The graph shows the total cost and revenue functions for a small business manufacturing widgets. The total cost includes both...

1. TRANSLATE the business question into a mathematical requirement

The question asks "when does the business break even?"

• TRANSLATE this business term:
  • Break even = Total Cost equals Revenue
  • Mathematically: \(\mathrm{Total\ Cost = Revenue}\)
  • Graphically: Where do these two lines intersect?

2. VISUALIZE by identifying the key elements on the graph

• Locate the two lines:
  • Solid line = Total Cost (starts at \(\mathrm{y = 4}\), showing fixed costs of $400)
  • Dashed line = Revenue (starts at origin)

• Find where they cross:
  • There's a marked point (shown with a dot) where both lines meet
  • This is the intersection point we need

3. VISUALIZE the coordinates by reading from the graph

• Read the x-coordinate (horizontal):
  • The intersection occurs at \(\mathrm{x = 8}\)
  • This means 8 hundreds = 800 units sold

• Read the y-coordinate (vertical):
  • The intersection occurs at \(\mathrm{y = 8}\)
  • This means 8 hundreds = $800 (both cost and revenue)

4. Verify the answer makes sense

• At 800 units sold:
  • Total Cost = $800
  • Revenue = $800
  • Profit = $0 ✓ (This confirms break-even)

Answer: \(\mathrm{(8, 8)}\) - Choice C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Student doesn't connect "break even" with "intersection point"

Some students may understand break-even conceptually but don't realize it means finding where the lines cross. Instead, they might:

• Look for a specific point on one line only
• Try to calculate something from the graph instead of just reading coordinates
• Confuse break-even with maximum profit or some other concept

This leads to confusion and random selection among answer choices.

Second Most Common Error:

Poor VISUALIZE execution: Misreading the graph coordinates

Students might:

• Confuse which line is which (reading from Total Cost instead of the intersection)
• Miscount the gridlines (reading 7 or 9 instead of 8)
• Read only one coordinate correctly but miss the other

For example, if they misread and think the intersection is at \(\mathrm{(4, 6)}\), they might select Choice A \(\mathrm{(4, 6)}\). Or if they read it as \(\mathrm{(6, 7)}\), they might select Choice B \(\mathrm{(6, 7)}\).

The Bottom Line:

This problem tests whether students can bridge business terminology with graphical analysis. Success requires both understanding what "break even" means mathematically and accurately reading coordinates from a visual representation. The actual mathematics is simple—the challenge is in the interpretation and precision of graph reading.