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Imagine you are on a road trip and you keep track of the distance you travel over time. In mathematics, the concept of a function change rate is somewhat analogous to measuring your average speed during a certain time period. When dealing with functions, the rate at which the function's output (or value) changes as the input varies is pivotal.

It's a measure of how one quantity changes in relation to another. In your road trip example, it's the change in distance (output) over time (input). Similarly, in functions, we look at how the y-values change as x-values change. A higher rate indicates a steeper change - like driving faster - while a lower rate suggests a gentler slope or slower speed.

Understanding the function change rate helps to identify trends and can be a stepping stone to more advanced concepts like derivatives, which measure this rate at a single point rather than over an interval.

Mathematical symbols and formulas allow us to succinctly express complex ideas. The average rate of change of a function can be mathematically represented using a simple formula: \[\frac{{f(b) - f(a)}}{{b - a}}\]. Here, \(a\) and \(b\) are two different input values from the domain of the function, and \(f(a)\) and \(f(b)\) are the corresponding output values. The difference \(f(b) - f(a)\) gives us the change in the function's output, while the difference \(b - a\) gives us the change in the input.

This ratio, which simplifies to the slope of the line connecting the points \((a, f(a))\) and \((b, f(b))\) on the graph of the function, is the essence of the average rate of change. It helps in visualizing functions in a spatial manner, further developing a student’s understanding of how mathematics can model real-world scenarios.

Now, let's illustrate the average rate of change with a practical example in mathematics. Suppose you're studying a linear function, such as \(f(x) = 2x + 3\). Choosing two points, say \(a=1\) and \(b=3\), you want to find out how quickly the function changes between these two points.

Using our formula, we get \[\frac{{f(3) - f(1)}}{{3 - 1}}\] which simplifies to \[\frac{{9 - 5}}{2}\]. The result is \(2\), meaning, on average, the function increases by \(2\) for each \(1\) unit increase in \(x\). This is reflective of the slope of the line. In real-life scenarios, such as physics or economics, understanding this rate could correspond to understanding changes in velocity or cost, making it a widely applicable and valuable concept.