Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
