Question

1. 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 steepness 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 (a line passing through the origin) is given by the slope $m=\frac{y}{x}$, where $(x,y)$ is a non - origin point on the line.

Step2: Analyze slopes for part a

For line $j$, if we take the point $(2,9)$, the slope $m_j=\frac{9}{2}=4.5$. For line $k$, if we take the point $(9,9)$, the slope $m_k = 1$. For line $\ell$, if we take the point $(9,4.5)$, the slope $m_{\ell}=\frac{4.5}{9}=0.5$. For line $m$, if we take the point $(9,2.2)$, the slope $m_m=\frac{2.2}{9}\approx0.24$. Lines $\ell$ and $m$ have unit rates less than 1. The steeper the line, the larger the absolute value of the slope. Lines $\ell$ and $m$ are less steep compared to $j$ and $k$ since their slopes are smaller.

Step3: Find unit - rate point for line $j$ in part b

For a proportional relationship $y = mx$, the unit rate is the value of $y$ when $x = 1$. For line $j$, the slope $m = 4.5$. The point $(1,4.5)$ on line $j$ represents the unit rate. In this context, the unit rate of 4.5 means that the object moves 4.5 miles per hour.

Answer:

a. Lines $\ell$ and $m$ have unit rates less than 1. Lines with smaller slopes are less steep.
b. The point $(1,4.5)$ on line $j$ represents the unit rate. The unit rate means the object moves 4.5 miles per hour.