Question

recognize proportional relationships in graphs

1. the graph shows the area y that can be covered by a given amount of paint x when using a paint sprayer.

a. does the graph show a proportional relationship? explain.

Explanation:

To determine if a graph shows a proportional relationship, we check two key points: 1) The graph is a straight line (linear), and 2) The line passes through the origin \((0,0)\). For the paint - area relationship, if the graph of area \(y\) (dependent variable) versus paint amount \(x\) (independent variable) is a straight line through the origin, then \(y\) is proportional to \(x\) (i.e., \(y = kx\) where \(k\) is the constant of proportionality). This is because in a proportional relationship, when \(x = 0\) (no paint), \(y=0\) (no area covered), and the ratio \(\frac{y}{x}\) (the rate of area covered per unit paint) is constant (which corresponds to a straight - line graph with slope \(k\)).

Answer:

To determine if the graph shows a proportional relationship, we use the properties of proportional relationships in graphs:

1. A proportional relationship's graph is a straight line (linear).
2. The line must pass through the origin \((0,0)\) (when \(x = 0\) (no paint), \(y = 0\) (no area covered)).

If the given graph for the area \(y\) (covered) and paint amount \(x\) (used) is a straight line and passes through the origin, then it shows a proportional relationship (since \(y=kx\) where \(k\) is the constant rate of area per unit paint). If it fails either of these two conditions (not a straight line or does not pass through the origin), then it is not a proportional relationship.

(Note: Since the actual graph is not provided, this is the general method. If we had the graph, we would visually inspect for the two conditions mentioned above.)