Enter the numerator and denominator polynomials to analyze their rational expression and identify key graphing features.
Analysis Results
Enter polynomials to begin.
Understanding Rational Expressions and Their Graphs
A rational expression is a function that can be written as the ratio of two polynomial functions, P(x) / Q(x), where Q(x) is not the zero polynomial. Graphing these functions involves identifying several key features:
Key Features for Graphing Rational Expressions:
Vertical Asymptotes: These occur at the values of x where the denominator Q(x) is zero, and the numerator P(x) is non-zero. They indicate where the function approaches infinity.
Horizontal Asymptotes: The behavior of the graph as x approaches positive or negative infinity is determined by comparing the degrees of the numerator (n) and the denominator (m).
If n < m, the horizontal asymptote is y = 0.
If n = m, the horizontal asymptote is y = (leading coefficient of P(x)) / (leading coefficient of Q(x)).
If n > m, there is no horizontal asymptote, but there might be a slant (oblique) asymptote if n = m + 1.
Holes: If a factor (x - c) can be canceled from both the numerator and the denominator, there is a hole at x = c. The y-coordinate of the hole can be found by substituting c into the simplified expression.
x-intercepts (Roots): These occur at the values of x where the numerator P(x) is zero, and the denominator Q(x) is non-zero. These are the roots of the numerator.
y-intercept: This is the value of the function when x = 0, i.e., f(0). It's found by substituting x = 0 into the original expression.
How this Calculator Works:
This calculator analyzes the provided numerator and denominator polynomials to identify:
Vertical Asymptotes: By finding the roots of the denominator polynomial.
Holes: By looking for common factors between the numerator and denominator polynomials.
x-intercepts: By finding the roots of the numerator polynomial that are not also roots of the denominator (after cancellation of common factors).
y-intercept: By evaluating the expression at x=0.
Horizontal Asymptote: By comparing the degrees of the polynomials and their leading coefficients.
Note: This calculator provides analytical information. For precise graphing, consider using a dedicated graphing tool or software.
// Helper function to parse a polynomial string into an object representing its coefficients.
// Example: "3x^2 – 5x + 2" -> { 2: 3, 1: -5, 0: 2 }
// Handles various formats like "x^3", "5", "2x", "-x^2 + 1"
function parsePolynomial(polyString) {
var coeffs = {};
if (!polyString || polyString.trim() === "") return coeffs;
// Normalize spacing and signs for easier parsing
polyString = polyString.replace(/\s+/g, "); // Remove all whitespace
polyString = polyString.replace(/-/g, '+-'); // Ensure subtractions are treated as adding negative terms
if (polyString.startsWith('+')) { // Remove leading '+' if it resulted from replacement
polyString = polyString.substring(1);
}
var terms = polyString.split('+');
for (var i = 0; i 1 && parts[1] !== "") { // Case like "x^3" or "2x^5"
exponent = parseInt(parts[1].substring(1)); // Remove '^'
} else { // Case like "2x"
exponent = 1;
}
} else { // Constant term
coefficient = parseFloat(term);
exponent = 0;
}
coeffs[exponent] = (coeffs[exponent] || 0) + (coefficient * sign);
}
return coeffs;
}
// Helper function to find roots of a polynomial
// This is a simplified approach and might not find all roots for complex polynomials.
// It focuses on integer and simple fractional roots that are common in textbook examples.
// For a robust solver, a numerical method (like Newton-Raphson) or a symbolic math library would be needed.
function findRoots(coeffs) {
var roots = [];
var degree = 0;
for (var exp in coeffs) {
if (coeffs.hasOwnProperty(exp)) {
degree = Math.max(degree, parseInt(exp));
}
}
// Special cases for low degree polynomials
if (degree === 0) { // Constant polynomial
if (coeffs[0] !== 0) return []; // Non-zero constant has no roots
// If it's 0, technically all x are roots, but this is usually not the intended case for rational expressions.
return [];
} else if (degree === 1) { // Linear: ax + b = 0 => x = -b/a
var a = coeffs[1] || 0;
var b = coeffs[0] || 0;
if (a !== 0) {
roots.push(-b / a);
}
} else if (degree === 2) { // Quadratic: ax^2 + bx + c = 0
var a = coeffs[2] || 0;
var b = coeffs[1] || 0;
var c = coeffs[0] || 0;
if (a === 0) { // Fallback to linear if 'a' is zero
if (b !== 0) {
roots.push(-c / b);
}
} else {
var discriminant = b * b – 4 * a * c;
if (discriminant >= 0) {
roots.push((-b + Math.sqrt(discriminant)) / (2 * a));
if (discriminant > 0) {
roots.push((-b – Math.sqrt(discriminant)) / (2 * a));
}
}
}
} else {
// For higher degrees, we'll try a limited set of rational roots (p/q)
// and a simple iteration for common factors. This is NOT exhaustive.
