Question

this summer, antonio fulfilled a lifelong dream and traveled to kenya. he took lots of photos and wants to put them all in his new photo album binder. first, he needs to buy album pages. this graph shows the relationship between the number of album pages antonio buys, x, and the number of photos the album will hold, 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 is a straight line passing through the origin \((0,0)\).

Step2: Analyze the Graph

The given graph is a straight line, and from the graph, when \( x = 0 \) (no album pages), \( y = 0 \) (no photos held), so it passes through the origin. Also, for a proportional relationship, the ratio \( \frac{y}{x} \) should be constant. Let's check some points. For example, when \( x = 5 \), from the graph (estimating), \( y = 20 \) (since the line seems to have a slope of 4? Wait, when \( x = 10 \), \( y = 40 \), so \( \frac{y}{x}=\frac{40}{10} = 4 \), when \( x = 5 \), \( y = 20 \), \( \frac{20}{5}=4 \), when \( x = 1 \), \( y = 4 \), \( \frac{4}{1}=4 \). So the ratio \( \frac{y}{x} \) is constant ( \( k = 4 \) here).

Answer:

Yes, \( x \) and \( y \) have a proportional relationship because the graph is a straight line passing through the origin \((0,0)\) and the ratio \(\frac{y}{x}\) is constant (e.g., \(\frac{40}{10} = \frac{20}{5} = 4\)).