Question

is the relationship proportional? the points lie on a? line that? through the origin. that means the relationship? proportional.

Explanation:

Step1: Check if points lie on a straight line

Looking at the table and graph, when time \(x = 1\), distance \(y=200\); \(x = 2\), \(y = 400\); \(x=3\), \(y = 600\); \(x = 4\), \(y=800\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \((x_1,y_1)=(1,200)\) and \((x_2,y_2)=(2,400)\), \(m=\frac{400 - 200}{2 - 1}=200\). For \((x_1,y_1)=(2,400)\) and \((x_2,y_2)=(3,600)\), \(m=\frac{600 - 400}{3 - 2}=200\). For \((x_1,y_1)=(3,600)\) and \((x_2,y_2)=(4,800)\), \(m=\frac{800 - 600}{4 - 3}=200\). The points lie on a straight line.

Step2: Check if the line passes through the origin

The equation of the line using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(1,200)\) and \(m = 200\) is \(y-200=200(x - 1)\), which simplifies to \(y=200x\). When \(x = 0\), \(y=0\). So the line passes through the origin \((0,0)\).

Step3: Determine if the relationship is proportional

A proportional relationship has the form \(y=kx\) (\(k\) is the constant of proportionality), where the graph is a straight line passing through the origin. Here \(y = 200x\), \(k = 200\).

Answer:

The points lie on a straight line that passes through the origin. That means the relationship is proportional.