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

Q: How do I find the y-intercept on a graph?

Look at where each line crosses the y-axis (the vertical axis). This happens when x = 0, so find the y-value at that point. That's your initial fee!

Q: Why does Jonathan's line go through the origin?

Since Jonathan has no initial fee ($0), his line starts at the origin (0,0). This means you only pay based on garden size with no upfront cost.

Q: What if two lines look very close together?

Read the y-axis scale carefully! Even small differences matter. Line II crosses at 200, while line III crosses at a much higher point around 700-800.

Q: Does the steepness of the line matter for initial fees?

No! The steepness (slope) shows the rate per square unit of garden area. Only the y-intercept tells you the initial fee amount.

Q: How can I double-check my matching?

Use the elimination method: Line I has y-intercept 0 (Jonathan), Line II has y-intercept 200 (Alex), Line III has y-intercept ~700 (Dany). Match the highest to lowest!

Y-Intercepts with Initial Fee Comparisons

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.

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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.

Step-by-step solution

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.

Final Answer

I = Jonathan
II = Alex
III = Dany

Key Points to Remember

Essential concepts to master this topic

Y-Intercept Rule: The y-intercept equals the initial fee amount
Line Matching: Higher y-intercept = higher initial fee (700 > 200 > 0)
Verification: Check where each line crosses the y-axis at x = 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing slope steepness with y-intercept values
Don't match lines based on how steep they look = wrong gardener assignments! Slope steepness shows rate per unit, not initial cost. Always focus on where each line crosses the y-axis to find the initial fee.

Practice Quiz

Test your knowledge with interactive questions

Look at the function shown in the figure.

When is the function positive?

FAQ

Everything you need to know about this question

How do I find the y-intercept on a graph?

Look at where each line crosses the y-axis (the vertical axis). This happens when x = 0, so find the y-value at that point. That's your initial fee!

Why does Jonathan's line go through the origin?

Since Jonathan has no initial fee ($0), his line starts at the origin (0,0). This means you only pay based on garden size with no upfront cost.

What if two lines look very close together?

Read the y-axis scale carefully! Even small differences matter. Line II crosses at 200, while line III crosses at a much higher point around 700-800.

Does the steepness of the line matter for initial fees?

No! The steepness (slope) shows the rate per square unit of garden area. Only the y-intercept tells you the initial fee amount.

How can I double-check my matching?

Use the elimination method: Line I has y-intercept 0 (Jonathan), Line II has y-intercept 200 (Alex), Line III has y-intercept ~700 (Dany). Match the highest to lowest!

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