Question

which relationship has a zero slope?

x	y		x	y
-3	2		-3	3
-1	2		-1	1
1	2		1	-1
3	2		3	-3

(and two graphs, one line through origin with positive slope, one 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 table

Take two points from the first table, e.g., \((-3, 2)\) and \((-1, 2)\). Here, \( y_1 = 2 \), \( y_2 = 2 \), so \( y_2 - y_1 = 0 \). The slope \( m=\frac{2 - 2}{-1 - (-3)}=\frac{0}{2}=0 \).

Step3: Analyze second table

Take points \((-3, 3)\) and \((-1, 1)\). \( y_2 - y_1 = 1 - 3=-2
eq0 \), so slope is not zero.

Step4: Analyze the line through origin

This line passes through \((0,0)\) and \((1,1)\). Slope \( m=\frac{1 - 0}{1 - 0}=1
eq0 \).

Step5: Analyze vertical line

A vertical line has undefined slope (since \( x \)-values are constant, \( x_2 - x_1 = 0 \), leading to division by zero), not zero.

Answer:

The relationship represented by the first table (with \( x \) values -3, -1, 1, 3 and \( y \) value 2 for all) has a zero slope.