Question

choose the intervals where the graph has a decreasing average rate of change.

• \\(x = 0\\) to \\(x = 13\\)
• \\(x = 3\\) to \\(x = 6\\)
• \\(x = 4\\) to \\(x = 8\\)
• \\(x = 6\\) to \\(x = 10\\)

Explanation:

Identify coordinates from the graph

Using the Graphical Analysis knowledge point, we locate the points on the graph corresponding to the \(x\)-values in the options:

• At \(x = 0\), \(y = 0\). Point: \((0, 0)\)
• At \(x = 3\), \(y = 2\). Point: \((3, 2)\)
• At \(x = 4\), \(y = 5\). Point: \((4, 5)\)
• At \(x = 6\), \(y = 1\). Point: \((6, 1)\)
• At \(x = 8\), \(y = 4\). Point: \((8, 4)\)
• At \(x = 10\), \(y = 6\). Point: \((10, 6)\)
• At \(x = 13\), \(y = 5\). Point: \((13, 5)\)

Calculate average rate of change

Using the Average Rate of Change knowledge point, we calculate the rate of change for each interval using the formula:

$$ \text{Rate} = \frac{y_2 - y_1}{x_2 - x_1} $$

• For \(x = 0\) to \(x = 13\):

$$ \frac{5 - 0}{13 - 0} = \frac{5}{13} \approx 0.38 \quad (\text{positive}) $$

• For \(x = 3\) to \(x = 6\):

$$ \frac{1 - 2}{6 - 3} = \frac{-1}{3} \approx -0.33 \quad (\text{negative}) $$

• For \(x = 4\) to \(x = 8\):

$$ \frac{4 - 5}{8 - 4} = \frac{-1}{4} = -0.25 \quad (\text{negative}) $$

• For \(x = 6\) to \(x = 10\):

$$ \frac{6 - 1}{10 - 6} = \frac{5}{4} = 1.25 \quad (\text{positive}) $$

Determine decreasing intervals

A decreasing average rate of change corresponds to a negative value.

• The interval from \(x = 3\) to \(x = 6\) has a rate of \(-\frac{1}{3}\).
• The interval from \(x = 4\) to \(x = 8\) has a rate of \(-\frac{1}{4}\).

Both of these intervals have a negative average rate of change, indicating a decrease over the interval.

Answer:

• \(x = 0\) to \(x = 13\)
• \(x = 3\) to \(x = 6\) (Correct answer)
• \(x = 4\) to \(x = 8\) (Correct answer)
• \(x = 6\) to \(x = 10\)