Find the average rate of change of f(x) = x^2 - 4x from x_1 = 5 to x_2 = 9.

Verified step by step guidance

1. Step 1: Recall the formula for the average rate of change of a function f(x) over an interval [x₁, x₂]. The formula is: (f(x₂) - f(x₁)) / (x₂ - x₁).
2. Step 2: Substitute the given values x₁ = 5 and x₂ = 9 into the formula. This gives: (f(9) - f(5)) / (9 - 5).
3. Step 3: Evaluate f(9) by substituting x = 9 into the function f(x) = x² - 4x. This results in f(9) = 9² - 4(9).
4. Step 4: Similarly, evaluate f(5) by substituting x = 5 into the function f(x) = x² - 4x. This results in f(5) = 5² - 4(5).
5. Step 5: Simplify the numerator (f(9) - f(5)) and the denominator (9 - 5), then divide the results to find the average rate of change.

Key Concepts

Average Rate of Change

The average rate of change of a function over an interval is calculated as the change in the function's value divided by the change in the input value. Mathematically, it is expressed as (f(x2) - f(x1)) / (x2 - x1). This concept is essential for understanding how a function behaves over a specific range.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For the function f(x) = x^2 - 4x, evaluating it at x = 5 and x = 9 will provide the necessary values to compute the average rate of change. This step is crucial for applying the average rate of change formula.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The function f(x) = x^2 - 4x is a quadratic function, and its graph is a parabola. Understanding the properties of quadratic functions, such as their shape and vertex, can provide insights into their behavior over different intervals.