Question

which relationship has a zero slope? \
(image of a line with positive slope) \
(table with x: -3, -1, 1, 3; y: 2, 2, 2, 2) \
(table with x: -3, -1, 1, 3; y: 3, 1, -1, -3) \
(image of a vertical line)

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). A zero slope means \( y_2 - y_1 = 0 \) (i.e., \( y \)-values are constant for different \( x \)-values).

Step2: Analyze first graph

The first graph is a line with a positive slope (since \( y \) increases as \( x \) increases), so slope is not zero.

Step3: Analyze second table

Take two points, e.g., \((-3, 2)\) and \((-1, 2)\). \( y_2 - y_1 = 2 - 2 = 0 \), \( x_2 - x_1=-1 - (-3)=2 \). Slope \( m=\frac{0}{2}=0 \). Check another pair: \((1, 2)\) and \((3, 2)\), \( y_2 - y_1 = 0 \), \( x_2 - x_1 = 2 \), slope \( 0 \). So this table has zero slope.

Step4: Analyze third table

Take \((-3, 3)\) and \((-1, 1)\). \( y_2 - y_1 = 1 - 3=-2 \), \( x_2 - x_1=-1 - (-3)=2 \). Slope \( m=\frac{-2}{2}=-1
eq0 \).

Step5: Analyze fourth graph

This is a vertical line, slope is undefined (since \( x \)-values are constant, \( x_2 - x_1 = 0 \), division by zero).

Answer:

The relationship in the second table (with \( x \) values \(-3, -1, 1, 3\) and \( y \) values all \( 2 \)) has a zero slope.