2.2 Determine the rule for finding y in the given table 4.1 2.3 Test the rule for when x = -2;-1;0;2;7

Based on the table from the previous question, we observe the following $(x, Y)$ pairs: $(-2, -5)$, $(-1, -2)$, $(0, 1)$, $(1, 4)$, $(2, 7)$.
For every unit increase in $x$, the value of $Y$ increases by 3 (e.g., from $x=0, Y=1$ to $x=1, Y=4$, $Y$ increases by $4-1=3$). This indicates a linear relationship of the form $Y = mx + c$, where $m$ is the slope.
The slope $m = 3$.
To find the y-intercept $c$, we can use any point from the table, for example, $(0, 1)$:

$$Y = mx + c$$ $$1 = 3(0) + c$$ $$1 = c$$

Thus, the rule for finding $Y$ is:

$$\boxed{Y = 3x + 1}$$

Substitute $x = -2$ into the rule $Y = 3x + 1$:

$$Y = 3(-2) + 1$$ $$Y = -6 + 1$$ $$Y = -5$$

Substitute $x = -1$ into the rule $Y = 3x + 1$:

$$Y = 3(-1) + 1$$ $$Y = -3 + 1$$ $$Y = -2$$

Substitute $x = 0$ into the rule $Y = 3x + 1$:

$$Y = 3(0) + 1$$ $$Y = 0 + 1$$ $$Y = 1$$

Substitute $x = 2$ into the rule $Y = 3x + 1$:

$$Y = 3(2) + 1$$ $$Y = 6 + 1$$ $$Y = 7$$

Substitute $x = 7$ into the rule $Y = 3x + 1$:

$$Y = 3(7) + 1$$ $$Y = 21 + 1$$ $$Y = 22$$