The graph shows the amount of air pollution removed by trees...

The graph shows the amount of air pollution removed by trees in the Chicago urban region for each month of the year. From the graph we see, for example, that the ozone level starting in May increases up to June, and then abruptly decreases. GRAPH CANNOT COPY a. Are these curves the graphs of functions? b. Look at the graph for particulates. Where is the function increasing? Decreasing? Constant? c. On what intervals do all four lower graphs indicate that the corresponding functions are constant? Why do you think the functions are constant on those intervals?

The graph shows the amount of air pollution removed by trees in the Chicago urban region for each month of the year. From the graph we see, for example, that the ozone level starting in May increases up to June, and then abruptly decreases.
GRAPH CANNOT COPY
a. Are these curves the graphs of functions?
b. Look at the graph for particulates. Where is the function increasing? Decreasing? Constant?
c. On what intervals do all four lower graphs indicate that the
corresponding functions are constant? Why do you think the functions are constant on those intervals?

Key Concepts

Function

A function is a relation that matches each input in its domain with exactly one output. In the context of a graph, this means that any vertical line drawn through the graph will intersect it at most once, ensuring that every input produces a unique output.

Vertical Line Test

The vertical line test is a simple visual check used to determine whether a curve in the plane represents a function. If any vertical line crosses the graph more than once, then the graph does not represent a function.

Monotonicity (Increasing and Decreasing Behavior)

Monotonicity refers to the property of functions to consistently increase or decrease over certain intervals. When the graph of a function rises as you move from left to right, the function is said to be increasing; when it falls, the function is decreasing. This concept is important for understanding the overall trends in a function's behavior over its domain.

Constant Functions

A constant function is one where the output does not change regardless of variations in the input. Graphically, it is represented by a horizontal line, indicating that over the interval in question, the function’s value remains unchanged.

Interval Analysis

Interval analysis involves examining segments of the domain to study specific characteristics of a function, such as whether it is increasing, decreasing, or constant. This is especially useful when interpreting graphs to understand periods of change or stability in the function's behavior.

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