To compare the rate of change of the three functions, we need to analyze each function carefully.
Function A: This function has points at (0, 2) and (6, 7). To find the rate of change (slope), we can use the formula:
\[ \text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 2}{6 - 0} = \frac{5}{6} \approx 0.8333 \]
Function B: Given the values in a table (0, 2.50), (2, 4.03), (4, 5.56), (6, 7.09), we can calculate the average rate of change over the interval from x = 0 to x = 6:
\[ \text{slope} = \frac{y(6) - y(0)}{6 - 0} = \frac{7.09 - 2.50}{6 - 0} = \frac{4.59}{6} \approx 0.765 \]
To check the rate of change over smaller intervals:
- From (0, 2.5) to (2, 4.03): \[ \text{slope} = \frac{4.03 - 2.50}{2 - 0} = \frac{1.53}{2} = 0.765 \]
- From (2, 4.03) to (4, 5.56): \[ \text{slope} = \frac{5.56 - 4.03}{4 - 2} = \frac{1.53}{2} = 0.765 \]
- From (4, 5.56) to (6, 7.09): \[ \text{slope} = \frac{7.09 - 5.56}{6 - 4} = \frac{1.53}{2} = 0.765 \]
Thus, the average rate of change for Function B also comes out to be approximately 0.765.
Function C: The equation \(y = 1.3x + 1\) clearly has a slope of 1.3.
Now that we have determined the rate of change for each function:
- Function A: Slope ≈ 0.8333
- Function B: Average slope ≈ 0.765
- Function C: Slope = 1.3
Ordering from least to greatest rate of change:
- Function B (approximately 0.765)
- Function A (approximately 0.8333)
- Function C (1.3)
So, the correct order is:
- Least: Function B
- Function A
- Greatest: Function C