Answer :
To determine which function has a constant additive rate of change of [tex]\(-\frac{1}{4}\)[/tex], we need to calculate the rate of change (slope) for each tabular data set. The rate of change is given by the difference in [tex]\(y\)[/tex]-values divided by the difference in [tex]\(x\)[/tex]-values between points.
Let's evaluate each dataset.
### First Dataset
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 20 & -1 \\ \hline 21 & -1.5 \\ \hline 22 & -2 \\ \hline 23 & -2.5 \\ \hline \end{array} \][/tex]
We calculate the rate of change between consecutive points:
1. Between [tex]\( (20, -1) \)[/tex] and [tex]\( (21, -1.5) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{-1.5 - (-1)}{21 - 20} = \frac{-1.5 + 1}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
2. To confirm this rate is consistent, check between [tex]\( (21, -1.5) \)[/tex] and [tex]\( (22, -2) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{-2 - (-1.5)}{22 - 21} = \frac{-2 + 1.5}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
3. Finally, check between [tex]\( (22, -2) \)[/tex] and [tex]\( (23, -2.5) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{-2.5 - (-2)}{23 - 22} = \frac{-2.5 + 2}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
The rate of change is consistently [tex]\(-0.5\)[/tex].
### Second Dataset
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -12 & 7 \\ \hline -11 & 11 \\ \hline -10 & 14 \\ \hline -9 & 17 \\ \hline \end{array} \][/tex]
We calculate the rate of change between consecutive points:
1. Between [tex]\( (-12, 7) \)[/tex] and [tex]\( (-11, 11) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{11 - 7}{-11 - (-12)} = \frac{11 - 7}{-11 + 12} = \frac{4}{1} = 4 \][/tex]
2. To confirm this rate is consistent, check between [tex]\( (-11, 11) \)[/tex] and [tex]\( (-10, 14) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{14 - 11}{-10 - (-11)} = \frac{14 - 11}{-10 + 11} = \frac{3}{1} = 3 \][/tex]
3. Finally, check between [tex]\( (-10, 14) \)[/tex] and [tex]\( (-9, 17) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{17 - 14}{-9 - (-10)} = \frac{17 - 14}{-9 + 10} = \frac{3}{1} = 3 \][/tex]
The rate of change in this dataset is not constant and varies.
### Conclusion
The first dataset has a constant rate of change of [tex]\(-0.5\)[/tex], and the second dataset does not demonstrate a constant rate of change of [tex]\(-\frac{1}{4}\)[/tex]. Thus, neither dataset has a constant rate of change of [tex]\(-\frac{1}{4}\)[/tex]. The function with a constant additive rate of change of [tex]\(-\frac{1}{4}\)[/tex] is not present in the given datasets.
Let's evaluate each dataset.
### First Dataset
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 20 & -1 \\ \hline 21 & -1.5 \\ \hline 22 & -2 \\ \hline 23 & -2.5 \\ \hline \end{array} \][/tex]
We calculate the rate of change between consecutive points:
1. Between [tex]\( (20, -1) \)[/tex] and [tex]\( (21, -1.5) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{-1.5 - (-1)}{21 - 20} = \frac{-1.5 + 1}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
2. To confirm this rate is consistent, check between [tex]\( (21, -1.5) \)[/tex] and [tex]\( (22, -2) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{-2 - (-1.5)}{22 - 21} = \frac{-2 + 1.5}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
3. Finally, check between [tex]\( (22, -2) \)[/tex] and [tex]\( (23, -2.5) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{-2.5 - (-2)}{23 - 22} = \frac{-2.5 + 2}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
The rate of change is consistently [tex]\(-0.5\)[/tex].
### Second Dataset
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -12 & 7 \\ \hline -11 & 11 \\ \hline -10 & 14 \\ \hline -9 & 17 \\ \hline \end{array} \][/tex]
We calculate the rate of change between consecutive points:
1. Between [tex]\( (-12, 7) \)[/tex] and [tex]\( (-11, 11) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{11 - 7}{-11 - (-12)} = \frac{11 - 7}{-11 + 12} = \frac{4}{1} = 4 \][/tex]
2. To confirm this rate is consistent, check between [tex]\( (-11, 11) \)[/tex] and [tex]\( (-10, 14) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{14 - 11}{-10 - (-11)} = \frac{14 - 11}{-10 + 11} = \frac{3}{1} = 3 \][/tex]
3. Finally, check between [tex]\( (-10, 14) \)[/tex] and [tex]\( (-9, 17) \)[/tex]:
[tex]\[ \text{Rate of change} = \frac{17 - 14}{-9 - (-10)} = \frac{17 - 14}{-9 + 10} = \frac{3}{1} = 3 \][/tex]
The rate of change in this dataset is not constant and varies.
### Conclusion
The first dataset has a constant rate of change of [tex]\(-0.5\)[/tex], and the second dataset does not demonstrate a constant rate of change of [tex]\(-\frac{1}{4}\)[/tex]. Thus, neither dataset has a constant rate of change of [tex]\(-\frac{1}{4}\)[/tex]. The function with a constant additive rate of change of [tex]\(-\frac{1}{4}\)[/tex] is not present in the given datasets.
