To analyze the functions based on the given criteria, let's break down each function.
Function Analysis:
Function A:
The points (8, 0), (4, -1), and (0, -2) can be used to determine the slope (rate of change). We can see that it moves from (4, -1) to (0, -2) which has a slope:
\[
\text{slope} = \frac{-2 - (-1)}{0 - 4} = \frac{-1}{-4} = \frac{1}{4}
\]
So, the average slope from (8, 0) to (0, -2) will be:
\[
\text{slope} = \frac{-2 - 0}{0 - 8} = \frac{-2}{-8} = \frac{1}{4}
\]
Function B:
The pairs (0, 1), (4, 9), (8, 17) show that it appears to be a quadratic function. By calculating the change in y for equal changes in x, we find:
- From 0 to 4: \( 9 - 1 = 8 \) (change in y for change in x of 4)
- From 4 to 8: \( 17 - 9 = 8 \) (change in y for change in x of 4)
This shows that while the increase is consistent, the rate of change is quadratic but not constant.
Function C:
This is a linear function represented as \( y = x + 20 \). The slope here is 1 (constant rate of change). The y-intercept is 20.
Evaluating the Criteria:
-
Function with the greatest rate of change:
- Function A: Ratio analysis gives a slope of \( \frac{1}{4} \) (upward but not steep).
- Function B: Appears quadratic. Around the range given, it looks to have varying steepness, particularly since it seems to increase quickly at higher values.
- Function C: Linear with a slope of 1.
Conclusion: Function C has the greatest constant rate of change.
-
Function with the smallest y-intercept:
- Function A: y-intercept is -2.
- Function B: y-intercept is 1.
- Function C: y-intercept is 20.
Conclusion: Function A has the smallest y-intercept.
-
Which function would benefit you the most if it represented your money earned per hour?
- Function A: Could imply negative earnings at specific intervals.
- Function B: Earnings seem to steadily increase, but they are not linear.
- Function C: Linear growth indicating positive earnings.
Conclusion: Function C would benefit the most as it represents consistent positive earnings.
-
Function with the highest initial amount of money:
- Function A: Starts at 0 at x=8 (negative initial).
- Function B: Starts at 1.
- Function C: Starts at 20.
Conclusion: Function C has the highest initial amount of money.
Final Answers:
- Function with the greatest rate of change: Function C
- Function with the smallest y-intercept: Function A
- Which function would benefit you the most if it represented your money earned per hour?: Function C
- Function with the highest initial amount of money: Function C