Let's examine the two tables to determine their respective rates of change.
### First table
[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]
To find the rate of change, use the formula:
[tex]\[
\text{Rate of change} = \frac{\Delta y}{\Delta x}
\][/tex]
Take two points: [tex]\((20, -1)\)[/tex] and [tex]\((21, -1.5)\)[/tex]:
[tex]\[
\Delta y = -1.5 - (-1) = -1.5 + 1 = -0.5
\][/tex]
[tex]\[
\Delta x = 21 - 20 = 1
\][/tex]
[tex]\[
\text{Rate of change} = \frac{-0.5}{1} = -0.5
\][/tex]
### Second table
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-12 & 7 \\
\hline
-11 & 11 \\
\hline
-10 & 14 \\
\hline
-9 & 17 \\
\hline
\end{array}
\][/tex]
Again, use the formula for the rate of change:
[tex]\[
\text{Rate of change} = \frac{\Delta y}{\Delta x}
\][/tex]
Take two points: [tex]\((-12, 7)\)[/tex] and [tex]\((-11, 11)\)[/tex]:
[tex]\[
\Delta y = 11 - 7 = 4
\][/tex]
[tex]\[
\Delta x = -11 - (-12) = -11 + 12 = 1
\][/tex]
[tex]\[
\text{Rate of change} = \frac{4}{1} = 4
\][/tex]
### Comparing Rates of Change
- The rate of change for the first table is [tex]\(-0.5\)[/tex].
- The rate of change for the second table is [tex]\(4.0\)[/tex].
To find which function has a constant additive rate of change of [tex]\(-\frac{1}{4}\)[/tex], compare the given constant rate with the computed rates:
- The first table has a rate of change of [tex]\(-0.5\)[/tex].
- The second table has a rate of change of [tex]\(4.0\)[/tex].
Neither [tex]\(-0.5\)[/tex] nor [tex]\(4.0\)[/tex] match the constant rate of [tex]\(-\frac{1}{4}\)[/tex]. Therefore, neither of the given tables has a constant additive rate of change of [tex]\(-\frac{1}{4}\)[/tex].
