91Ó°ÊÓ

Linear functions are fundamental in understanding relationships where variables change at a constant rate relative to each other. A linear function can be expressed in the form:

• \( f(x) = mx + b \)

Here, \(m\) represents the slope of the line, which indicates how steep the line is, and \(b\) is the y-intercept, or where the line crosses the y-axis.
A linear function is characterized by its straight-line graph. This simplicity makes it easy to predict future values, a useful feature in different fields such as physics, economics, and statistics. The particular linear function we are discussing has a negative slope, which we'll explore further in another section.

Change in variables is a common concept in algebra and calculus, describing how changes in one variable affect another. We often denote these changes as \( \Delta x \) for changes in the independent variable \( x \) and \( \Delta y \) for changes in the dependent variable \( y \).
When evaluating the average rate of change, we consider both how \( y \) responds as \( x \) changes. The formula for the average rate of change is:

• \( \dfrac{f(x + \Delta x) - f(x)}{\Delta x} \)

In this particular problem, the change in \( y \) is given by \( \Delta y = -\Delta x \). This relation suggests a negative correlation. As \( x \) increases by some amount, \( y \) decreases by that same amount. This stark relationship implies linearity with a particular inclination, leading us directly to a specific type of linear function.

The concept of a negative slope is vital in understanding how the independent and dependent variables relate inversely. In mathematical terms, when a function has a negative slope, it indicates that as \( x \) increases, \( y \) decreases. The slope is essentially the ratio of change in \( y \) to the change in \( x \). A linear function with a slope of

• -1, for instance, tells us that for every single unit increase in \( x \), \( y \) decreases by one unit.

Such behavior is visible in the function found in this exercise, where the relationship \( \Delta y = -\Delta x \) simplifies to a slope of -1, forming the line described by \( f(x) = -x + b \).
This negative slope vividly reflects the inverse relationship and further solidifies the concept of how moving in the positive direction on the x-axis results in a linear decrease along the y-axis.