Answer :
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].
