How to Calculate Vertical Asymptote

Vertical Asymptote Calculator

Enter the coefficients for the denominator of your rational function: f(x) = N(x) / (ax² + bx + c)

Calculation Result:

function calculateAsymptote() { var a = parseFloat(document.getElementById('coeffA').value); var b = parseFloat(document.getElementById('coeffB').value); var c = parseFloat(document.getElementById('coeffC').value); var resultDiv = document.getElementById('vaResult'); var output = document.getElementById('vaOutput'); if (isNaN(a) || isNaN(b) || isNaN(c)) { output.innerHTML = "Please enter valid numbers for all coefficients."; resultDiv.style.display = "block"; return; } var resultHtml = ""; if (a === 0) { if (b === 0) { resultHtml = "Since the denominator is a constant (" + c + "), there are no vertical asymptotes."; } else { var x = -c / b; resultHtml = "This is a linear denominator (bx + c = 0).Setting " + b + "x + " + c + " = 0 gives:x = " + x.toFixed(4) + ""; } } else { var discriminant = (b * b) – (4 * a * c); if (discriminant > 0) { var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); resultHtml = "The denominator has two real roots:x = " + x1.toFixed(4) + "x = " + x2.toFixed(4) + ""; } else if (discriminant === 0) { var x = -b / (2 * a); resultHtml = "The denominator has one repeated real root:x = " + x.toFixed(4) + ""; } else { resultHtml = "The discriminant is negative (b² – 4ac < 0). There are no real vertical asymptotes for this function."; } } output.innerHTML = resultHtml; resultDiv.style.display = "block"; }

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A vertical asymptote is a vertical line that a graph approaches but never touches or crosses as the input (x) approaches a specific value. In rational functions, vertical asymptotes typically occur at values that make the denominator equal to zero, provided those values do not also make the numerator zero (which might indicate a "hole").

Step-by-Step Guide: How to Find Vertical Asymptotes

To calculate the vertical asymptote of a rational function f(x) = N(x) / D(x), follow these three essential steps:

1. Simplify the Function: Factor both the numerator and the denominator. If there are common factors, cancel them out. Note: The values of x that make these canceled factors zero represent "holes" (removable discontinuities), not asymptotes.
2. Set the Denominator to Zero: Take the simplified denominator and set it equal to zero (D(x) = 0).
3. Solve for x: The solutions to this equation are the equations of the vertical asymptotes.

A Practical Example

Let's find the vertical asymptotes for the function: f(x) = (x + 2) / (x² – 5x + 6)

Step 1: Factor
Numerator: (x + 2)
Denominator: (x – 2)(x – 3)
Function: f(x) = (x + 2) / [(x – 2)(x – 3)]

Step 2: Check for Common Factors
There are no common factors between the numerator and denominator, so nothing can be canceled. No holes exist.

Step 3: Solve Denominator = 0
(x – 2)(x – 3) = 0
Setting each factor to zero gives:
x – 2 = 0 → x = 2
x – 3 = 0 → x = 3

Therefore, the vertical asymptotes for this function are the lines x = 2 and x = 3.

Important Considerations

• Holes vs. Asymptotes: If a value makes both the numerator and denominator zero, it is a hole. If it makes only the denominator zero, it is a vertical asymptote.
• Logarithmic Functions: For functions like f(x) = log(x), a vertical asymptote occurs where the argument is zero (e.g., x = 0).
• Trigonometric Functions: Functions like tan(x) have infinitely many vertical asymptotes because the denominator (cos(x)) is zero at intervals of π (e.g., x = π/2, 3π/2).