4.3 Level Points
We'll say that a graph has a level point wherever the slope is zero. Most examples of level points are either low points or high points. This graph has three level points: one low point and two high points.
A high point does not need to be the highest point on the graph, but rather just higher than other points on the graph nearby. Similarly, a low point just needs to be lower than any nearby points on the graph. Transition diagrams allowed us to analyze the height of a graph, and they will be helpful for analyzing slope, too!
Example.
Let's analyze the slope of the following graph.
We'll divide the \( x \)-axis up into regions according to whether the graph has direct variance or indirect variance. This graph has two level points: a low point at \( x = -2 \) and a high point at \( x = 1 \).
The variance transitions at both the low point and the high point.
Let's see one more example.
Example.
Now let's take a look at the following graph. We'll draw a transition diagram for slope.
Our graph has a high point at \( x = 0 \) and no other level points. The graph has direct variance for negative \( x \)-values, and the graph has indirect variance for positive \( x \)-values.
We see that the slope transitions at the level point.
At the start of this section, we said, "most examples of level points are either low points or high points." Low points and high points are always level points, but not every level point is either a low point or a high point! In this chapter and the next we'll see many more examples of level points. We'll learn to use the derivative to both locate and classify level points.