Question

Use the image to answer the question.

A coordinate plane for f of x versus x. it starts from -2, then up to 1, then back down to 3, then goes down again to 4 and 5 then goes back up to 6.

Given the graph of f(x) , on which interval is the average rate of change the greatest?

(1 point)
Responses

The function has the greatest average rate of change over the interval [0,1].
The function has the greatest average rate of change over the interval left bracket 0 comma 1 right bracket .

The function has the greatest average rate of change over the interval [4,5].
The function has the greatest average rate of change over the interval left bracket 4 comma 5 right bracket .

The function has the greatest average rate of change over the interval [5,6].
The function has the greatest average rate of change over the interval left bracket 5 comma 6 right bracket .

The function has the greatest average rate of change over the interval [1,4].
The function has the greatest average rate of change over the interval left bracket 1 comma 4 right bracket .
Skip to navigation

Answer 1

To determine the interval where the average rate of change is the greatest based on the described graph of \( f(x) \), we calculate the average rate of change for each interval mentioned in the options:

1. **Interval [0, 1]**:
- Calculate \( f(0) \) and \( f(1) \).
- Average rate of change = \( \frac{f(1) - f(0)}{1 - 0} \).

2. **Interval [1, 4]**:
- Calculate \( f(1) \) and \( f(4) \).
- Average rate of change = \( \frac{f(4) - f(1)}{4 - 1} \).

3. **Interval [4, 5]**:
- Calculate \( f(4) \) and \( f(5) \).
- Average rate of change = \( \frac{f(5) - f(4)}{5 - 4} \).

4. **Interval [5, 6]**:
- Calculate \( f(5) \) and \( f(6) \).
- Average rate of change = \( \frac{f(6) - f(5)}{6 - 5} \).

To identify where the average rate of change is greatest, the two key intervals likely experiencing rapid vertical changes are [4, 5] and [5, 6]. Intervals with steeper slopes will yield greater average rates of change.

Based on your description that the function goes up notably to 1, dips back down notably to 4 and then rises again to 6, we can infer:

- The interval [4, 5] likely experiences a large increase in \( f(x) \) relative to a small increase in \( x\).
- The interval [5, 6] would continue that increase.

Given the options, it is most likely that the greatest average rate of change is for the interval **[5, 6]**, where the function appears to rise sharply.

So, the correct response would be:

**The function has the greatest average rate of change over the interval [5, 6]**.