A solar energy company installs photovoltaic panels on residential rooftops to generate electricity. The table shows the relationship between the...

1. TRANSLATE the problem information

• Given information:
  • Table showing area x (square feet) and energy f(x) (kilowatt-hours)
  • Three data points: \((5,6), (10,12), (15,18)\)
  • Need to find which equation defines f

2. INFER the relationship pattern

• Since we have input-output pairs, look for a pattern
• Check if there's a constant rate by calculating energy per square foot
• Strategy: Divide f(x) by x for each data point to find the unit rate

3. SIMPLIFY to find the constant rate

• Point (5, 6): \(6 ÷ 5 = 1.2\) kWh per sq ft
• Point (10, 12): \(12 ÷ 10 = 1.2\) kWh per sq ft
• Point (15, 18): \(18 ÷ 15 = 1.2\) kWh per sq ft

4. INFER the function equation

• Since the rate is constant at 1.2, this is a linear function
• The equation is \(\mathrm{f(x) = 1.2x}\)

5. Verify by checking one point

• \(\mathrm{f(5) = 1.2(5) = 6}\) ✓ (matches the table)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find the constant rate of change. Instead, they might try to use individual data points directly with the answer choices or look for additive patterns rather than multiplicative relationships.

This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when dividing energy by area, such as calculating \(6 ÷ 5 = 1.5\) instead of 1.2, or mixing up the division (calculating \(5 ÷ 6\) instead of \(6 ÷ 5\)).

This may lead them to select Choice A (\(\mathrm{f(x) = 0.6x}\)) or another incorrect option.

The Bottom Line:

This problem requires recognizing that a linear relationship means constant rate of change, then systematically calculating that rate rather than trying to work backwards from the answer choices.