Question

1. the graph below shows the amount of money a diner can make from selling milkshakes.

a. is the number of milkshakes sold proportional to the profit? how do you know?
b. what does the point (0,0) mean in the context of the situation?
c. what is the unit rate?
d. what point represents the unit rate on the graph?

1. for any proportional relationship, what does the point (1,r) represent?

Explanation:

Step1: Check proportionality

A proportional relationship has a straight - line graph passing through the origin \((0,0)\). The given graph is a straight line passing through \((0,0)\), so the number of milkshakes sold is proportional to the profit.

Step2: Interpret \((0,0)\)

In the context of selling milkshakes, when \(x = 0\) (number of milkshakes sold is \(0\)), \(y=0\) (profit is \(0\)). So, it means if no milkshakes are sold, there is no profit.

Step3: Calculate unit rate

The unit rate \(r\) (slope of the line) can be calculated using the formula \(r=\frac{y}{x}\). Let's take a point \((x,y)=(10,15)\). Then \(r = \frac{15}{10}=1.5\).

Step4: Find unit - rate point

The unit rate is the value of \(y\) when \(x = 1\). Since the unit rate \(r = 1.5\), the point is \((1,1.5)\)

Step5: Explain \((1,r)\)

For a proportional relationship \(y=rx\) (where \(r\) is the constant of proportionality). When \(x = 1\), \(y=r\). So the point \((1,r)\) represents the constant of proportionality (unit rate).

Answer:

a. Yes, because the graph is a straight - line passing through the origin.
b. If no milkshakes are sold (\(x = 0\)), there is no profit (\(y = 0\)).
c. \(1.5\) (dollars per milkshake)
d. \((1,1.5)\)

1. The point \((1,r)\) represents the constant of proportionality (unit rate) for the proportional relationship.