Increasing, Decreasing, and Constant Intervals Lesson

In this lesson, we want to learn how to determine where a function is increasing, decreasing, or constant from its graph. Let's begin with something simple, the linear function. We know when we look at the graph of a line with a positive slope, the graph rises as we move from left to right. In other words, the y-values are always increasing as the x-values are increasing. $$y=2x - 1$$ $$m=+2$$ Additionally, if the line has a negative slope, the graph falls as we move from left to right. In this case, the y-values are always decreasing as the x-values are increasing. $$y=-5x + 2$$ $$m=-5$$ When working with lines, the process is very easy, for a positively sloped line, we will increase across the entire domain. On the other hand, for a negatively sloped line, we will decrease across the entire domain. In most cases, we will deal with a graph that is much more complex than a simple line. To deal with harder situations, let's think about a simple rule:
A function f is increasing on any interval if for any (x₁) and (x₂), we have: $$x_1 < x_2$$ $$f(x_1) < f(x_2)$$ This tells us as we move right from x₁ to x₂, the y-values are getting larger. A function f is decreasing on any interval if for any (x₁) and (x₂), we have: $$x_1 < x_2$$ $$f(x_1) > f(x_2)$$ This tells us as we move right from x₁ to x₂, the y-values are getting smaller. A function f is constant on any interval if for any (x₁) and (x₂), we have: $$x_1 < x_2$$ $$f(x_1)=f(x_2)$$ This tells us as we move right from x₁ to x₂, the y-values are constant.