The table below shows the average annual temperature for the city of Riverside for the years 2005 and 2015. If...

1. TRANSLATE the problem information

• Given information:
  • Year 2005: Temperature = 58.2°F
  • Year 2015: Temperature = 60.6°F
  • Relationship is linear
  • \(\mathrm{T(t)}\) models temperature t years after 2005

• What this tells us: We need to set up coordinates where \(\mathrm{t = 0}\) corresponds to year 2005

2. INFER the coordinate system and data points

• Since t represents years after 2005:
  • 2005 corresponds to \(\mathrm{t = 0}\), so we have point \(\mathrm{(0, 58.2)}\)
  • 2015 is 10 years after 2005, so \(\mathrm{t = 10}\), giving us point \(\mathrm{(10, 60.6)}\)

3. INFER what components we need for the linear function

• For \(\mathrm{T(t) = mt + b}\), we need:
  • Slope m using our two points
  • Y-intercept b (which is the temperature when \(\mathrm{t = 0}\))

4. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = (y_2 - y_1)/(x_2 - x_1)}\)
• \(\mathrm{m = (60.6 - 58.2)/(10 - 0)}\)
  \(\mathrm{m = 2.4/10}\)
  \(\mathrm{m = 0.24}\)

5. INFER the complete function

• Since \(\mathrm{t = 0}\) gives \(\mathrm{T = 58.2}\), the y-intercept \(\mathrm{b = 58.2}\)
• Therefore: \(\mathrm{T(t) = 58.2 + 0.24t}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what 'years after 2005' means and use the actual calendar years as their t-values instead of years elapsed since 2005.

They might set up points as \(\mathrm{(2005, 58.2)}\) and \(\mathrm{(2015, 60.6)}\), then calculate slope as \(\mathrm{2.4/10 = 0.24}\), but try to use the point-slope form incorrectly. This confusion about the coordinate system leads them to select Choice D (\(\mathrm{T(t) = 58.2 + 0.24(t - 2005)}\)), which represents a function where t is the actual calendar year rather than years after 2005.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the coordinate system but make an arithmetic error when calculating the slope, getting 2.4 instead of 0.24.

They correctly recognize that \(\mathrm{(0, 58.2)}\) and \(\mathrm{(10, 60.6)}\) are the data points, but calculate slope as \(\mathrm{(60.6 - 58.2)/10 = 2.4}\) without dividing by 10 properly. This leads them to select Choice C (\(\mathrm{T(t) = 58.2 + 2.4t}\)).

The Bottom Line:

This problem tests whether students can properly set up a coordinate system based on a word problem description and distinguish between 'years after a reference point' versus 'actual calendar years' - a crucial skill for real-world modeling problems.