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. Identify the formula for the average rate of change, which is \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).
2. Substitute \( x_1 = 5 \) and \( x_2 = 9 \) into the formula.
3. Calculate \( f(x_1) = f(5) \) by substituting 5 into the function \( f(x) = x^2 - 4x \).
4. Calculate \( f(x_2) = f(9) \) by substituting 9 into the function \( f(x) = x^2 - 4x \).
5. Substitute \( f(5) \) and \( f(9) \) into the average rate of change formula and simplify.

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 values. 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 specific values 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 given, f(x) = x^2 - 4x, is a quadratic function where the 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.