which relationship has a zero slope?\n| x | y |\n| -3 | 2 |\n| -1 | 2 |\n| 1 | 2 |\n| 3 | 2 |\n| x | y |\n|…

Explanation:

Step1: Recall slope - formula

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. A zero - slope means $y_2 - y_1 = 0$ (since the denominator $x_2 - x_1\neq0$ for a well - defined slope). That is, the $y$ - values are constant for different $x$ - values.

Step2: Analyze the first table

In the first table, when $x=-3,y = 2$; when $x=-1,y = 2$; when $x = 1,y = 2$; when $x = 3,y = 2$. The $y$ - value is always 2 regardless of the $x$ - value. Using the slope formula with any two points $(x_1,y_1)$ and $(x_2,y_2)$ from this table, say $(x_1=-3,y_1 = 2)$ and $(x_2 = 1,y_2 = 2)$, then $m=\frac{2 - 2}{1-(-3)}=\frac{0}{4}=0$.

Step3: Analyze the second table

In the second table, when $x=-3,y = 3$; when $x=-1,y = 1$; when $x = 1,y=-1$; when $x = 3,y=-3$. For example, if we take $(x_1=-3,y_1 = 3)$ and $(x_2=-1,y_2 = 1)$, then $m=\frac{1 - 3}{-1-(-3)}=\frac{-2}{2}=-1\neq0$.

Step4: Analyze the first graph

The first graph is a non - vertical and non - horizontal line. If we take two points on the line, say $(0,0)$ and $(1,1)$, then $m=\frac{1 - 0}{1 - 0}=1\neq0$.

Step5: Analyze the second graph

The second graph is a vertical line. The slope of a vertical line is undefined because for a vertical line $x_2 - x_1=0$, and the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ has a zero denominator.

Answer:

The first table (where $x=-3,y = 2$; $x=-1,y = 2$; $x = 1,y = 2$; $x = 3,y = 2$) has a zero slope.