To determine which of the two functions has a constant additive rate of change of [tex]\(-\frac{1}{4}\)[/tex], we need to find the rate of change for each function and compare it to [tex]\(-\frac{1}{4}\)[/tex].
### Function 1:
Given points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
20 & -1 \\
21 & -1.5 \\
22 & -2 \\
23 & -2.5 \\
\hline
\end{array}
\][/tex]
First, we calculate the rate of change between each pair of consecutive points.
The rate of change is calculated as:
[tex]\[
\text{Rate of change} = \frac{\Delta y}{\Delta x} = \frac{(y_2 - y_1)}{(x_2 - x_1)}
\][/tex]
Let's calculate the rate of change using the first two points:
[tex]\[
\text{Rate of change (Function 1)} = \frac{-1.5 - (-1)}{21 - 20} = \frac{-1.5 + 1}{1} = \frac{-0.5}{1} = -0.5
\][/tex]
### Function 2:
Given points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-12 & 7 \\
-11 & 11 \\
-10 & 14 \\
-9 & 17 \\
\hline
\end{array}
\][/tex]
Again, we calculate the rate of change between each pair of consecutive points.
Let's calculate the rate of change using the first two points:
[tex]\[
\text{Rate of change (Function 2)} = \frac{11 - 7}{-11 - (-12)} = \frac{11 - 7}{-11 + 12} = \frac{4}{1} = 4.0
\][/tex]
### Comparison with Constant Rate of Change:
We are given that the constant additive rate of change we are checking for is [tex]\( -\frac{1}{4} \)[/tex].
- For Function 1, the rate of change is [tex]\( -0.5 \)[/tex].
- For Function 2, the rate of change is [tex]\( 4.0 \)[/tex].
Neither of these rates of change match the given constant rate of change of [tex]\( -\frac{1}{4} \)[/tex].
### Conclusion:
Therefore, neither function has a constant additive rate of change of [tex]\( -\frac{1}{4} \)[/tex].