// A more complete solver would use algorithms like Jenkins-Traub or numerical methods.
var rationalRootCandidates = [];
var constantTerm = coeffs[0] || 0;
var leadingCoeff = 0;
var maxDegree = 0;
for(var exp in coeffs) {
if (coeffs.hasOwnProperty(exp)) {
if(parseInt(exp) > maxDegree) {
maxDegree = parseInt(exp);
leadingCoeff = coeffs[exp];
}
}
}
if (constantTerm !== 0) {
// Try factors of the constant term
for (var i = 1; i <= Math.abs(constantTerm); i++) {
if (constantTerm % i === 0) {
rationalRootCandidates.push(i);
rationalRootCandidates.push(-i);
}
}
}
if (leadingCoeff !== 0) {
// Try factors of the leading coefficient
for (var i = 1; i <= Math.abs(leadingCoeff); i++) {
if (leadingCoeff % i === 0) {
// Also add potential divisors of the constant term divided by these factors
for (var j = 1; j a – b);
for (var j = 0; j < rationalRootCandidates.length; j++) {
var rootCandidate = rationalRootCandidates[j];
var sum = 0;
for (var exp in coeffs) {
if (coeffs.hasOwnProperty(exp)) {
sum += coeffs[exp] * Math.pow(rootCandidate, parseInt(exp));
}
}
// Check if the candidate is close to zero (due to floating point inaccuracies)
if (Math.abs(sum) 0) {
var sumOfCoeffs = 0;
var alternatingSumOfCoeffs = 0;
for(var exp in coeffs) {
if (coeffs.hasOwnProperty(exp)) {
sumOfCoeffs += coeffs[exp];
alternatingSumOfCoeffs += coeffs[exp] * Math.pow(-1, parseInt(exp));
}
}
if (Math.abs(sumOfCoeffs) x=1 is a root
roots.push(1);
}
if (Math.abs(alternatingSumOfCoeffs) x=-1 is a root
roots.push(-1);
}
}
}
// Filter out roots that are effectively zero for denominator checks to avoid division by zero issues later
// And round for cleaner output.
var uniqueRoots = [];
for(var i = 0; i < roots.length; i++) {
var root = roots[i];
if (Math.abs(root) maxDegree) {
maxDegree = currentExp;
leadingCoeff = coeffs[exp];
}
}
}
return { degree: maxDegree, coefficient: leadingCoeff };
}
// Function to evaluate the polynomial at a specific x value
function evaluatePolynomial(coeffs, x) {
var result = 0;
for (var exp in coeffs) {
if (coeffs.hasOwnProperty(exp)) {
result += coeffs[exp] * Math.pow(x, parseInt(exp));
}
}
return result;
}
// Function to find common factors (represented by their roots)
// Returns roots that are common to both numerator and denominator.
