For the given function \(\mathrm{f(x) = x^5 + 9x + 17}\), the graph of \(\mathrm{y = f(x)}\) in the xy-plane...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = x^5 + 9x + 17}\)
  • The graph passes through point \(\mathrm{(0, b)}\)
  • Need to find the value of b

• What this tells us: If a graph passes through point \(\mathrm{(0, b)}\), then when we input \(\mathrm{x = 0}\), the output is b

2. INFER the mathematical relationship

• Since the point \(\mathrm{(0, b)}\) is on the graph of \(\mathrm{y = f(x)}\), we know that \(\mathrm{f(0) = b}\)
• To find b, we need to evaluate the function when \(\mathrm{x = 0}\)

3. SIMPLIFY by substituting x = 0

• \(\mathrm{f(0) = (0)^5 + 9(0) + 17}\)
• \(\mathrm{f(0) = 0 + 0 + 17}\)
• \(\mathrm{f(0) = 17}\)

Therefore, \(\mathrm{b = 17}\).

Answer: 17

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand what 'the graph passes through point \(\mathrm{(0, b)}\)' means mathematically. They might think this is asking about the general form of the function rather than a specific point evaluation.

This leads to confusion about what the question is actually asking, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need to find \(\mathrm{f(0)}\), but make arithmetic errors when evaluating \(\mathrm{0^5}\) or forget that \(\mathrm{9(0) = 0}\).

For example, they might incorrectly think \(\mathrm{0^5 = 1}\) (confusing it with the rule that any non-zero number to the power of 0 equals 1), leading them to calculate \(\mathrm{f(0) = 1 + 0 + 17 = 18}\).

The Bottom Line:

This problem tests whether students can connect the geometric concept of a point on a graph with the algebraic concept of function evaluation. The key insight is recognizing that 'passes through \(\mathrm{(0, b)}\)' translates directly to '\(\mathrm{f(0) = b}\).'