Graphing Calculator for Domain and Range

Quadratic Function Domain & Range Calculator

Enter the coefficients for a quadratic function in the form: f(x) = ax² + bx + c

function calculateDomainRange() { var a = parseFloat(document.getElementById("coeffA").value); var b = parseFloat(document.getElementById("coeffB").value); var c = parseFloat(document.getElementById("coeffC").value); var domainResultElement = document.getElementById("domainResult"); var rangeResultElement = document.getElementById("rangeResult"); if (isNaN(a) || isNaN(b) || isNaN(c)) { domainResultElement.innerHTML = "Domain: Please enter valid numbers for all coefficients."; rangeResultElement.innerHTML = "Range:"; return; } var domain = "(-∞, ∞)"; // Domain for all polynomial functions is all real numbers var range; if (a === 0) { // This is a linear function: f(x) = bx + c if (b === 0) { // This is a constant function: f(x) = c range = "[" + c + ", " + c + "]"; // Range is just the constant value } else { // This is a non-horizontal linear function range = "(-∞, ∞)"; // Range for non-horizontal lines is all real numbers } } else { // This is a quadratic function: f(x) = ax^2 + bx + c var vertexX = -b / (2 * a); var vertexY = a * Math.pow(vertexX, 2) + b * vertexX + c; if (a > 0) { // Parabola opens upwards range = "[" + vertexY.toFixed(4) + ", ∞)"; } else { // Parabola opens downwards range = "(-∞, " + vertexY.toFixed(4) + "]"; } } domainResultElement.innerHTML = "Domain: " + domain; rangeResultElement.innerHTML = "Range: " + range; }

Understanding Domain and Range for Quadratic Functions

The concepts of domain and range are fundamental in mathematics, especially when working with functions. They describe the set of all possible input values (domain) and the set of all possible output values (range) for a given function.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree two. It is typically written in the standard form: f(x) = ax² + bx + c, where 'a', 'b', and 'c' are real numbers, and 'a' is not equal to zero. The graph of a quadratic function is a parabola, which is a U-shaped curve. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.

If 'a' happens to be zero, the function simplifies to f(x) = bx + c, which is a linear function. If both 'a' and 'b' are zero, it becomes f(x) = c, a constant function.

The Domain of a Quadratic Function

For any polynomial function, including quadratic functions, there are no restrictions on the values that 'x' can take. You can square any real number, multiply it by any coefficient, and add or subtract other numbers without encountering mathematical impossibilities (like dividing by zero or taking the square root of a negative number). Therefore, the domain of any quadratic function is always all real numbers.

In interval notation, this is expressed as (-∞, ∞), meaning 'x' can be any value from negative infinity to positive infinity.

The Range of a Quadratic Function

The range of a quadratic function is more specific and depends on two key factors: the direction the parabola opens and the location of its vertex. The vertex is the turning point of the parabola – either its lowest point (minimum) if it opens upwards, or its highest point (maximum) if it opens downwards.

The y-coordinate of the vertex determines the boundary of the range. The x-coordinate of the vertex can be found using the formula: x = -b / (2a). Once you have the x-coordinate, you can substitute it back into the original function f(x) = ax² + bx + c to find the y-coordinate of the vertex, which we'll call y_vertex.

• If 'a' > 0 (parabola opens upwards): The vertex is the minimum point. The function's output values (y-values) will be y_vertex or greater. The range is [y_vertex, ∞).
• If 'a' < 0 (parabola opens downwards): The vertex is the maximum point. The function's output values (y-values) will be y_vertex or less. The range is (-∞, y_vertex].

Special Cases (when a = 0):

• If a = 0 and b ≠ 0 (Linear Function): For a non-horizontal line like f(x) = bx + c, the graph extends infinitely in both positive and negative y-directions. The range is (-∞, ∞).
• If a = 0 and b = 0 (Constant Function): For a horizontal line like f(x) = c, the function always outputs the same value 'c'. The range is simply [c, c] or {c}.

How to Use the Calculator

This calculator simplifies the process of finding the domain and range for quadratic functions. Simply input the coefficients 'a', 'b', and 'c' from your function f(x) = ax² + bx + c into the respective fields. Click "Calculate Domain & Range," and the calculator will instantly display the domain and range in interval notation, taking into account the special cases for linear and constant functions.

Examples:

1. Function: f(x) = x² - 4x + 3
   • Coefficients: a=1, b=-4, c=3
   • Domain: (-∞, ∞) (always for quadratic)
   • Vertex x: -(-4) / (2*1) = 2
   • Vertex y: (1)*(2)² - (4)*(2) + 3 = 4 - 8 + 3 = -1
   • Since a > 0, the parabola opens upwards.
   • Range: [-1, ∞)
2. Function: f(x) = -2x² + 8x - 5
   • Coefficients: a=-2, b=8, c=-5
   • Domain: (-∞, ∞)
   • Vertex x: -(8) / (2*-2) = -8 / -4 = 2
   • Vertex y: (-2)*(2)² + (8)*(2) - 5 = -8 + 16 - 5 = 3
   • Since a < 0, the parabola opens downwards.
   • Range: (-∞, 3]
3. Function: f(x) = 5x + 10 (Here, a=0, b=5, c=10)
   • Coefficients: a=0, b=5, c=10
   • Domain: (-∞, ∞)
   • Since a=0 and b≠0, it's a linear function.
   • Range: (-∞, ∞)
4. Function: f(x) = 7 (Here, a=0, b=0, c=7)
   • Coefficients: a=0, b=0, c=7
   • Domain: (-∞, ∞)
   • Since a=0 and b=0, it's a constant function.
   • Range: [7, 7]