The function f is defined by \(\mathrm{f(x) = 2(x + 5)^2 - 8}\). The y-intercept of the graph of \(\mathrm{y...

1. INFER the strategy from the problem setup

• Given: \(\mathrm{f(x) = 2(x + 5)^2 - 8}\) and we need the y-intercept
• Key insight: The y-intercept occurs where the graph crosses the y-axis, which happens when \(\mathrm{x = 0}\)
• Strategy: Substitute \(\mathrm{x = 0}\) into the function to find \(\mathrm{f(0)}\)

2. SIMPLIFY by substituting x = 0

• \(\mathrm{f(0) = 2(0 + 5)^2 - 8}\)
• \(\mathrm{f(0) = 2(5)^2 - 8}\)

3. SIMPLIFY using order of operations

• Handle the exponent first: \(\mathrm{5^2 = 25}\)
• \(\mathrm{f(0) = 2(25) - 8}\)
• Multiply: \(\mathrm{2(25) = 50}\)
• \(\mathrm{f(0) = 50 - 8}\)
• Subtract: \(\mathrm{f(0) = 42}\)

Answer: D) 42

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the calculation steps.

The most frequent mistakes include:

• Calculating \(\mathrm{5^2 = 10}\) instead of 25
• Computing \(\mathrm{2(25) = 40}\) instead of 50
• Making sign errors with the subtraction

These arithmetic errors typically lead them to select Choice A) -42 (if they get -50 - 8), Choice B) 10 (if they use \(\mathrm{5^2 = 10}\)), or Choice C) 25 (if they stop after finding \(\mathrm{5^2}\) and don't complete the calculation).

Second Most Common Error:

Missing conceptual knowledge: Not understanding what a y-intercept represents.

Some students may try to set \(\mathrm{f(x) = 0}\) and solve for x (finding x-intercepts instead), or become confused about the relationship between the y-intercept and function evaluation. This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can connect the geometric concept of y-intercept to the algebraic process of function evaluation, then execute the arithmetic correctly. The calculation itself is straightforward, but small arithmetic errors can easily lead to wrong answer choices that appear plausible.