A linear function f gives a company's profit, in dollars, for selling x items. The company's profit is $220 when...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x)}\) = company's profit for selling \(\mathrm{x}\) items
  • When 8 items sold: profit = $220
  • When 10 items sold: profit = $320
  • \(\mathrm{f(x)}\) is linear

• What this tells us: We have two coordinate points: \(\mathrm{(8, 220)}\) and \(\mathrm{(10, 320)}\)

2. INFER the solution strategy

• Since \(\mathrm{f(x)}\) is linear, it has the form \(\mathrm{f(x) = mx + b}\)
• Strategy: Find slope \(\mathrm{m}\) first, then find y-intercept \(\mathrm{b}\)
• We'll use the slope formula, then substitute one point to find \(\mathrm{b}\)

3. SIMPLIFY to find the slope

• Use slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• With points \(\mathrm{(8, 220)}\) and \(\mathrm{(10, 320)}\):
  \(\mathrm{m = \frac{320 - 220}{10 - 8}}\)
  \(\mathrm{m = \frac{100}{2}}\)
  \(\mathrm{m = 50}\)

4. SIMPLIFY to find the y-intercept

• Substitute \(\mathrm{m = 50}\) and point \(\mathrm{(8, 220)}\) into \(\mathrm{f(x) = mx + b}\):
  \(\mathrm{220 = 50(8) + b}\)
  \(\mathrm{220 = 400 + b}\)
  \(\mathrm{b = 220 - 400 = -180}\)

5. Write the final function

• \(\mathrm{f(x) = 50x - 180}\)

Answer: D. \(\mathrm{f(x) = 50x - 180}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up the slope calculation but make arithmetic errors, such as calculating the slope as 32 instead of 50 (perhaps confusing 100/2 with some other calculation), or they assume the function passes through the origin (\(\mathrm{b = 0}\)).

For example, if they think the slope is 32 and \(\mathrm{b = 0}\), they get \(\mathrm{f(x) = 32x}\). This actually works for one point: \(\mathrm{f(10) = 320}\), which might seem to confirm their answer, but \(\mathrm{f(8) = 256 \neq 220}\).

This may lead them to select Choice B (\(\mathrm{f(x) = 32x}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misinterpret which coordinate represents which value, or they might try to use the answer choices to work backwards without systematic calculation.

Some students see Choice C (\(\mathrm{f(x) = 50x - 10x}\)) and think it looks promising because it has the correct slope coefficient of 50, not realizing it simplifies to \(\mathrm{f(x) = 40x}\), which doesn't work for either given point.

This may lead them to select Choice C (\(\mathrm{f(x) = 50x - 10x}\)).

The Bottom Line:

This problem tests whether students can systematically apply the linear function process rather than taking shortcuts or making calculation errors. The key is recognizing that you must find both \(\mathrm{m}\) and \(\mathrm{b}\) through careful calculation, not just guess from answer choices.