The function g is defined by \(\mathrm{g(x) = (x + 14)(t - x)}\), where t is a constant. In the...

1. TRANSLATE the coordinate information

• Given information:
  • \(\mathrm{g(x) = (x + 14)(t - x)}\) where \(\mathrm{t}\) is unknown
  • Graph passes through point \(\mathrm{(24, 0)}\)
• What this tells us: Since the graph passes through \(\mathrm{(24, 0)}\), we know that \(\mathrm{g(24) = 0}\)

2. INFER how to find the unknown constant

• Key insight: We can substitute \(\mathrm{x = 24}\) into our function and set it equal to 0
• This will give us an equation we can solve for \(\mathrm{t}\)
• Strategy: Use the zero product property once we have our equation

3. SIMPLIFY to solve for t

• Substitute \(\mathrm{x = 24}\):
  \(\mathrm{g(24) = (24 + 14)(t - 24) = 0}\)
• This becomes:
  \(\mathrm{38(t - 24) = 0}\)
• By zero product property: Since \(\mathrm{38 ≠ 0}\), we need \(\mathrm{t - 24 = 0}\)
• Therefore:
  \(\mathrm{t = 24}\)

4. SIMPLIFY to find g(0)

• Now we know our complete function:
  \(\mathrm{g(x) = (x + 14)(24 - x)}\)
• Substitute \(\mathrm{x = 0}\):
  \(\mathrm{g(0) = (0 + 14)(24 - 0)}\)
• Calculate:
  \(\mathrm{g(0) = 14 × 24 = 336}\)

Answer: 336

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not realize that "the graph passes through \(\mathrm{(24, 0)}\)" means \(\mathrm{g(24) = 0}\). Instead, they might try to substitute random values or attempt to expand the original expression without using the given point information.

This leads to confusion and guessing since they can't determine the value of \(\mathrm{t}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{g(24) = 0}\) but make calculation errors. They might incorrectly solve \(\mathrm{38(t - 24) = 0}\) or make arithmetic mistakes when calculating \(\mathrm{14 × 24}\).

This may cause them to arrive at an incorrect final answer even though their approach was sound.

The Bottom Line:

This problem tests whether students can connect coordinate geometry (points on a graph) with function notation and algebraic manipulation. The key breakthrough moment is recognizing that a point on the graph gives you a specific function value to work with.