Question

use the graph to answer questions 2–5.

1. which lines represent proportional relationships?

explain how you know.

1. find the rate of change for each line.

a: 4
b: 0.25
c: 1
d: 0.5
e: 0.125

1. find the constant of proportionality for each proportional relationship.
2. write a function rule for each proportional relationship.

(graph: a coordinate plane with lines a, b, c, d, e. line a passes through (0,0) and (1,4). line b has a y-intercept of 4. line c passes through (0,0) and (1,1). line d passes through (0,0) and (4,2). line e passes through (0,0) and (6,1).)

Explanation:

Step1: Recall Proportional Relationship Rule

A proportional relationship is a linear relationship (a straight line) that passes through the origin \((0,0)\). So we check which lines pass through \((0,0)\).
Looking at the graph: Line A, Line C, Line D, Line E pass through the origin. Line B does not (it starts at \((0,4)\)).

Step2: Confirm with Definition

The equation of a proportional relationship is \(y = kx\), where \(k\) is the constant of proportionality, and when \(x = 0\), \(y=0\). So lines passing through \((0,0)\) are proportional. From the graph, A, C, D, E pass through \((0,0)\), B does not (y - intercept is 4).

Answer:

Lines A, C, D, and E represent proportional relationships because they are straight lines that pass through the origin \((0,0)\), which is a key characteristic of proportional relationships (their equations are of the form \(y = kx\) where \(k\) is the constant of proportionality, and when \(x = 0\), \(y = 0\)).