Question

every month, midtown grocery store donates 8-ounce cans of vegetables to a food bank. the food bank’s volunteers fill bags with these cans and deliver the bags to families in need. this graph shows the relationship between the number of bags the food bank’s volunteers can fill, x, and the total number of cans they deliver, y. do x and y have a proportional relationship?

Explanation:

Step1: Recall proportional relationship

A proportional relationship between two variables \( x \) and \( y \) means \( y = kx \) (where \( k \) is a constant), and the graph should pass through the origin \((0,0)\) and have a constant slope (constant rate of change).

Step2: Check the graph

The graph shown passes through the origin \((0,0)\) (since when \( x = 0 \) bags, \( y = 0 \) cans). Now check the slope (rate of change). For example, when \( x = 1 \), \( y = 4 \); \( x = 2 \), \( y = 8 \); \( x = 3 \), \( y = 12 \); \( x = 4 \), \( y = 16 \); \( x = 5 \), \( y = 20 \). The slope \( k=\frac{y}{x}\) for each pair: \(\frac{4}{1} = 4\), \(\frac{8}{2}=4\), \(\frac{12}{3}=4\), \(\frac{16}{4}=4\), \(\frac{20}{5}=4\). The slope is constant (\( k = 4 \)) and the graph passes through the origin. So \( y = 4x \), which is a proportional relationship.

Answer:

Yes, \( x \) and \( y \) have a proportional relationship.