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Use the image to answer the question.

it is a graph with points at (0,-1), (1, 0), (2,-1), (3,1), (5, 1)

Describe the behavior of the function by determining over which interval the function has a negative average rate of change.

The average rate of change is negative over the interval ______.

(1 point)
Responses

[3, 5]
[3, 5]

[0, 1]
[0, 1]

[1, 2]
[1, 2]

[2, 3]

1 answer

To determine where the function has a negative average rate of change, we need to look for intervals where the function's values decrease.

Let's evaluate the points within each provided interval:

  1. Interval [0, 1]:

    • At \( x = 0 \), \( y = -1 \)
    • At \( x = 1 \), \( y = 0 \)
    • The average rate of change: \[ \text{AROC} = \frac{0 - (-1)}{1 - 0} = \frac{1}{1} = 1 \] (Positive)

  2. Interval [1, 2]:

    • At \( x = 1 \), \( y = 0 \)
    • At \( x = 2 \), \( y = -1 \)
    • The average rate of change: \[ \text{AROC} = \frac{-1 - 0}{2 - 1} = \frac{-1}{1} = -1 \] (Negative)

  3. Interval [2, 3]:

    • At \( x = 2 \), \( y = -1 \)
    • At \( x = 3 \), \( y = 1 \)
    • The average rate of change: \[ \text{AROC} = \frac{1 - (-1)}{3 - 2} = \frac{2}{1} = 2 \] (Positive)

  4. Interval [3, 5]:

    • At \( x = 3 \), \( y = 1 \)
    • At \( x = 5 \), \( y = 1 \)
    • The average rate of change: \[ \text{AROC} = \frac{1 - 1}{5 - 3} = \frac{0}{2} = 0 \] (Neither positive nor negative)

From this analysis, we see that the only interval with a negative average rate of change is:

[1, 2].