How does the graph of \( f(x)=|x| \) compare with the graph of \( g(x)=3|x| \) ?

The graph of \( f(x) = |x| \) is a simple V-shape that opens upwards with its vertex at the origin (0, 0). It has two linear segments: one rising from the left for negative x-values and another rising from the right for positive x-values, both with a slope of 1. In contrast, the graph of \( g(x) = 3|x| \) also forms a V-shape but is steeper. Its vertex is also at the origin, but the slopes of the two segments are now 3, meaning for every unit increase in x, the output rises three units. This scaling effect gives the graph of \( g(x) \) a more pronounced height compared to \( f(x) \). When it comes to real-world applications, absolute value functions like \( f(x) \) and \( g(x) \) are often used in situations where we want to measure distance, since distance cannot be negative. For example, in economics, \( f(x) \) might model profit or loss scenarios where both gains and losses need to be visualized regardless of their direction. On the other hand, \( g(x) \), being three times steeper, might represent situations where quantity or impact is amplified, like measuring the surface area of a material where thickness scales up costs significantly.