A city's population \(\mathrm{P(y)}\), in persons, is modeled by a linear function of y, where y is the number of...

1. TRANSLATE the problem information

• Given information:
  • Population function P(y) is linear in terms of y (years after 2000)
  • 2010 population: 62,000 (this means when \(\mathrm{y = 10}\), \(\mathrm{P = 62{,}000}\))
  • 2016 population: 80,000 (this means when \(\mathrm{y = 16}\), \(\mathrm{P = 80{,}000}\))
• This gives us two coordinate points: \(\mathrm{(10, 62000)}\) and \(\mathrm{(16, 80000)}\)

2. INFER the solution approach

• Since we have two points and need a linear equation, we should:
  • First calculate the slope using the slope formula
  • Then use point-slope form to find the complete equation

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{80000 - 62000}{16 - 10}}\)
• \(\mathrm{m = \frac{18000}{6}}\)
• \(\mathrm{m = 3000}\)

4. SIMPLIFY to find the complete equation

• Using point-slope form with \(\mathrm{(10, 62000)}\):
• \(\mathrm{P(y) = m(y - y_1) + P_1}\)
• \(\mathrm{P(y) = 3000(y - 10) + 62000}\)
• \(\mathrm{P(y) = 3000y - 30000 + 62000}\)
• \(\mathrm{P(y) = 3000y + 32000}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students correctly identify the slope as 3000 but then incorrectly assume one of the given population values (62,000 or 80,000) is the y-intercept without considering what y-intercept actually represents.

Since y-intercept occurs when \(\mathrm{y = 0}\) (year 2000), neither given data point represents the y-intercept. Students who make this error might select Choice B (\(\mathrm{P(y) = 3000y + 62000}\)) or Choice D (\(\mathrm{P(y) = 3000y + 80000}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating the slope, getting 3200 instead of 3000 (possibly from calculation errors with 18000/6).

This leads them to select Choice C (\(\mathrm{P(y) = 3200y + 30000}\)).

The Bottom Line:

This problem requires understanding that the y-intercept (when \(\mathrm{y = 0}\)) represents the population in year 2000, which is different from the given data points for 2010 and 2016. Students must use algebraic methods rather than assuming given values directly correspond to the y-intercept.