Question
What is the average rate of change on what interval for the graph of f(x) with the following points: (0, -1), (1, 1), (3, 0), (5, 2), (6, 2), (7, 0)
Answers
GPT-4o mini
To find the average rate of change of the function \( f(x) \) on an interval between two points, we can use the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( (a, f(a)) \) and \( (b, f(b)) \) are two points on the graph.
Let's evaluate this for some intervals using the provided points:
1. **Between (0, -1) and (1, 1)**:
- \( a = 0 \), \( b = 1 \)
- \( f(0) = -1 \), \( f(1) = 1 \)
- Average Rate of Change:
\[
\frac{1 - (-1)}{1 - 0} = \frac{2}{1} = 2
\]
2. **Between (1, 1) and (3, 0)**:
- \( a = 1 \), \( b = 3 \)
- \( f(1) = 1 \), \( f(3) = 0 \)
- Average Rate of Change:
\[
\frac{0 - 1}{3 - 1} = \frac{-1}{2} = -0.5
\]
3. **Between (3, 0) and (5, 2)**:
- \( a = 3 \), \( b = 5 \)
- \( f(3) = 0 \), \( f(5) = 2 \)
- Average Rate of Change:
\[
\frac{2 - 0}{5 - 3} = \frac{2}{2} = 1
\]
4. **Between (5, 2) and (6, 2)**:
- \( a = 5 \), \( b = 6 \)
- \( f(5) = 2 \), \( f(6) = 2 \)
- Average Rate of Change:
\[
\frac{2 - 2}{6 - 5} = \frac{0}{1} = 0
\]
5. **Between (6, 2) and (7, 0)**:
- \( a = 6 \), \( b = 7 \)
- \( f(6) = 2 \), \( f(7) = 0 \)
- Average Rate of Change:
\[
\frac{0 - 2}{7 - 6} = \frac{-2}{1} = -2
\]
These calculations give you the average rates of change for each of the noted intervals on the graph of \( f(x) \). If you need the average rate of change over a specific interval, you can apply the above formula using the coordinates of the endpoints of that interval.
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( (a, f(a)) \) and \( (b, f(b)) \) are two points on the graph.
Let's evaluate this for some intervals using the provided points:
1. **Between (0, -1) and (1, 1)**:
- \( a = 0 \), \( b = 1 \)
- \( f(0) = -1 \), \( f(1) = 1 \)
- Average Rate of Change:
\[
\frac{1 - (-1)}{1 - 0} = \frac{2}{1} = 2
\]
2. **Between (1, 1) and (3, 0)**:
- \( a = 1 \), \( b = 3 \)
- \( f(1) = 1 \), \( f(3) = 0 \)
- Average Rate of Change:
\[
\frac{0 - 1}{3 - 1} = \frac{-1}{2} = -0.5
\]
3. **Between (3, 0) and (5, 2)**:
- \( a = 3 \), \( b = 5 \)
- \( f(3) = 0 \), \( f(5) = 2 \)
- Average Rate of Change:
\[
\frac{2 - 0}{5 - 3} = \frac{2}{2} = 1
\]
4. **Between (5, 2) and (6, 2)**:
- \( a = 5 \), \( b = 6 \)
- \( f(5) = 2 \), \( f(6) = 2 \)
- Average Rate of Change:
\[
\frac{2 - 2}{6 - 5} = \frac{0}{1} = 0
\]
5. **Between (6, 2) and (7, 0)**:
- \( a = 6 \), \( b = 7 \)
- \( f(6) = 2 \), \( f(7) = 0 \)
- Average Rate of Change:
\[
\frac{0 - 2}{7 - 6} = \frac{-2}{1} = -2
\]
These calculations give you the average rates of change for each of the noted intervals on the graph of \( f(x) \). If you need the average rate of change over a specific interval, you can apply the above formula using the coordinates of the endpoints of that interval.