According to data provided by the US Department of Energy, the average price per gallon of regular gasoline in the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{F(x) = 2.74 - 0.19(x - 3)}\) models gas prices
  • \(\mathrm{F(x)}\) = average price per gallon x months after September 1
  • Need to find what the constant 2.74 estimates

2. INFER the approach

• Key insight: Since 2.74 is a constant (not a coefficient of x), it represents an actual price, not a rate of change
• To find which price it represents, we need to determine when \(\mathrm{F(x) = 2.74}\)
• This means setting up the equation \(\mathrm{2.74 = 2.74 - 0.19(x - 3)}\)

3. SIMPLIFY the equation

• Start with: \(\mathrm{2.74 = 2.74 - 0.19(x - 3)}\)
• Subtract 2.74 from both sides: \(\mathrm{0 = -0.19(x - 3)}\)
• Divide by -0.19: \(\mathrm{0 = x - 3}\)
• Therefore: \(\mathrm{x = 3}\)

4. TRANSLATE the time reference

• \(\mathrm{x = 3}\) means 3 months after September 1, 2014
• September 1 + 3 months = December 1, 2014
• So 2.74 represents the average price per gallon on December 1, 2014

Answer: D. The average price per gallon on December 1, 2014

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategic approach of setting \(\mathrm{F(x)}\) equal to the constant to find what it represents. Instead, they might think 2.74 is automatically the September 1 price (since it appears first in the function) or confuse it with a rate of change.

This may lead them to select Choice C (The average price per gallon on September 1, 2014) without doing any calculations.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly set up \(\mathrm{F(x) = 2.74}\) and solve to get \(\mathrm{x = 3}\), but then miscount the months. They might think \(\mathrm{x = 3}\) means "the third month" rather than "3 months after September 1," leading them to conclude it represents November 1 instead of December 1.

This causes confusion since November 1 isn't an answer choice, leading to guessing or incorrectly selecting a different choice.

The Bottom Line:

This problem requires recognizing that constants in function contexts represent specific output values, and the key is determining the input that produces that output. Students who try to interpret function components without this systematic approach often guess incorrectly.