Enter the coefficients for the numerator and denominator polynomials to analyze and identify key features of the rational expression.
Your analysis will appear here.
Understanding Rational Expressions and Their Graphs
A rational expression is a function that can be written as the ratio of two polynomials, P(x)/Q(x), where Q(x) is not the zero polynomial. Graphing these expressions involves understanding several key features that determine their visual behavior on a coordinate plane.
Key Features for Graphing:
Vertical Asymptotes: Occur at the real values of x where the denominator Q(x) equals zero, provided that the numerator P(x) is not zero at those same values. These are vertical lines that the graph approaches but never touches.
Horizontal Asymptotes: Determined by comparing the degrees of the numerator polynomial (deg(P)) and the denominator polynomial (deg(Q)):
If deg(P) < deg(Q), the horizontal asymptote is the line y = 0.
If deg(P) = deg(Q), the horizontal asymptote is the line y = a/b, where a is the leading coefficient of P(x) and b is the leading coefficient of Q(x).
If deg(P) > deg(Q), there is no horizontal asymptote. There might be a slant (oblique) asymptote if deg(P) = deg(Q) + 1.
Holes (Removable Discontinuities): Occur when a factor (x – c) can be canceled from both the numerator and the denominator. A hole exists at x = c, and its y-coordinate is found by substituting c into the simplified expression.
x-intercepts (Roots): Occur at the real values of x where the numerator P(x) equals zero, provided the denominator Q(x) is not zero at those values. These are the points where the graph crosses the x-axis.
y-intercept: Occurs at the value of the function when x = 0. This is found by calculating P(0)/Q(0), assuming Q(0) is not zero.
How This Calculator Works:
This calculator attempts to parse your input polynomials and identify the presence of vertical asymptotes, horizontal asymptotes, and potential x-intercepts. Due to the complexity of parsing arbitrary polynomial strings and finding roots analytically, this calculator focuses on common cases and provides a simplified analysis. It handles basic polynomial forms like ax^n + bx^(n-1) + ... + c. For more complex expressions or those requiring advanced root-finding algorithms, graphical tools or dedicated mathematical software are recommended.
Numerator: $P(x) = x^2 – 1$. Roots are $x = 1$ and $x = -1$.
Denominator: $Q(x) = x^2 – 4$. Roots are $x = 2$ and $x = -2$.
Vertical Asymptotes: Since the denominator is zero at $x=2$ and $x=-2$, and the numerator is non-zero at these points, vertical asymptotes are at $x = 2$ and $x = -2$.
Horizontal Asymptote: The degree of the numerator (2) equals the degree of the denominator (2). The leading coefficients are both 1. So, the horizontal asymptote is $y = 1/1 = 1$.
x-intercepts: Occur when $x^2 – 1 = 0$, which are $x = 1$ and $x = -1$.
Inputting 1x^2+0x+-1 for the numerator and 1x^2+0x+-4 for the denominator will provide an analysis based on these principles.
// Helper function to parse polynomial string into an object {degree: coefficient}
function parsePolynomial(polyString) {
var terms = polyString.replace(/-\s*/g, '+-').split('+');
var coefficients = {};
var maxDegree = 0;
for (var i = 0; i 1 && parts[1] !== ") {
degree = parseInt(parts[1].replace('^', "));
} else {
degree = 1; // x term
}
} else {
coefficient = parseFloat(term);
degree = 0; // constant term
}
if (!isNaN(coefficient)) {
coefficients[degree] = (coefficients[degree] || 0) + coefficient;
if (degree > maxDegree) {
maxDegree = degree;
}
}
}
// Ensure all degrees up to maxDegree have an entry, even if 0
for (var d = 0; d x = -b/a
var a = coeffs[1] || 0;
var b = coeffs[0] || 0;
if (a !== 0) roots.push(-b / a);
} else if (degree === 2) { // ax^2 + bx + c = 0
var a = coeffs[2] || 0;
var b = coeffs[1] || 0;
var c = coeffs[0] || 0;
if (a !== 0) {
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 if (degree === 0) {
// Constant polynomial, no roots unless it's 0
} else {
// Higher degrees are complex to solve analytically here
}
return roots.map(function(root) { return parseFloat(root.toFixed(4)); });
}
// Helper function to evaluate polynomial at a point
function evaluatePolynomial(polyObj, x) {
var result = 0;
for (var degree in polyObj.coefficients) {
result += polyObj.coefficients[degree] * Math.pow(x, parseInt(degree));
}
return result;
}
function calculateRationalExpressionFeatures() {
var numeratorInput = document.getElementById("numeratorPolynomial").value;
var denominatorInput = document.getElementById("denominatorPolynomial").value;
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = "Calculating…";
try {
var numeratorPoly = parsePolynomial(numeratorInput);
var denominatorPoly = parsePolynomial(denominatorInput);
var VA = []; // Vertical Asymptotes
var HA = null; // Horizontal Asymptote
var xIntercepts = [];
var yIntercept = null;
var holes = []; // For simplicity, we won't explicitly calculate holes without factor analysis
// Find roots of denominator for VA
var denominatorRoots = findRoots(denominatorPoly);
for (var i = 0; i 1e-9) { // Use tolerance for float comparison
VA.push("x = " + root);
} else {
// Simplified hole detection: if numerator root == denominator root
// A more robust method involves factoring.
var numeratorRootsForHoleCheck = findRoots(numeratorPoly);
for(var j=0; j < numeratorRootsForHoleCheck.length; j++){
if(Math.abs(numeratorRootsForHoleCheck[j] – root) h.includes(root.toString()))) { // If not already marked as hole
VA.push("x = " + root);
}
}
}
// Determine Horizontal Asymptote
var numDegree = numeratorPoly.degree;
var denDegree = denominatorPoly.degree;
if (numDegree denDegree
HA = "None (degree of numerator is greater than denominator)";
// Could add logic for slant asymptote here if numDegree === denDegree + 1
}
// Find x-intercepts
var numeratorRoots = findRoots(numeratorPoly);
for (var i = 0; i 1e-9) {
xIntercepts.push("x = " + root);
} else {
// If denominator is zero, it might be a hole or undefined point, not an intercept
}
}
// Find y-intercept
if (Math.abs(evaluatePolynomial(denominatorPoly, 0)) > 1e-9) {
yIntercept = evaluatePolynomial(numeratorPoly, 0) / evaluatePolynomial(denominatorPoly, 0);
yIntercept = yIntercept.toFixed(4);
} else {
yIntercept = "Undefined (denominator is zero at x=0)";
}
var output = "