Question:The function g is defined by \(\mathrm{g(x) = (x + 16)(t - 2x)}\), where t is a constant. In the...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = (x + 16)(t - 2x)}\) where \(\mathrm{t}\) is constant
  • The graph passes through point \(\mathrm{(22, 0)}\)

• What this tells us: Since \(\mathrm{(22, 0)}\) is on the graph, we know \(\mathrm{g(22) = 0}\)

2. INFER the approach to find the unknown constant

• We have a function with an unknown parameter \(\mathrm{t}\)
• Since we know \(\mathrm{g(22) = 0}\), we can substitute \(\mathrm{x = 22}\) and solve for \(\mathrm{t}\)
• Once we find \(\mathrm{t}\), we can evaluate \(\mathrm{g(0)}\)

3. SIMPLIFY to solve for t

• Substitute \(\mathrm{x = 22}\) into the function:
  \(\mathrm{g(22) = (22 + 16)(t - 2(22)) = 0}\)
  \(\mathrm{g(22) = (38)(t - 44) = 0}\)

• Apply Zero Product Property: Since the product equals 0, at least one factor must be 0
• Since \(\mathrm{38 ≠ 0}\), we must have: \(\mathrm{t - 44 = 0}\)
• Therefore: \(\mathrm{t = 44}\)

4. SIMPLIFY to find g(0)

• Now substitute \(\mathrm{x = 0}\) and \(\mathrm{t = 44}\):
  \(\mathrm{g(0) = (0 + 16)(44 - 2(0))}\)
  \(\mathrm{g(0) = (16)(44)}\)
  \(\mathrm{g(0) = 704}\)

Answer: 704

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might not recognize that "the graph passes through \(\mathrm{(22, 0)}\)" means \(\mathrm{g(22) = 0}\). They may try to work directly with the general form of the function without using the given point.

Without this key insight, they cannot determine the value of \(\mathrm{t}\) and get stuck early in the problem. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{g(22) = 0}\) but make algebraic errors when solving for \(\mathrm{t}\). For example, they might incorrectly solve \(\mathrm{(38)(t - 44) = 0}\) and get a wrong value for \(\mathrm{t}\).

With an incorrect value of \(\mathrm{t}\), their final calculation of \(\mathrm{g(0)}\) will be wrong, leading them to an incorrect numerical answer.

The Bottom Line:

This problem tests whether students understand the connection between points on a graph and function values, then requires careful algebraic manipulation. The key insight is recognizing that a given point provides the constraint needed to find the unknown parameter.