For the function \(\mathrm{g(x) = x^3 + 2x^2 + kx - 3}\), where k is a constant, the graph of...

1. TRANSLATE the point condition

• Given information:
  • Function: \(\mathrm{g(x) = x^3 + 2x^2 + kx - 3}\)
  • The graph passes through point \(\mathrm{(2, 21)}\)

• What this tells us: Since \(\mathrm{(2, 21)}\) is on the graph, when we input \(\mathrm{x = 2}\), we must get output \(\mathrm{y = 21}\). Therefore: \(\mathrm{g(2) = 21}\)

2. SIMPLIFY by substituting x = 2

• Substitute \(\mathrm{x = 2}\) into the function:

\(\mathrm{g(2) = (2)^3 + 2(2)^2 + k(2) - 3}\)

• Evaluate the powers and products:

\(\mathrm{g(2) = 8 + 2(4) + 2k - 3}\)
\(\mathrm{g(2) = 8 + 8 + 2k - 3}\)
\(\mathrm{g(2) = 13 + 2k}\)

3. SIMPLIFY by solving the equation

• Since \(\mathrm{g(2) = 21}\), we have:

\(\mathrm{13 + 2k = 21}\)

• Subtract 13 from both sides:

\(\mathrm{2k = 8}\)

• Divide by 2:

\(\mathrm{k = 4}\)

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't connect "passes through \(\mathrm{(2, 21)}\)" to the condition \(\mathrm{g(2) = 21}\). They might try to substitute the point coordinates incorrectly, like setting \(\mathrm{x^3 + 2x^2 + kx - 3 = 2 + 21}\), or they might not know how to use the point information at all. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when evaluating the polynomial. Common mistakes include computing \(\mathrm{(2)^3}\) as 6 instead of 8, or \(\mathrm{2(2)^2}\) as 6 instead of 8, leading to incorrect expressions like \(\mathrm{11 + 2k = 21}\), which gives \(\mathrm{k = 5}\). This may lead them to select an incorrect answer.

The Bottom Line:

This problem tests whether students understand the fundamental relationship between points on a graph and function values. The key insight is recognizing that geometric information (a point on the graph) translates directly to algebraic information (an equation to solve).