Question
Graph starts at 0,1.5 and goes through 7,12.5 what is the average rate of change of a function on the interval x:4,8
Answers
GPT-4o mini
To find the average rate of change of a function on the interval \( x = 4 \) to \( x = 8 \), we need to use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a \) and \( b \) are the endpoints of the interval, and \( f(a) \) and \( f(b) \) are the corresponding values of the function at those points.
From the information provided, we have two points on the graph: \( (0, 1.5) \) and \( (7, 12.5) \). However, we do not have the function values \( f(4) \) and \( f(8) \) needed to calculate the average rate of change over that interval.
To estimate or determine the function values at \( x = 4 \) and \( x = 8 \), we can utilize a linear interpolation method between the two known points.
1. **Find the slope (rate of change) between the points (0, 1.5) and (7, 12.5):**
\[
\text{slope} = \frac{12.5 - 1.5}{7 - 0} = \frac{11}{7} \approx 1.57
\]
2. **Equation of line using point-slope form (using the point (0, 1.5)):**
\[
y - 1.5 = \frac{11}{7}(x - 0)
\]
Rearranging gives us:
\[
y = \frac{11}{7}x + 1.5
\]
3. **Calculate \( f(4) \) and \( f(8) \):**
- For \( x = 4 \):
\[
f(4) = \frac{11}{7}(4) + 1.5 = \frac{44}{7} + \frac{10.5}{7} = \frac{54.5}{7} \approx 7.7857 \text{ (exactly } \frac{109}{14}\text{)}
\]
- For \( x = 8 \):
\[
f(8) = \frac{11}{7}(8) + 1.5 = \frac{88}{7} + \frac{10.5}{7} = \frac{98.5}{7} \approx 14.0714 \text{ (exactly } \frac{207}{14}\text{)}
\]
4. **Now calculate the average rate of change from 4 to 8:**
\[
\text{Average Rate of Change} = \frac{f(8) - f(4)}{8 - 4} = \frac{\frac{98.5}{7} - \frac{54.5}{7}}{4} = \frac{\frac{44}{7}}{4} = \frac{44}{28} = \frac{11}{7} \approx 1.5714
\]
So, the average rate of change of the function on the interval from \( x = 4 \) to \( x = 8 \) is \( \frac{11}{7} \) or approximately \( 1.57 \).
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a \) and \( b \) are the endpoints of the interval, and \( f(a) \) and \( f(b) \) are the corresponding values of the function at those points.
From the information provided, we have two points on the graph: \( (0, 1.5) \) and \( (7, 12.5) \). However, we do not have the function values \( f(4) \) and \( f(8) \) needed to calculate the average rate of change over that interval.
To estimate or determine the function values at \( x = 4 \) and \( x = 8 \), we can utilize a linear interpolation method between the two known points.
1. **Find the slope (rate of change) between the points (0, 1.5) and (7, 12.5):**
\[
\text{slope} = \frac{12.5 - 1.5}{7 - 0} = \frac{11}{7} \approx 1.57
\]
2. **Equation of line using point-slope form (using the point (0, 1.5)):**
\[
y - 1.5 = \frac{11}{7}(x - 0)
\]
Rearranging gives us:
\[
y = \frac{11}{7}x + 1.5
\]
3. **Calculate \( f(4) \) and \( f(8) \):**
- For \( x = 4 \):
\[
f(4) = \frac{11}{7}(4) + 1.5 = \frac{44}{7} + \frac{10.5}{7} = \frac{54.5}{7} \approx 7.7857 \text{ (exactly } \frac{109}{14}\text{)}
\]
- For \( x = 8 \):
\[
f(8) = \frac{11}{7}(8) + 1.5 = \frac{88}{7} + \frac{10.5}{7} = \frac{98.5}{7} \approx 14.0714 \text{ (exactly } \frac{207}{14}\text{)}
\]
4. **Now calculate the average rate of change from 4 to 8:**
\[
\text{Average Rate of Change} = \frac{f(8) - f(4)}{8 - 4} = \frac{\frac{98.5}{7} - \frac{54.5}{7}}{4} = \frac{\frac{44}{7}}{4} = \frac{44}{28} = \frac{11}{7} \approx 1.5714
\]
So, the average rate of change of the function on the interval from \( x = 4 \) to \( x = 8 \) is \( \frac{11}{7} \) or approximately \( 1.57 \).