To determine which option represents a function with a lesser rate of change than the function \(y = -\frac{4}{5}x + 5\), we first need to establish the rate of change (slope) of the given function. The slope here is \(-\frac{4}{5}\).
A function has a lesser rate of change if its slope (rate of change) is greater than \(-\frac{4}{5}\). This means we are looking for a relationship where the slope is less negative (or more positive) than \(-\frac{4}{5}\).
Let’s analyze each option:
Option A
We have:
- \( (-6, -6) \)
- \( (-2, 9) \)
- \( (2, -11) \)
- \( (6, -31) \)
Calculate the slope between two points:
- Between \((-6, -6)\) and \((-2, 9)\): \[ \text{slope} = \frac{9 - (-6)}{-2 - (-6)} = \frac{15}{4} \]
- Between \((-2, 9)\) and \((2, -11)\): \[ \text{slope} = \frac{-11 - 9}{2 - (-2)} = \frac{-20}{4} = -5 \]
- Between \((2, -11)\) and \((6, -31)\): \[ \text{slope} = \frac{-31 - (-11)}{6 - 2} = \frac{-20}{4} = -5 \]
The slopes calculated above are both much less than \(-\frac{4}{5}\), so Option A does not have a lesser rate of change.
Option B
We have:
- \( (-6, -6) \)
- \( (-2, 1) \)
- \( (2, 9) \)
- \( (6, 14) \)
Calculate the slope:
- Between \((-6, -6)\) and \((-2, 1)\): \[ \text{slope} = \frac{1 - (-6)}{-2 - (-6)} = \frac{7}{4} \]
- Between \((-2, 1)\) and \((2, 9)\): \[ \text{slope} = \frac{9 - 1}{2 - (-2)} = \frac{8}{4} = 2 \]
- Between \((2, 9)\) and \((6, 14)\): \[ \text{slope} = \frac{14 - 9}{6 - 2} = \frac{5}{4} \]
The slopes in this option are all greater than \(-\frac{4}{5}\), so Option B does not represent a lesser rate of change.
Option C
We have:
- \( (-6, -6) \)
- \( (-2, 9) \)
- \( (2, -11) \)
- \( (6, -31) \)
Already calculated slopes show that they are negative and generally large negative, validating that Option C has not a lesser rate of change.
Option D
We have:
- \( (-6, -4) \)
- \( (-2, 2) \)
- \( (2, 8) \)
- \( (6, 14) \)
Calculate the slope:
- Between \((-6, -4)\) and \((-2, 2)\): \[ \text{slope} = \frac{2 - (-4)}{-2 - (-6)} = \frac{6}{4} = \frac{3}{2} \]
- Between \((-2, 2)\) and \((2, 8)\): \[ \text{slope} = \frac{8 - 2}{2 - (-2)} = \frac{6}{4} = \frac{3}{2} \]
- Between \((2, 8)\) and \((6, 14)\): \[ \text{slope} = \frac{14 - 8}{6 - 2} = \frac{6}{4}= \frac{3}{2} \]
All slopes in Option D seem valid because they are all positive, confirming they are greater than \(-4/5\).
The correct choice representing a function with a lesser rate of change than \(y = -\frac{4}{5}x + 5\) is Option D.