To determine which function has a lesser rate of change than the linear function represented by the data in the table, we first need to find the rate of change (slope) for the original function.
Let’s analyze the given points from the table:
- \((-4, 10)\)
- \((0, 3)\)
- \((4, -4)\)
- \((8, -11)\)
We can calculate the slope (rate of change) \( m \) using two points, for example, using the points \((0, 3)\) and \((4, -4)\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 3}{4 - 0} = \frac{-7}{4} \]
The slope of the given function is \(-\frac{7}{4}\).
Now, we compare this slope with the slopes of the given answer choices. A lesser rate of change means a slope that is closer to zero than \(-\frac{7}{4}\):
- Option A: \(y = -5x - 5\) has a slope of \(-5\).
- Option B: Insufficient information is provided to evaluate.
- Option C: \(y = -\frac{7}{4}x + 2\) has a slope of \(-\frac{7}{4}\).
- Option D: Insufficient information is provided to evaluate.
To compare:
- A slope of \(-5\) is less than a slope of \(-\frac{7}{4}\) (since \(-5 < -1.75\)).
- The slope of \(-\frac{7}{4}\) is equal to the slope in option C.
Thus, the only option that has a lesser rate of change (slope that is closer to zero than \(-\frac{7}{4}\)) is:
Option A: \(y = -5x - 5\).