After Reading all the information on this page, please answer questions at the bottom of the page.
A Rational Function is a ratio of two polynomials. It is also known as a Rational Expression. It is called these things because one is being divided by the other like a "ratio." These are all examples of Rational Functions:
The General form of a Rational Function is : f(x)= p(x)/g(x), where both p(x) and g(x) are polynomial functions of some sort, and g(x) is never equal to 0.
Behavior of Rational Functions:
Domain: The DOMAIN of Rational Functions includes everything from negative infinity to positive infinity, EXCEPT where the DENOMINATOR equals zero. This is also where there is a VERTICAL ASYMPTOTE.
Range: The RANGE of Rational Functions includes everything from negative infinity to positive infinity, EXCEPT where there is a HORIZONTAL ASYMPTOTE.
ZEROS: These are the values when the graph goes through a point on the x-axis and when the coordinate makes y=0. In a Rational Function, ZEROS can be found when the NUMERATOR is equal to zero. [example: In f2(x) above, the zero would be found like this:
Y-INTERCEPTS: These are the values when the graph goes through a point on the y-axis and when the coordinate makes x=0. In a Rational Function, the Y-INTERCEPT can be found by substituting x for 0. [example: In f4(x) above, the y-int. would be found like this:
Range: The RANGE of Rational Functions includes everything from negative infinity to positive infinity, EXCEPT where there is a HORIZONTAL ASYMPTOTE.
ZEROS: These are the values when the graph goes through a point on the x-axis and when the coordinate makes y=0. In a Rational Function, ZEROS can be found when the NUMERATOR is equal to zero. [example: In f2(x) above, the zero would be found like this:
Y-INTERCEPTS: These are the values when the graph goes through a point on the y-axis and when the coordinate makes x=0. In a Rational Function, the Y-INTERCEPT can be found by substituting x for 0. [example: In f4(x) above, the y-int. would be found like this:
Asymptotes:
HORIZONTAL ASYMPTOTES: These are the horizontal lines that the functions get closer and closer to on a graph, but never touch. HORIZONTAL ASYMPTOTES of a function can be found when the degree in the DENOMINATOR is greater than the degree in the NUMERATOR or when the degree is the same in both the NUMERATOR and DENOMINATOR.
When Denominator > Numerator: asymptote at y=o, or the x-axis When Numerator=Denominator: asymptote at a number found by taking the quotient of the first two terms in denominator and numerator. [example in f2(x) above, the asymptote would be found like this: |
VERTICAL ASYMPTOTES: These are the vertical lines that the functions get closer and closer to on a graph, but never touch. VERTICAL ASYMPTOTES of a function can be found when the DENOMINATOR is equal to zero. If when solving for the variable and the answer you get is not a real number (sqrt(negative number)), then there is no vertical asymptote. [example in f1(x) above, the asymptote would be found like this:
OBLIQUE/SLANT ASYMPTOTES: These are the slanted lines that the functions get closer and closer to on a graph, but never touch. OBLIQUE or SLANT ASYMPTOTES of a function can be found when the degree in the NUMERATOR is greater then degree of the DENOMINATOR.
To find the asymptote, long division or factoring has to be used.[example in f3(x) above, the asymptote would be found like this: |
The slant asymptote of this function would be y=2x-3.
How Can We Use Rational Functions in the Real World????
Rational Functions are actually used more than you think they are. They are used in architecture when building houses and buildings so that angles, and measurements are correct.
They are used when dealing with loans, such as interest rates, compound rates, annual rates and things of this nature.
They are also used in medicines so that each patient is prescribed the correct dosage of medicine.
I use rational functions without even realizing it every time I manage my time. I think about how much work I can do in a certain time frame, and "work" problems are commonly used with rational functions.
How do Rational Functions Relate to the Common Core?
Questions
1) Find the Domain, Range, y-intercepts and zeros (if any) of the following function: 1/x
2) Find the asymptotes (if any) of the following function: