Average Rate of Change of a Quadratic Function Calculator
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Understanding the Average Rate of Change of a Quadratic Function
The average rate of change of a function over an interval represents how much the function's output changes, on average, for each unit change in its input over that interval. For a quadratic function of the form f(x) = ax² + bx + c, the average rate of change between two points x₁ and x₂ is calculated as the change in the function's value (Δy) divided by the change in the x-values (Δx):
Formula:
Average Rate of Change = (f(x₂) - f(x₁)) / (x₂ - x₁)
Where:
f(x₂) = a(x₂)² + b(x₂) + cf(x₁) = a(x₁)² + b(x₁) + c
This calculator helps you quickly find this value. Simply input the two x-values defining your interval and the coefficients (a, b, and c) of your quadratic function.
Example:
Let's find the average rate of change for the function f(x) = 2x² - 3x + 5 between x₁ = 1 and x₂ = 4.
Identify coefficients and x-values:
a = 2
b = -3
c = 5
x₁ = 1
x₂ = 4
Calculate f(x₁):
f(1) = 2(1)² - 3(1) + 5 = 2 - 3 + 5 = 4
Calculate f(x₂):
f(4) = 2(4)² - 3(4) + 5 = 2(16) - 12 + 5 = 32 - 12 + 5 = 25
Calculate the average rate of change:
(f(x₂) - f(x₁)) / (x₂ - x₁) = (25 - 4) / (4 - 1) = 21 / 3 = 7
Therefore, the average rate of change of f(x) = 2x² - 3x + 5 between x = 1 and x = 4 is 7.
function calculateAverageRateOfChange() {
var x1 = parseFloat(document.getElementById("x1").value);
var x2 = parseFloat(document.getElementById("x2").value);
var a = parseFloat(document.getElementById("a").value);
var b = parseFloat(document.getElementById("b").value);
var c = parseFloat(document.getElementById("c").value);
var resultDiv = document.getElementById("result");
if (isNaN(x1) || isNaN(x2) || isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
return;
}
if (x1 === x2) {
resultDiv.innerHTML = "The two x-values cannot be the same (x₁ ≠ x₂).";
return;
}
// Calculate f(x1) and f(x2)
var fx1 = a * Math.pow(x1, 2) + b * x1 + c;
var fx2 = a * Math.pow(x2, 2) + b * x2 + c;
// Calculate the average rate of change
var averageRateOfChange = (fx2 – fx1) / (x2 – x1);
resultDiv.innerHTML = "The average rate of change between x = " + x1 + " and x = " + x2 + " is:
" + averageRateOfChange + " ";
}
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