Final answer:
Determining increasing and decreasing intervals of functions involves looking at the slope in different intervals. Downward slopes indicate decreasing intervals, while upward slopes indicate increasing intervals. Constructing a histogram can help visualize these intervals and the overall behavior of the function.
Step-by-step explanation:
To determine the increasing and decreasing intervals of various functions, you need to look at the slope of the function in different intervals. For a function described by parts, such as in Part A, a downward slope that levels off indicates a decreasing interval until the slope reaches zero. At zero, the function is neither increasing nor decreasing.
In Part B, the function starts at zero with an upward slope, which means it is increasing. However, as the slope decreases in magnitude and eventually levels off, the function's rate of increase is slowing down.
In Part C, we see a function that begins at zero with an upward slope that increases in magnitude until it becomes a positive constant, indicating a continuously increasing interval with a steady growth rate once it reaches the constant slope.
To visualize this, constructing a histogram can be helpful. When creating a histogram, it is important to make five to six intervals, use a ruler and pencil to sketch the graph accurately, and scale the axes properly to reflect the data range and intervals.
To summarize the behavior of these functions, consider the gradient of each section:
- Part A: Decreasing interval until the slope reaches zero
- Part B: Increasing interval with a decreasing rate of increase
- Part C: Continuously increasing interval with a constant rate once the slope becomes constant
When reading and manipulating graphs, it's crucial to understand the function's properties, including the equation of a line (slope and intercept), how to change the slope or intercept, how to compute and interpret growth rates, and how to annotate the graph to highlight different intervals of behavior.