Problem 104 - College Algebra

Question

What is the average rate of change of a function?

Step-by-Step Solution

Answer

The average rate of change of a function, from a point \((x_1, f(x_1))\) to a point \((x_2, f(x_2))\), is given by \(\frac{f(x_2)-f(x_1)}{x_2-x_1}\). This measures how much the function is changing per unit increase in the input.

Step 1: Understanding the Concept

The average rate of change of a function between two points is the change in the y-value divided by the change in the x-value. It measures how much the function is changing per unit increase in the input.

Step 2: The Mathematical Representation

Mathematically, the average rate of change of a function \(f(x)\) from a point \((x_1, f(x_1))\) to a point \((x_2, f(x_2))\) is given by \(\frac{f(x_2)-f(x_1)}{x_2-x_1}\). This formula is similar to the one used to find the slope of a line.

Step 3: Interpretation of the Average rate of change

A positive average rate of change indicates an increasing function, a negative one indicates a decreasing function, and zero indicates the function is constant between the two points.