Given input and output values for three mappings, determine the rule (equation) for each mapping.

The inputs are $-3, -2, -1, 0, 1$. The corresponding outputs are $14, 9, 4, -1, -6$.

Observe the change in $y$ for each unit increase in $x$:
$9 - 14 = -5$
$4 - 9 = -5$
$-1 - 4 = -5$
$-6 - (-1) = -5$
The output decreases by 5 for every 1 unit increase in the input. This means the slope $m = -5$.

Using the linear equation $y = mx + c$ and a point, for example $(0, -1)$:

The rule for this mapping is $y = -5x - 1$.

Mapping 2:

The inputs are $-10, -9, -8, -7, -6$. The corresponding outputs are $16, 14, 12, 10, 8$.

Observe the change in $y$ for each unit increase in $x$:
$14 - 16 = -2$
$12 - 14 = -2$
$10 - 12 = -2$
$8 - 10 = -2$
The output decreases by 2 for every 1 unit increase in the input. This means the slope $m = -2$.

Using the linear equation $y = mx + c$ and a point, for example $(-6, 8)$:

The rule for this mapping is $y = -2x - 4$.

Mapping 3:

The inputs are $1, 2, 3, 4, 5$. The corresponding outputs are $-6, -8, -10, -12, -14$.

Observe the change in $y$ for each unit increase in $x$:
$-8 - (-6) = -2$
$-10 - (-8) = -2$
$-12 - (-10) = -2$
$-14 - (-12) = -2$
The output decreases by 2 for every 1 unit increase in the input. This means the slope $m = -2$.

Using the linear equation $y = mx + c$ and a point, for example $(1, -6)$:

The rule for this mapping is $y = -2x - 4$.

Mapping 4:

The inputs are $1, 2, 3, 4, 5$. The corresponding outputs are $8, 15, 22, 29, 30$.

Observe the change in $y$ for each unit increase in $x$:
$15 - 8 = 7$
$22 - 15 = 7$
$29 - 22 = 7$
The output increases by 7 for every 1 unit increase in the input. This means the slope $m = 7$.

Using the linear equation $y = mx + c$ and a point, for example $(1, 8)$:

The rule for this mapping is $y = 7x + 1$.

Using the rule $y = 7x + 1$ for $x=5$:

The value written in the image for $x=5$ is 30. Based on the established pattern, the value should be 36.

The rules for the mappings are:

• Mapping 1: $\boxed{y = -5x - 1}$
• Mapping 2: $\boxed{y = -2x - 4}$
• Mapping 3: $\boxed{y = -2x - 4}$
• Mapping 4: $\boxed{y = 7x + 1}$ (with the last value 30 likely being a mistake, it should be 36)