Question: The function g is defined by \(\mathrm{g(x) = n - m(x - 2)^2}\), where m and n are real...

1. TRANSLATE the y-intercept information

• Given information:
  • \(\mathrm{g(x) = n - m(x-2)^2}\) where \(\mathrm{m \gt 0}\)
  • \(\mathrm{y = g(x) + 3}\) has y-intercept of 11
  • \(\mathrm{mn = 320}\)

• What this tells us: At \(\mathrm{x = 0}\), the value of \(\mathrm{g(x) + 3}\) equals 11

2. SIMPLIFY to find g(0)

• Substitute \(\mathrm{x = 0}\) into g(x):
  \(\mathrm{g(0) = n - m(0-2)^2}\)
  \(\mathrm{= n - m(4)}\)
  \(\mathrm{= n - 4m}\)

• So the y-intercept condition becomes:
  \(\mathrm{g(0) + 3 = 11}\)
  \(\mathrm{(n - 4m) + 3 = 11}\)
  \(\mathrm{n - 4m = 8}\)
  \(\mathrm{n = 4m + 8}\)

3. INFER the substitution strategy

• We now have n in terms of m: \(\mathrm{n = 4m + 8}\)
• We also have the constraint \(\mathrm{mn = 320}\)
• Key insight: Substitute the expression for n into the product equation

4. SIMPLIFY the resulting equation

• Substitute \(\mathrm{n = 4m + 8}\) into \(\mathrm{mn = 320}\):
  \(\mathrm{m(4m + 8) = 320}\)
  \(\mathrm{4m^2 + 8m = 320}\)
  \(\mathrm{4m^2 + 8m - 320 = 0}\)

• Divide everything by 4:
  \(\mathrm{m^2 + 2m - 80 = 0}\)

5. SIMPLIFY by factoring

• Need two numbers that multiply to -80 and add to 2
• Those numbers are 10 and -8: \(\mathrm{(10)(-8) = -80}\) and \(\mathrm{10 + (-8) = 2}\)
• Factor: \(\mathrm{(m + 10)(m - 8) = 0}\)
• Solutions: \(\mathrm{m = -10}\) or \(\mathrm{m = 8}\)

6. APPLY CONSTRAINTS to select final answer

• Since we're given that \(\mathrm{m \gt 0}\), we reject \(\mathrm{m = -10}\)
• Therefore: \(\mathrm{m = 8}\)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to convert "y-intercept of 11" into the mathematical condition. They might try to use \(\mathrm{y = 11}\) directly without recognizing that the y-intercept occurs specifically when \(\mathrm{x = 0}\), requiring them to evaluate \(\mathrm{g(0) + 3 = 11}\).

This confusion leads to setting up incorrect equations from the start, making the entire solution path invalid and causing them to get stuck and guess randomly.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve the quadratic to get \(\mathrm{m = -10}\) or \(\mathrm{m = 8}\), but forget to apply the given constraint \(\mathrm{m \gt 0}\). Without this final step, they might select the wrong value or become uncertain about which solution is correct.

This oversight could lead them to incorrectly choose \(\mathrm{m = -10}\) or cause confusion about the final answer.

The Bottom Line:

This problem tests whether students can systematically work through multiple algebraic constraints while maintaining awareness of all given conditions. The key challenge is translating the y-intercept condition correctly and then methodically combining all three pieces of information (the function definition, y-intercept, and product constraint) into a solvable system.