Question

the graphs of several proportional relationships are shown. each line shows the distance in miles that an object moves based on the number of hours it is in motion.
a. using the graph, identify each relationship that has a unit rate less than 1. discuss slopes of the lines in your response.
b. identify the point on the graph that best demonstrates the unit rate of the relationship represented by line j. what does the unit rate mean in this context?

Explanation:

Step1: Recall unit - rate formula

The unit rate of a proportional relationship represented by a line on a graph is the slope of the line. The slope formula for a line passing through the origin \((0,0)\) and a point \((x,y)\) is \(m=\frac{y}{x}\).

Step2: Analyze the lines for unit - rate less than 1

For a unit rate (slope) \(m < 1\), we need \(\frac{y}{x}<1\) or \(y < x\) for non - zero \(x\). Looking at the graph, lines \(e\) and \(m\) have the property that for \(x>0\), the \(y\) - values of the points on these lines are less than the \(x\) - values.

Step3: Find the unit rate of line \(j\)

For line \(j\), if we take a point on the line, say \((1,4)\) (since the line is proportional and passes through the origin \((0,0)\) and \((1,4)\)), the unit rate (slope) \(m=\frac{y}{x}\). Substituting \(x = 1\) and \(y = 4\), the unit rate is 4. In this context, the unit rate represents the distance (in miles) the object moves per hour.

Answer:

a. Lines \(e\) and \(m\) have a unit rate less than 1.
b. A point on line \(j\) like \((1,4)\) shows the unit rate. The unit rate of line \(j\) is 4, which means the object represented by line \(j\) moves 4 miles per hour.