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Create A Rational Function with The Following Characteristics Calculator

Reviewed by Calculator Editorial Team

This calculator helps you create a rational function with specific characteristics such as roots, asymptotes, and vertical shifts. Rational functions are ratios of two polynomials and are fundamental in algebra and calculus.

Introduction

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is:

f(x) = (anxn + an-1xn-1 + ... + a0) / (bmxm + bm-1xm-1 + ... + b0)

Where the numerator and denominator are polynomials, and the denominator is not identically zero. Rational functions have important properties such as roots (where the function equals zero), asymptotes (lines the function approaches but never touches), and vertical shifts (changes in the y-intercept).

How to Use This Calculator

  1. Enter the roots of the rational function (values of x where f(x) = 0).
  2. Enter the vertical asymptotes (values of x where the function approaches infinity).
  3. Enter the horizontal asymptote (the line the function approaches as x goes to ±∞).
  4. Enter any vertical shift (how much the function is moved up or down).
  5. Click "Calculate" to generate the rational function.

The calculator will display the function in its simplest form and provide a graph for visualization.

The Formula

The rational function is constructed using the following steps:

  1. Create the numerator polynomial from the given roots.
  2. Create the denominator polynomial from the given vertical asymptotes.
  3. Adjust the horizontal asymptote by scaling the numerator or denominator.
  4. Apply any vertical shift.

Numerator: (x - r1)(x - r2)...(x - rn)

Denominator: (x - v1)(x - v2)...(x - vm)

Final function: f(x) = [Numerator] / [Denominator] + C (vertical shift)

Worked Example

Example 1

Create a rational function with:

  • Roots at x = 2 and x = -3
  • Vertical asymptotes at x = 1 and x = -2
  • Horizontal asymptote at y = 2
  • Vertical shift of 1

The resulting function is:

f(x) = [(x - 2)(x + 3)] / [(x - 1)(x + 2)] + 1

Simplified: f(x) = (x² + x - 6) / (x² + x - 2) + 1

FAQ

What is a rational function?
A rational function is a ratio of two polynomials. It has roots (where the function equals zero) and asymptotes (lines the function approaches but never touches).
How do I find the roots of a rational function?
The roots are the values of x that make the numerator equal to zero. You can find them by solving the numerator polynomial equation.
What is the difference between vertical and horizontal asymptotes?
Vertical asymptotes occur where the denominator is zero (and the numerator is not zero). Horizontal asymptotes describe the behavior of the function as x approaches ±∞.