Question
A graph is first pointed at (0,2) the next point is at (3, 20) the last one is at (5, 90).
What is the average rate of change between the first point and last?
Explain hiw mych it is increasing over time.
What is the average rate of change between the first point and last?
Explain hiw mych it is increasing over time.
Answers
GPT-4o mini
To find the average rate of change of a function between two points, we can use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
In your case, the first point is (0, 2), and the last point is (5, 90). Here, \(a = 0\) and \(b = 5\), and the corresponding function values are \(f(a) = 2\) and \(f(b) = 90\).
Now, let's plug these values into the formula:
\[
\text{Average Rate of Change} = \frac{90 - 2}{5 - 0} = \frac{88}{5} = 17.6
\]
So, the average rate of change from the first point (0, 2) to the last point (5, 90) is \(17.6\).
### Explanation of Change Over Time:
This average rate of change of \(17.6\) indicates that, on average, the value of the function increases by \(17.6\) units for every 1 unit increase in \(x\) (or over time). Over the interval from \(x = 0\) to \(x = 5\), the graph is showing a significant increase from \(2\) to \(90\).
To understand how much it is increasing over time:
1. **Initial Value at \(x = 0\)**: The value starts at \(2\).
2. **Final Value at \(x = 5\)**: The value ends at \(90\).
3. **Total Increase**: The total increase over the interval is \(90 - 2 = 88\).
This large increase indicates that not only does the function rise steeply, but also that the rate of increase is significant as the function progresses through its domain from \(x = 0\) to \(x = 5\).
Thus, the function exhibits a strong upward trend in this interval, reflecting that it is increasing rapidly over time.
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
In your case, the first point is (0, 2), and the last point is (5, 90). Here, \(a = 0\) and \(b = 5\), and the corresponding function values are \(f(a) = 2\) and \(f(b) = 90\).
Now, let's plug these values into the formula:
\[
\text{Average Rate of Change} = \frac{90 - 2}{5 - 0} = \frac{88}{5} = 17.6
\]
So, the average rate of change from the first point (0, 2) to the last point (5, 90) is \(17.6\).
### Explanation of Change Over Time:
This average rate of change of \(17.6\) indicates that, on average, the value of the function increases by \(17.6\) units for every 1 unit increase in \(x\) (or over time). Over the interval from \(x = 0\) to \(x = 5\), the graph is showing a significant increase from \(2\) to \(90\).
To understand how much it is increasing over time:
1. **Initial Value at \(x = 0\)**: The value starts at \(2\).
2. **Final Value at \(x = 5\)**: The value ends at \(90\).
3. **Total Increase**: The total increase over the interval is \(90 - 2 = 88\).
This large increase indicates that not only does the function rise steeply, but also that the rate of increase is significant as the function progresses through its domain from \(x = 0\) to \(x = 5\).
Thus, the function exhibits a strong upward trend in this interval, reflecting that it is increasing rapidly over time.