Question

the graph at the right shows the relationship between rainfall during the growing season and the growth of a type of plant. which statements about the graph are true?
the point (1, 10) shows the constant of proportionality
the constant of proportionality is 5/7
the graph does not show a proportional relationship
the graph is a straight line through the origin
the point (28, 20) means the type of plant grows 20 mm when it rains 28 cm

Explanation:

Step1: Recall the properties of proportional relationship

A proportional relationship has a graph that is a straight - line through the origin. The equation is \(y = kx\), where \(k=\frac{y}{x}\) (constant of proportionality) and \((x,y)\) is a non - origin point on the line.

Step2: Check each statement

• For the point \((1,10)\):

If \(x = 1\) and \(y = 10\), from the graph, when \(x = 14\), \(y = 10\). So the point \((1,10)\) is not on the graph. The constant of proportionality \(k=\frac{y}{x}\). When \(x = 14\), \(y = 10\), \(k=\frac{10}{14}=\frac{5}{7}\), not related to the point \((1,10)\). So the statement “The point \((1,10)\) shows the constant of proportionality” is false.

• For the constant of proportionality \(k=\frac{5}{7}\):

Using the point \((14,10)\), \(k=\frac{y}{x}=\frac{10}{14}=\frac{5}{7}\).

• For the proportional relationship:

Since the graph is a straight - line through the origin (\((0,0)\) is on the line), it shows a proportional relationship. So the statement “The graph does not show a proportional relationship” is false.

• For the graph being a straight - line through the origin:

Visually, the graph is a straight - line and passes through \((0,0)\).

• For the point \((28,20)\):

When \(x = 28\) (rainfall in cm), \(y = 20\) (plant growth in mm). Using \(y=kx\) with \(k = \frac{5}{7}\), \(y=\frac{5}{7}\times28=20\).

Answer:

The constant of proportionality is \(\frac{5}{7}\), the graph is a straight line through the origin, and the point \((28,20)\) means the type of plant grows \(20\) mm when it rains \(28\) cm. So the correct statements are: “The constant of proportionality is \(\frac{5}{7}\)”, “The graph is a straight line through the origin”, “The point \((28,20)\) means the type of plant grows \(20\) mm when it rains \(28\) cm”.