Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To determine which function has a constant additive rate of change of -14, we need to analyze the changes in the [tex]\( y \)[/tex]-values with respect to the changes in the [tex]\( x \)[/tex]-values for the provided data tables. A constant rate of change means that the difference between consecutive [tex]\( y \)[/tex]-values divided by the difference between consecutive [tex]\( x \)[/tex]-values remains the same.
Let's analyze each dataset step-by-step.
Dataset 1:
[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. From [tex]\( x = 20 \)[/tex] to [tex]\( x = 21 \)[/tex]:
[tex]\[ \frac{-1.5 - (-1)}{21 - 20} = \frac{-1.5 + 1}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
2. From [tex]\( x = 21 \)[/tex] to [tex]\( x = 22 \)[/tex]:
[tex]\[ \frac{-2 - (-1.5)}{22 - 21} = \frac{-2 + 1.5}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
3. From [tex]\( x = 22 \)[/tex] to [tex]\( x = 23 \)[/tex]:
[tex]\[ \frac{-2.5 - (-2)}{23 - 22} = \frac{-2.5 + 2}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
The rate of change in Dataset 1 is consistently [tex]\(-0.5\)[/tex]. This is not equal to [tex]\(-14\)[/tex].
Dataset 2:
[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. From [tex]\( x = -12 \)[/tex] to [tex]\( x = -11 \)[/tex]:
[tex]\[ \frac{11 - 7}{-11 - (-12)} = \frac{11 - 7}{-11 + 12} = \frac{4}{1} = 4 \][/tex]
2. From [tex]\( x = -11 \)[/tex] to [tex]\( x = -10 \)[/tex]:
[tex]\[ \frac{14 - 11}{-10 - (-11)} = \frac{14 - 11}{-10 + 11} = \frac{3}{1} = 3 \][/tex]
3. From [tex]\( x = -10 \)[/tex] to [tex]\( x = -9 \)[/tex]:
[tex]\[ \frac{17 - 14}{-9 - (-10)} = \frac{17 - 14}{-9 + 10} = \frac{3}{1} = 3 \][/tex]
The rate of change in Dataset 2 is [tex]\( [4.0, 3.0, 3.0] \)[/tex]. We see that the rate of change is not consistent and certainly not [tex]\(-14\)[/tex].
Conclusion:
Based on the analysis, neither dataset has a constant additive rate of change of [tex]\(-14\)[/tex]. Dataset 1 has a consistent rate of change of [tex]\(-0.5\)[/tex], and Dataset 2 does not have a consistent rate of change at all (4.0, 3.0, 3.0). Therefore, none of the given functions have a constant additive rate of change of [tex]\(-14\)[/tex].
Let's analyze each dataset step-by-step.
Dataset 1:
[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. From [tex]\( x = 20 \)[/tex] to [tex]\( x = 21 \)[/tex]:
[tex]\[ \frac{-1.5 - (-1)}{21 - 20} = \frac{-1.5 + 1}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
2. From [tex]\( x = 21 \)[/tex] to [tex]\( x = 22 \)[/tex]:
[tex]\[ \frac{-2 - (-1.5)}{22 - 21} = \frac{-2 + 1.5}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
3. From [tex]\( x = 22 \)[/tex] to [tex]\( x = 23 \)[/tex]:
[tex]\[ \frac{-2.5 - (-2)}{23 - 22} = \frac{-2.5 + 2}{1} = \frac{-0.5}{1} = -0.5 \][/tex]
The rate of change in Dataset 1 is consistently [tex]\(-0.5\)[/tex]. This is not equal to [tex]\(-14\)[/tex].
Dataset 2:
[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. From [tex]\( x = -12 \)[/tex] to [tex]\( x = -11 \)[/tex]:
[tex]\[ \frac{11 - 7}{-11 - (-12)} = \frac{11 - 7}{-11 + 12} = \frac{4}{1} = 4 \][/tex]
2. From [tex]\( x = -11 \)[/tex] to [tex]\( x = -10 \)[/tex]:
[tex]\[ \frac{14 - 11}{-10 - (-11)} = \frac{14 - 11}{-10 + 11} = \frac{3}{1} = 3 \][/tex]
3. From [tex]\( x = -10 \)[/tex] to [tex]\( x = -9 \)[/tex]:
[tex]\[ \frac{17 - 14}{-9 - (-10)} = \frac{17 - 14}{-9 + 10} = \frac{3}{1} = 3 \][/tex]
The rate of change in Dataset 2 is [tex]\( [4.0, 3.0, 3.0] \)[/tex]. We see that the rate of change is not consistent and certainly not [tex]\(-14\)[/tex].
Conclusion:
Based on the analysis, neither dataset has a constant additive rate of change of [tex]\(-14\)[/tex]. Dataset 1 has a consistent rate of change of [tex]\(-0.5\)[/tex], and Dataset 2 does not have a consistent rate of change at all (4.0, 3.0, 3.0). Therefore, none of the given functions have a constant additive rate of change of [tex]\(-14\)[/tex].
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
