According to data from the National Weather Service, the average temperature in degrees Fahrenheit in Riverside during a spring day...

1. TRANSLATE the problem information

• Given function: \(\mathrm{T(h) = 70 + 1.2(h - 6)}\)
• \(\mathrm{h}\) represents hours after midnight
• \(\mathrm{T(h)}\) represents average temperature at time h
• Question asks: What does the constant 70 represent?

2. INFER the approach

• To understand what a constant in a function represents, I need to find when that constant value appears as the output
• The constant 70 will be the output \(\mathrm{T(h)}\) when the variable part equals zero
• This happens when \(\mathrm{(h - 6) = 0}\), so when \(\mathrm{h = 6}\)

3. SIMPLIFY by evaluating the function

• When \(\mathrm{h = 6}\): \(\mathrm{T(6) = 70 + 1.2(6 - 6)}\)
• \(\mathrm{T(6) = 70 + 1.2(0)}\)
• \(\mathrm{T(6) = 70 + 0 = 70}\)

4. TRANSLATE the time back to real-world context

• Since \(\mathrm{h = 6}\) means 6 hours after midnight
• \(\mathrm{h = 6}\) corresponds to 6 AM
• Therefore, the constant 70 represents the average temperature at 6 AM

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse the constant 70 with the y-intercept of the function, thinking it represents the temperature at midnight \(\mathrm{(h = 0)}\).

They calculate \(\mathrm{T(0) = 70 + 1.2(0 - 6)}\)
\(\mathrm{= 70 + 1.2(-6)}\)
\(\mathrm{= 70 - 7.2}\)
\(\mathrm{= 62.8}\), but then incorrectly think the constant 70 itself represents the midnight temperature.

This may lead them to select Choice C (The average temperature at midnight).

Second Most Common Error:

Poor INFER skill: Students don't realize they need to find when the constant appears as the function output. Instead, they might confuse the roles of the coefficient 1.2 and the constant 70.

They might think 70 represents the hourly rate of change, confusing it with the coefficient.

This may lead them to select Choice A (The average hourly increase in temperature).

The Bottom Line:

The key insight is recognizing that constants in linear functions often represent the output value when the variable expression equals zero. This requires understanding both the algebraic structure and the real-world context of the variables.