The function g is defined by \(\mathrm{g(x) = px^2 + r}\), where p and r are positive constants. The graph...

1. TRANSLATE the given information into equations

• Given information:
  • Function: \(\mathrm{g(x) = px^2 + r}\)
  • Point \(\mathrm{(1, 9)}\) lies on the graph
  • Point \(\mathrm{(3, 41)}\) lies on the graph
  • Need to find: \(\mathrm{pr}\)

• What this tells us: If a point \(\mathrm{(x, y)}\) lies on the graph, then \(\mathrm{g(x) = y}\)

2. TRANSLATE each point into an equation

• From point \(\mathrm{(1, 9)}\): \(\mathrm{g(1) = 9}\)
  Substituting: \(\mathrm{p(1)^2 + r = 9}\)
  Simplifying: \(\mathrm{p + r = 9}\)

• From point \(\mathrm{(3, 41)}\): \(\mathrm{g(3) = 41}\)
  Substituting: \(\mathrm{p(3)^2 + r = 41}\)
  Simplifying: \(\mathrm{9p + r = 41}\)

3. SIMPLIFY by solving the system of equations

• We now have:
  \(\mathrm{p + r = 9}\) ... equation (1)
  \(\mathrm{9p + r = 41}\) ... equation (2)

• Subtract equation (1) from equation (2):
  \(\mathrm{(9p + r) - (p + r) = 41 - 9}\)
  \(\mathrm{8p = 32}\)
  \(\mathrm{p = 4}\)

4. SIMPLIFY to find r and calculate pr

• Substitute \(\mathrm{p = 4}\) back into equation (1):
  \(\mathrm{4 + r = 9}\)
  \(\mathrm{r = 5}\)

• Calculate the final answer:
  \(\mathrm{pr = 4 \times 5 = 20}\)

Answer: C (20)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to connect the geometric concept "point lies on graph" with the algebraic requirement \(\mathrm{g(x) = y}\)

Instead of recognizing that \(\mathrm{(1, 9)}\) means \(\mathrm{g(1) = 9}\), they might attempt to work with the coordinates directly without substituting into the function. This leads to confusion about how to use the given points and often results in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equations correctly but make arithmetic errors when solving the system

For example, when subtracting \(\mathrm{(p + r = 9)}\) from \(\mathrm{(9p + r = 41)}\), they might get \(\mathrm{8p = 31}\) instead of \(\mathrm{8p = 32}\), leading to \(\mathrm{p = 31/8}\). This cascades into an incorrect value for r and ultimately pr, potentially leading them to select Choice A (9) or Choice B (18).

The Bottom Line:

This problem tests whether students can bridge the gap between graphical and algebraic representations. The key insight is recognizing that points on a graph translate directly into function equations, then systematically solving the resulting system.