To determine which function has a constant rate of change equal to [tex]\(-3\)[/tex], we need to check the rate of change (or the slope) for each given function.
1. Table of points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 2 \\
\hline
1 & 5 \\
\hline
2 & 8 \\
\hline
3 & 11 \\
\hline
\end{array}
\][/tex]
Calculate the change in [tex]\(y\)[/tex] for each unit increase in [tex]\(x\)[/tex]:
- From [tex]\(x = 0\)[/tex] to [tex]\(x = 1\)[/tex]: [tex]\(5 - 2 = 3\)[/tex]
- From [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex]: [tex]\(8 - 5 = 3\)[/tex]
- From [tex]\(x = 2\)[/tex] to [tex]\(x = 3\)[/tex]: [tex]\(11 - 8 = 3\)[/tex]
The rate of change is constant and equal to 3, not -3.
2. Set of points:
[tex]\(\{(1, 5), (2, 2), (3, -5), (4, -8)\}\)[/tex]
Calculate the change in [tex]\(y\)[/tex] for each unit increase in [tex]\(x\)[/tex]:
- From [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex]: [tex]\(2 - 5 = -3\)[/tex]
- From [tex]\(x = 2\)[/tex] to [tex]\(x = 3\)[/tex]: [tex]\(-5 - 2 = -7\)[/tex]
- From [tex]\(x = 3\)[/tex] to [tex]\(x = 4\)[/tex]: [tex]\(-8 - (-5) = -3\)[/tex]
The rate of change is not constant, so this does not meet our criteria.
3. Set of points:
[tex]\(\{(1, 5), (2, 2), (3, -1), (4, -4)\}\)[/tex]
Calculate the change in [tex]\(y\)[/tex] for each unit increase in [tex]\(x\)[/tex]:
- From [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex]: [tex]\(2 - 5 = -3\)[/tex]
- From [tex]\(x = 2\)[/tex] to [tex]\(x = 3\)[/tex]: [tex]\(-1 - 2 = -3\)[/tex]
- From [tex]\(x = 3\)[/tex] to [tex]\(x = 4\)[/tex]: [tex]\(-4 - (-1) = -3\)[/tex]
The rate of change is constant and equal to -3.
4. Linear function:
[tex]\[2y = -6x + 10\][/tex]
Simplify the equation to find the slope:
[tex]\[y = -3x + 5\][/tex]
The slope (rate of change) here is -3.
Therefore, the functions that have a constant rate of change equal to -3 are:
3) [tex]\(\{(1, 5), (2, 2), (3, -1), (4, -4)\}\)[/tex]
4) [tex]\(2y = -6x + 10\)[/tex]
Thus, the answer is [tex]\(3\)[/tex] and [tex]\(4\)[/tex].