function findCommonFactors(numCoeffs, denCoeffs) {
var numRoots = findRoots(numCoeffs);
var denRoots = findRoots(denCoeffs);
var common = [];
var epsilon = 1e-9; // Tolerance for floating point comparison
for (var i = 0; i < numRoots.length; i++) {
for (var j = 0; j < denRoots.length; j++) {
if (Math.abs(numRoots[i] – denRoots[j]) < epsilon) {
if (common.indexOf(numRoots[i]) === -1) {
common.push(numRoots[i]);
}
}
}
}
return common.map(function(r) { return parseFloat(r.toFixed(4)); });
}
function calculateFeatures() {
var numeratorPolyString = document.getElementById('numeratorPoly').value;
var denominatorPolyString = document.getElementById('denominatorPoly').value;
var resultDiv = document.getElementById('displayResult');
resultDiv.innerHTML = ""; // Clear previous results
var numCoeffs = parsePolynomial(numeratorPolyString);
var denCoeffs = parsePolynomial(denominatorPolyString);
var numInfo = getLeadingTermInfo(numCoeffs);
var denInfo = getLeadingTermInfo(denCoeffs);
var verticalAsymptotes = [];
var holes = [];
var xIntercepts = [];
var yIntercept = null;
var horizontalAsymptote = null;
var slantAsymptote = null;
// — Calculate Vertical Asymptotes —
var denRoots = findRoots(denCoeffs);
var commonFactorRoots = findCommonFactors(numCoeffs, denCoeffs);
for (var i = 0; i < denRoots.length; i++) {
var root = denRoots[i];
var isCommon = false;
for (var j = 0; j < commonFactorRoots.length; j++) {
if (Math.abs(root – commonFactorRoots[j]) < 1e-9) {
isCommon = true;
break;
}
}
if (!isCommon) {
verticalAsymptotes.push(root);
}
}
// — Calculate Holes —
holes = commonFactorRoots;
// — Calculate x-intercepts —
var numRoots = findRoots(numCoeffs);
for (var i = 0; i < numRoots.length; i++) {
var root = numRoots[i];
var isCommon = false;
for (var j = 0; j < commonFactorRoots.length; j++) {
if (Math.abs(root – commonFactorRoots[j]) < 1e-9) {
isCommon = true;
break;
}
}
if (!isCommon) {
xIntercepts.push(root);
}
}
// — Calculate y-intercept —
// Evaluate original expression at x=0. If denominator is zero at x=0, then no y-intercept.
var numAtZero = evaluatePolynomial(numCoeffs, 0);
var denAtZero = evaluatePolynomial(denCoeffs, 0);
if (denAtZero !== 0) {
yIntercept = numAtZero / denAtZero;
}
// — Calculate Asymptotes —
var numDegree = numInfo.degree;
var denDegree = denInfo.degree;
var numLeadingCoeff = numInfo.coefficient;
var denLeadingCoeff = denInfo.coefficient;
if (numDegree denDegree
if (numDegree === denDegree + 1 && denLeadingCoeff !== 0) {
// Slant asymptote calculation requires polynomial long division.
// This is complex to implement directly here.
// A simplified check for existence:
slantAsymptote = "Exists (degree of numerator is one more than denominator)";
} else {
slantAsymptote = "None"; // Or there might be a curvilinear asymptote if degree difference > 1
}
}
// — Display Results —
var output = "";
output += "
Analysis Complete
";
if (verticalAsymptotes.length > 0) {
output += "Vertical Asymptotes: " + verticalAsymptotes.map(function(va) { return "x = " + va.toFixed(4); }).join(", ") + "";
} else {
output += "Vertical Asymptotes: None identified.";
}
if (holes.length > 0) {
output += "Holes: Occur at x = " + holes.map(function(h) { return h.toFixed(4); }).join(", ") + "";
// Optionally, calculate y-coordinates of holes if a simplified form was available.
// For now, just indicate their presence.
} else {
output += "Holes: None identified.";
}
if (xIntercepts.length > 0) {
output += "x-intercepts: " + xIntercepts.map(function(xi) { return "x = " + xi.toFixed(4); }).join(", ") + "";
} else {
output += "x-intercepts: None identified.";
}
if (yIntercept !== null) {
output += "y-intercept: y = " + yIntercept.toFixed(4) + "";
} else {
output += "y-intercept: None (or undefined at x=0).";
}
if (horizontalAsymptote) {
output += "Horizontal Asymptote: " + horizontalAsymptote + "";
} else {
output += "Horizontal Asymptote: None identified.";
}
if (slantAsymptote) {
output += "Slant Asymptote: " + slantAsymptote + "";
}
if (numDegree > denDegree && numDegree !== denDegree + 1) {
output += "Note: Since the degree of the numerator is more than one greater than the denominator, there might be a curvilinear asymptote, not a linear slant asymptote.";
}
if (denDegree === 0 && numDegree === 0 && numCoeffs[0] === 0 && denCoeffs[0] === 0) {
output += "Note: Both numerator and denominator are the zero polynomial, resulting in an indeterminate form 0/0.";
} else if (denDegree === 0 && denCoeffs[0] === 0) {
output += "Error: Denominator is the zero polynomial. This is not a valid rational function.";
}
resultDiv.innerHTML = output;
}