On January 1, 2015, a city's minimum hourly wage was $9.25. It will increase by $0.50 on the first day...

1. TRANSLATE the problem information

• Given information:
  • Starting wage on January 1, 2015: \(\$9.25\)
  • Annual increase: \(\$0.50\) each year for 5 years
  • Need function for wage x years after January 1, 2015

2. INFER the mathematical relationship

• This describes a linear relationship where:
  • The wage starts at a fixed amount (\(\$9.25\))
  • The wage increases by a constant amount (\(\$0.50\)) each year
• This matches the linear function form \(\mathrm{f(x) = mx + b}\) where:
  • \(\mathrm{b}\) = initial value = \(\$9.25\)
  • \(\mathrm{m}\) = rate of change = \(\$0.50\) per year

3. Build and verify the function

• Construct: \(\mathrm{f(x) = 9.25 + 0.50x}\)
• Test with a few values:
  • \(\mathrm{x = 1}\): \(\mathrm{f(1) = 9.25 + 0.50(1) = \$9.75}\) ✓
  • \(\mathrm{x = 2}\): \(\mathrm{f(2) = 9.25 + 0.50(2) = \$10.25}\) ✓

4. Check against answer choices

• Choice A: \(\mathrm{f(x) = 9.25 - 0.50x}\) shows decreasing wage
• Choice B: \(\mathrm{f(x) = 9.25x - 0.50}\) multiplies wage by years
• Choice C: \(\mathrm{f(x) = 9.25 + 0.50x}\) matches our model ✓
• Choice D: \(\mathrm{f(x) = 9.25x + 0.50}\) multiplies wage by years

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret "increase by \(\$0.50\)" and think the wage gets multiplied by something involving \(\$0.50\), or they confuse which number represents the starting amount versus the change amount.

For example, they might think since the wage "increases," they need to multiply \(\$9.25\) by something, leading them to incorrectly select Choice B (\(\mathrm{f(x) = 9.25x - 0.50}\)) or Choice D (\(\mathrm{f(x) = 9.25x + 0.50}\)).

Second Most Common Error:

Poor TRANSLATE execution: Students correctly identify that wages are increasing but accidentally use subtraction instead of addition, perhaps because they're thinking about the calculation backwards or making a sign error.

This may lead them to select Choice A (\(\mathrm{f(x) = 9.25 - 0.50x}\)), which would show wages decreasing over time.

The Bottom Line:

Success requires carefully translating the word problem to identify what stays constant (the \(\$9.25\) starting wage) versus what changes (the \(\$0.50\) annual increase), then correctly building the linear function using these components.