Sure, let's examine how the [tex]\( y \)[/tex]-values are increasing over each interval step-by-step.
Given the input-output table for the linear function [tex]\( y = 3x \)[/tex]:
[tex]\[
\begin{array}{|r|r|}
\hline
x & y \\
\hline
-2 & -6 \\
\hline
-1 & -3 \\
\hline
0 & 0 \\
\hline
1 & 3 \\
\hline
2 & 6 \\
\hline
3 & 9 \\
\hline
\end{array}
\][/tex]
We need to determine how the [tex]\( y \)[/tex]-values increase as [tex]\( x \)[/tex] changes.
### Step-by-Step Analysis:
1. From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]:
[tex]\[
y\text{-value changes from } -6 \text{ to } -3.
\][/tex]
The difference is:
[tex]\[
-3 - (-6) = 3
\][/tex]
2. From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]:
[tex]\[
y\text{-value changes from } -3 \text{ to } 0.
\][/tex]
The difference is:
[tex]\[
0 - (-3) = 3
\][/tex]
3. From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]:
[tex]\[
y\text{-value changes from } 0 \text{ to } 3.
\][/tex]
The difference is:
[tex]\[
3 - 0 = 3
\][/tex]
4. From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
y\text{-value changes from } 3 \text{ to } 6.
\][/tex]
The difference is:
[tex]\[
6 - 3 = 3
\][/tex]
5. From [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[
y\text{-value changes from } 6 \text{ to } 9.
\][/tex]
The difference is:
[tex]\[
9 - 6 = 3
\][/tex]
### Summary:
After calculating the differences for each interval, we observe that the [tex]\( y \)[/tex]-value increases by 3 in each interval.
Thus, the best description of how the [tex]\( y \)[/tex]-values are increasing over each interval is:
[tex]\[
\textbf{Add 3 over each interval.}
\][/tex]
