matematicas visuales visual math

We have already study the end behavior of several rational functions.

If the degree of the numerator is less than the degree of the numerator, the rational function has a horizontal asymptote y=0;

If the degree of the numerator is equal than the degree of the numerator, the rational function has a horizontal asymptote y=k;

Rational Functions (1): Linear rational functions
Rational functions can be writen as the quotient of two polynomials. Linear rational functions are the simplest of this kind of functions.
Rational Functions (2): degree 2 denominator
When the denominator of a rational function has degree 2 the function can have two, one or none real singularities.

If the degree of the numerator is exactly one more than the degree of the numerator, the rational function has a oblique or slant asymptote.

Rational Functions (3): Oblique Asymptote
For large absolute values of x, some rational functions behave like an oblique straight line, we call this line an oblique or slant asymptote.

If you divide the numerator of a rational function by its denominator you get a polynomial quotient Q(x) plus a remainder term whose degree is less than of the denominator. The remainder, when divided by the denominator, contributes very little to the y-values fo the rational function for large values of |x|.

Rational functions: formula, polynomial plus a proper rational function | matematicasVisuales

We say that the rational function is asymptotic to the polynomial quotient. The end behavior of the rational function follow the end behavior of the polynomial quotient. f(x) will behave like Q(x) for large values of x.

We can start with a rational function that has a polynomial of degree 2 as a polynomial quotient, a number as a remainder and a degree 1 polynomial as a denominator.

Rational functions: formula, polynomial of degree 2 plus a proper rational function | matematicasVisuales

In the first mahtlet you could play with the three elements of this kind of rational function: a quotient polynomial (in green, a parabola), the numerator (a number, in blue a horizontal line) and the denominator (a straight line, in orange)

For example:

Rational functions: graph of a rational function with asymptotic behavior like a parabola | matematicasVisuales

The formula for the previous graph is:

The equation of the quotient polynomial (asymptotic function) is:

Another example:

Rational functions: graph of a rational function with asymptotic behavior like a parabola | matematicasVisuales

In the next mathlet you can play with rational functions that has a degree 3 polynomial as a polynomial quotient, a number as a remainder and a degree 1 polynomial as a denominator.

Rational functions: formula, polynomial of degree 3 plus a proper rational function | matematicasVisuales

The three elements of this kind of rational function are: a quotient polynomial (in green, a cubic function), the numerator (a number, in blue a horizontal line) and the denominator (a straight line, in orange)

For example:

Rational functions: graph of a rational function with asymptotic behavior like a cubic function | matematicasVisuales

The formula for the previous graph is:

The equation of the quotient polynomial (asymptotic function) is:

Another example:

Rational functions: graph of a rational function with asymptotic behavior like a cubic function | matematicasVisuales

The next mathlet is similar.

Rational functions: formula, polynomial of degree 2 plus a proper rational function with a degree 2 polynomial in the denominator | matematicasVisuales

For example:

Rational functions: graph of a polynomial of degree 2 plus a proper rational function with a degree 2 polynomial in the denominator, asymptotic behavior like a parabola with two singularities | matematicasVisuales

The formula for the previous graph is:

The equation of the quotient polynomial (asymptotic function) is:

Another example:

Rational functions: graph of a polynomial of degree 2 plus a proper rational function with a degree 2 polynomial in the denominator, asymptotic behavior like a parabola without real singularities | matematicasVisuales

In the last mathlet of this series you can play with rational functions that has a degree 3 polynomial as a polynomial quotient, a degree 1 polynomial as a remainder and a degree 2 polynomial as a denominator.

Rational functions: formula, rational functions that has a degree 3 polynomial as a polynomial quotient, a degree 1 polynomial as a remainder and a degree 2 polynomial as a denominator | matematicasVisuales

For example:

Rational functions: graph of a rational function with asymptotic behavior like a cubic with two singularities | matematicasVisuales

The formula for the previous graph is:

The equation of the quotient polynomial (asymptotic function) is:

Another example:

Rational functions: graph of a rational function with asymptotic behavior like a cubic with two singularities | matematicasVisuales

REFERENCES

G.E. Shilov, Calculus of Rational Functions, Mir Publishers, Moscow.
I.M. Gelfand, E.G. Glagoleva, E.E. Shnol, Functions and Graphs, Dover Publications, Inc., Mineola, New York.

MORE LINKS

Rational Functions (1): Linear rational functions
Rational functions can be writen as the quotient of two polynomials. Linear rational functions are the simplest of this kind of functions.
Rational Functions (2): degree 2 denominator
When the denominator of a rational function has degree 2 the function can have two, one or none real singularities.
Polynomial Functions (1): Linear functions
Two points determine a stright line. As a function we call it a linear function. We can see the slope of a line and how we can get the equation of a line through two points. We study also the x-intercept and the y-intercept of a linear equation.
Polynomial Functions (2): Quadratic functions
Polynomials of degree 2 are quadratic functions. Their graphs are parabolas. To find the x-intercepts we have to solve a quadratic equation. The vertex of a parabola is a maximum of minimum of the function.
Powers with natural exponents (and positive rational exponents)
Power with natural exponents are simple and important functions. Their inverse functions are power with rational exponents (a radical or a nth root)
Polynomial Functions (3): Cubic functions
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the x-axis at least once.
Polynomial Functions (4): Lagrange interpolating polynomial
We can consider the polynomial function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.
Taylor polynomials (4): Rational function 1
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (5): Rational function 2
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (6): Rational function with two real singularities
This function has two real singularities at -1 and 1. Taylor polynomials approximate the function in an interval centered at the center of the series. Its radius is the distance to the nearest singularity.
Taylor polynomials (7): Rational function without real singularities
This is a continuos function and has no real singularities. However, the Taylor series approximates the function only in an interval. To understand this behavior we should consider a complex function.
Taylor polynomials: Rational function with two complex singularities
We will see how Taylor polynomials approximate the function inside its circle of convergence.
Complex Polynomial Functions(1): Powers with natural exponent
Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.
Complex Polynomial Functions(2): Polynomial of degree 2
A polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.
Complex Polynomial Functions(3): Polynomial of degree 3
A complex polinomial of degree 3 has three roots or zeros.
Complex Polynomial Functions(4): Polynomial of degree n
Every complex polynomial of degree n has n zeros or roots.
Polynomial functions and derivative (2): Quadratic functions
The derivative of a quadratic function is a linear function, it is to say, a straight line.
Polynomial functions and derivative (3): Cubic functions
The derivative of a cubic function is a quadratic function, a parabola.
Definite integral
The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the x-axis between two vertical lines.