The problem describes a function $g(x)$ with the following characteristics:

1. A positive change in the average rate of change over the interval $[-6, 0]$.
2. An average rate of change of 0 over the interval $[0, 3]$.
3. A positive average rate of change over the interval $[3, 6]$.

To determine the correct graph, let's analyze these properties step by step:

Key Observations:

• Positive change in the average rate of change $[-6, 0]$: This means $g(x)$ is increasing overall during this interval (the slope of the secant line is positive).

• Average rate of change = 0 over $[0, 3]$: This implies $g(x)$ must be constant (a flat line segment) over this interval.

• Positive average rate of change $[3, 6]$: This indicates $g(x)$ is increasing again in this interval.

Step-by-Step Matching with the Graphs:

• The graph must show:
  1. An upward (concave or linear) trend on $[-6, 0]$,
  2. A flat, horizontal segment on $[0, 3]$,
  3. An increasing trend on $[3, 6]$.

Correct Answer:

The first graph satisfies these conditions:

1. The segment from $[-6, 0]$ shows an increasing trend.
2. The segment from $[0, 3]$ is flat.
3. The segment from $[3, 6]$ rises again.

Would you like me to break this analysis down further or clarify any concepts?

Related Questions:

1. What is the difference between the average rate of change and instantaneous rate of change?
2. How can a function's concavity affect the interpretation of its average rate of change?
3. What does a horizontal line on a graph indicate about the function's rate of change?
4. How would the graph change if the average rate of change on $[3, 6]$ were negative instead of positive?
5. Can a function have a constant average rate of change over one interval and a varying rate on another?

Tip:

When analyzing graphs of functions, always focus on the slope of secant lines over the specified intervals to determine the average rate of change!