Question

3 for each graph, explain whether the relationship could be proportional. a b height of triangular wall in inches weight of coins in grams number of bricks number of coins

Explanation:

Graph A

Step1: Check if points lie on a straight line through origin

A proportional relationship has a graph that is a straight line passing through the origin \((0,0)\). In graph A, if we assume the points follow a pattern. Let's check the ratio of \(y\) (height) to \(x\) (number of bricks). For the first non - origin point \((3,5)\), the ratio is \(\frac{5}{3}\approx1.67\). For the point \((6,7)\), the ratio is \(\frac{7}{6}\approx1.17\). Since the ratios are not constant, and if we were to connect the points (assuming they are part of a line), we can see that the points do not lie on a straight line (because the rate of change is not uniform). A proportional relationship must have a constant rate of change (slope) and pass through the origin. Here, the rate of change is not constant.

Graph B

Step1: Check if points lie on a straight line through origin

For a proportional relationship \(y = kx\) (where \(k\) is the constant of proportionality). Let's take the points \((3,10)\), \(k_1=\frac{10}{3}\approx3.33\); \((6,20)\), \(k_2 = \frac{20}{6}=\frac{10}{3}\approx3.33\); \((9,30)\), \(k_3=\frac{30}{9}=\frac{10}{3}\approx3.33\); \((12,40)\), \(k_4=\frac{40}{12}=\frac{10}{3}\approx3.33\); \((15,50)\), \(k_5=\frac{50}{15}=\frac{10}{3}\approx3.33\). The ratio of \(y\) (weight of coins) to \(x\) (number of coins) is constant (\(k = \frac{10}{3}\)). Also, the graph passes through the origin \((0,0)\). A relationship is proportional if \(y=kx\) (\(k
eq0\)), which is equivalent to \(\frac{y}{x}=k\) (constant) and passes through \((0,0)\)

Answer:

• Graph A: The relationship is not proportional. The points do not lie on a straight line (the rate of change between the number of bricks and the height of the triangular wall is not constant).
• Graph B: The relationship is proportional. The ratio of the weight of coins to the number of coins is constant (\(k=\frac{10}{3}\)), and the graph passes through the origin.