91Ó°ÊÓ

Let's consider a linear function of the form \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. For a function to be decreasing, the output must decrease as the input increases. In other words, for any given pair of points (x1, y1) and (x2, y2) on the graph, if x2 > x1, then y2 < y1. Following the linear function form, we have \(y1 = mx1 + b\) and \(y2 = mx2 + b\). Now, let's subtract the first equation from the second equation: \(y2 - y1 = m(x2 - x1)\) We know that for a decreasing function, if x2 > x1, then y2 < y1. Therefore, \(x2 - x1 > 0\) and \(y2 - y1 < 0\). To satisfy this inequality, the slope \(m\) must be negative, and so it proves that if a linear function is decreasing, then the slope of its graph is negative.

Now, let's assume that the slope of the linear function's graph is negative, meaning \(m < 0\). We will show that if \(m < 0\), then the function is decreasing. Consider two points, (x1, y1) and (x2, y2) on the graph. Using the linear function form, we have \(y1 = mx1 + b\) and \(y2 = mx2 + b\). Subtract the first equation from the second equation: \(y2 - y1 = m(x2 - x1)\) For a function to be decreasing, we want to show that if \(x2 > x1\), then \(y2 < y1\). Since the slope \(m\) is negative, \(m(x2 - x1) < 0\) when \(x2 > x1\). Thus, \(y2 - y1 < 0\) when \(x2 > x1\), which implies that \(y2 < y1\) when \(x2 > x1\). This proves that if the slope of a linear function's graph is negative, then the function is decreasing. To conclude, we have shown that a linear function is decreasing if and only if the slope of its graph is negative.