of a square and its area is a function. The reasoning behind this is that every square that has the same side length will have the same area. (The input, in this case the side length will always have the same single output, in this case- the area)

The relationship between the height of a rectangle and the area of a rectangle, though, is not a function. In this example, the area depends upon the width of the rectangle for the area, not just the height. Let’s say that one rectangle’s height is 3 and the width is 5, then that rectangle’s area is 15. If the second rectangle’s height is still 3 but it has a width of 7, then its area will be 21 not 15. That means that the input (3), has more than just one output.

However, a function is still a function if an output has more than one input. For example, the set of points: {(-5,2), (-2, 1), (0,0), (3,7), (4,2)} is a function even though the -5, and 4 both result in 2.

A set of points that is not a function is: {(-5,2), (-2, 1), (0,0), (3,7), (4,2), (3,-3)} because one element in the domain, 3, corresponds with more than one range 7 & -3.

Today, we also talked about Function Notation or f (x)

An extremely important lesson that we were taught about function notation is this: y = f (x) ---- (x, y). This means that the point (x, y) is on the graph of f.

For example:

f (x)= 2x +1

Then...

f (2)=2(2)+ 1=5

and

f (a)= 2(a) +1

and

f (!)=2(!) +1

* All these examples are assuming that y (the output) is a function of x (the input).