Linear Functions: Comparing Garden Renovation Quotes with $700, $200, and $0 Initial Fees

The chart shows various quotes for the garden renovation. The price consists of an initial fee and a fixed payment according to the garden size.

Dany proposes an initial fee of 700 dollars.

Alex proposes an initial fee of 200 dollars.

Jonathan proposes taking no initial fee at all.

Match each gardener to their corresponding lines.

To solve this problem, we'll follow these steps:

• Identify the initial fees: According to the problem, Dany's initial fee is 700 dollars, Alex's is 200 dollars, and Jonathan's has no initial fee (0 dollars).
• Analyze the chart: On the chart, the y-intercept of a line represents the initial fee charged before calculating based on garden size.
• Match each line: The line with the highest y-intercept matches Dany's 700 dollars, the next (200 dollars) matches Alex, and the zero y-intercept matches Jonathan.

According to the analysis, the lines match the gardeners as follows:
$$I = Jonathan (no initial fee, line passing through the origin)
$$II = Alex (200 dollars y-intercept, moderate slope)
$$III = Dany (700 dollars y-intercept, highest slope)

Therefore, the solution to the problem is: I = Jonathan, II = Alex, III = Dany.