xy142183bThe table shows three values of x and their corresponding values of y for the equation y = 3x^2 +...

1. TRANSLATE the problem information

• Given information:
  • Table shows x and y values for the equation \(\mathrm{y = 3x^2 + 5x - 4}\)
  • When \(\mathrm{x = 3}\), \(\mathrm{y = b}\) (unknown constant)
  • Need to find the value of b

• What this tells us: The value b represents the y-coordinate when \(\mathrm{x = 3}\) in the given quadratic function.

2. INFER the solution approach

• Since we know the equation and the x-value (\(\mathrm{x = 3}\)), we can substitute directly into the equation to find the corresponding y-value.
• This y-value will equal b.

3. SIMPLIFY by substituting and calculating

Substitute \(\mathrm{x = 3}\) into \(\mathrm{y = 3x^2 + 5x - 4}\):

\(\mathrm{y = 3(3)^2 + 5(3) - 4}\)

Following order of operations:

• First, calculate the exponent: \(\mathrm{3^2 = 9}\)
• Then multiply: \(\mathrm{3(9) = 27}\) and \(\mathrm{5(3) = 15}\)
• Finally, combine: \(\mathrm{27 + 15 - 4 = 38}\)

Therefore, \(\mathrm{b = 38}\).

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Order of operations confusion, specifically calculating \(\mathrm{3x^2}\) incorrectly.

Some students mistakenly calculate \(\mathrm{3(3)^2}\) as \(\mathrm{(3 \times 3)^2 = 9^2 = 81}\), instead of correctly computing \(\mathrm{3 \times (3^2) = 3 \times 9 = 27}\). This leads to \(\mathrm{y = 81 + 15 - 4 = 92}\), which doesn't match any answer choice and causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors in the final calculation steps.

Students correctly get to \(\mathrm{3(9) + 15 - 4 = 27 + 15 - 4}\), but then make mistakes in the addition and subtraction. For example, calculating \(\mathrm{27 + 15 = 32}\) instead of 42, leading to \(\mathrm{32 - 4 = 28}\), or getting \(\mathrm{27 + 15 = 42}\) but then calculating \(\mathrm{42 - 4 = 36}\). This may lead them to select Choice B (36).

The Bottom Line:

This problem tests whether students can systematically apply order of operations and perform accurate arithmetic in a function evaluation context. The calculations are straightforward once the correct approach is identified, making computational accuracy the key challenge.