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:
b:
c:
d:
e:

Explanation:

Question 2

Step1: Recall Proportional Relationship Rule

A proportional relationship is a linear relationship (a straight line) that passes through the origin \((0,0)\) and can be represented by \(y = kx\) (where \(k\) is the constant of proportionality). So we check which lines pass through \((0,0)\).
Looking at the graph: Lines A, C, D, E pass through the origin \((0,0)\), while line B has a \(y\)-intercept of 4 (it starts at \((0,4)\)), so it does not pass through the origin.

Step2: Confirm Linear and Origin - Passing

Proportional relationships are linear (straight lines) and pass through \((0,0)\). Lines A, C, D, E are straight lines and pass through \((0,0)\), so they represent proportional relationships. Line B is straight but does not pass through \((0,0)\) (has a \(y\)-intercept of 4), so it is not proportional.

Answer:

Lines A, C, D, and E represent proportional relationships. We know this because a proportional relationship is a linear relationship (a straight line) that passes through the origin \((0,0)\), and lines A, C, D, E are straight lines passing through \((0,0)\), while line B does not pass through \((0,0)\) (it has a \(y\)-intercept of 4).

Question 3

The rate of change (slope) of a line \(y=mx + b\) (for proportional relationships \(b = 0\), so \(y=mx\)) is calculated as \(m=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}\). We can use two points on each line, with one point being \((0,0)\) for proportional lines (A, C, D, E) and for line B, we can use two points (e.g., \((0,4)\) and another point).

Line A:

Step1: Identify Two Points

From the graph, line A passes through \((0,0)\) and \((1,4)\) (approximate, looking at the grid).

Step2: Calculate Slope

Slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 0}{1 - 0}=4\).

Line B:

Step1: Identify Two Points

Line B passes through \((0,4)\) and \((4,5)\) (approximate, looking at the grid).

Step2: Calculate Slope

Slope \(m=\frac{5 - 4}{4 - 0}=\frac{1}{4}=0.25\).

Line C:

Step1: Identify Two Points

Line C passes through \((0,0)\) and \((4,4)\) (approximate, looking at the grid).

Step2: Calculate Slope

Slope \(m=\frac{4 - 0}{4 - 0}=1\).

Line D:

Step1: Identify Two Points

Line D passes through \((0,0)\) and \((4,2)\) (approximate, looking at the grid).

Step2: Calculate Slope

Slope \(m=\frac{2 - 0}{4 - 0}=\frac{1}{2}=0.5\).

Line E:

Step1: Identify Two Points

Line E passes through \((0,0)\) and \((4,0.5)\) (approximate, looking at the grid).

Step2: Calculate Slope

Slope \(m=\frac{0.5 - 0}{4 - 0}=\frac{1}{8}=0.125\).