A marketing analyst uses function p defined by \(\mathrm{p(x) = mx + n}\), where m and n are constants, to...

1. TRANSLATE the problem information

• Given information:
  • Function \(\mathrm{p(x) = mx + n}\) (where m and n are constants)
  • Graph of \(\mathrm{y = -p(x) + 12}\) has x-intercept at \(\mathrm{(3, 0)}\)
  • Sum \(\mathrm{m + n = \frac{5}{2}}\)
  • Need to find: value of n
• What the x-intercept tells us: When \(\mathrm{x = 3}\), \(\mathrm{y = 0}\)

2. TRANSLATE the x-intercept condition

• At the x-intercept \(\mathrm{(3, 0)}\): \(\mathrm{-p(3) + 12 = 0}\)
• This means: \(\mathrm{p(3) = 12}\)

3. INFER what p(3) equals in terms of our unknowns

• Since \(\mathrm{p(x) = mx + n}\), we have: \(\mathrm{p(3) = m(3) + n = 3m + n}\)
• Therefore: \(\mathrm{3m + n = 12}\)

4. INFER the solution strategy

• We now have two equations with two unknowns:
  • \(\mathrm{3m + n = 12}\) (from x-intercept condition)
  • \(\mathrm{m + n = \frac{5}{2}}\) (given sum constraint)
• Strategy: Solve for one variable, then substitute

5. SIMPLIFY to solve for m

• From the sum constraint: \(\mathrm{n = \frac{5}{2} - m}\)
• Substitute into the first equation: \(\mathrm{3m + (\frac{5}{2} - m) = 12}\)
• Combine like terms: \(\mathrm{3m + \frac{5}{2} - m = 12}\)
• This gives us: \(\mathrm{2m + \frac{5}{2} = 12}\)
• Subtract \(\mathrm{\frac{5}{2}}\):
  \(\mathrm{2m = 12 - \frac{5}{2}}\)
  \(\mathrm{2m = \frac{24}{2} - \frac{5}{2}}\)
  \(\mathrm{2m = \frac{19}{2}}\)
• Therefore: \(\mathrm{m = \frac{19}{4}}\)

6. SIMPLIFY to find n

• Using \(\mathrm{n = \frac{5}{2} - m}\): \(\mathrm{n = \frac{5}{2} - \frac{19}{4}}\)
• Convert to common denominator:
  \(\mathrm{n = \frac{10}{4} - \frac{19}{4}}\)
  \(\mathrm{n = -\frac{9}{4}}\)

Answer: \(\mathrm{-\frac{9}{4}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what the x-intercept condition means mathematically. They might think that since the graph passes through \(\mathrm{(3, 0)}\), then \(\mathrm{p(3) = 0}\), forgetting that the actual graph is \(\mathrm{y = -p(x) + 12}\), not \(\mathrm{y = p(x)}\).

This leads to the incorrect equation \(\mathrm{3m + n = 0}\) instead of \(\mathrm{3m + n = 12}\). Following through with this error, they would get \(\mathrm{m = -\frac{5}{2}}\) and \(\mathrm{n = 5}\) as their final answer, which doesn't match any reasonable expectation for the problem.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up both equations but make arithmetic errors when working with fractions. Common mistakes include:

• Incorrectly converting 12 to twenths when finding \(\mathrm{12 - \frac{5}{2}}\)
• Sign errors when subtracting fractions
• Forgetting to convert to a common denominator

This leads to various incorrect numerical answers and causes confusion about which approach to trust.

The Bottom Line:

This problem requires careful interpretation of what an x-intercept means for a transformed function, combined with systematic algebraic manipulation of two equations. Students must resist the urge to rush through the fraction arithmetic and instead work methodically through each step.