For the quadratic function g, the y-intercept is k, where k is a constant. Of the following equations that define...

1. TRANSLATE the problem requirements

• Given information:
  • The y-intercept of function g equals some constant k
  • We need the equation where k appears "directly" without algebraic manipulation
• What this means: We're looking for k to be visible as a written coefficient or constant in the equation itself

2. INFER the key insight about "directly visible"

• The question isn't asking us to calculate the y-intercept (which would be the same for all options)
• It's asking which form shows the y-intercept value without any computation
• This points us toward standard form: \(\mathrm{ax^2 + bx + c}\), where c is automatically the y-intercept

3. TRANSLATE each equation form and check visibility

Let's examine where k would appear in each form:

• Option A: \(\mathrm{g(x) = (x - 4)^2 - 4}\) (vertex form)
  • To find y-intercept, must expand or substitute \(\mathrm{x = 0}\)
  • k is hidden and requires calculation
• Option B: \(\mathrm{g(x) = x^2 - 8x + 12}\) (standard form)
  • The constant term 12 is immediately visible
  • No calculation needed: \(\mathrm{k = 12}\) appears directly
• Option C: \(\mathrm{g(x) = (x - 2)(x - 6)}\) (factored form)
  • Must substitute \(\mathrm{x = 0}\) to find y-intercept
  • k is hidden in the multiplication
• Option D: \(\mathrm{g(x) = 2(x^2 - 4x + 6)}\) (factored with coefficient)
  • Must substitute \(\mathrm{x = 0}\) and multiply by 2
  • k is hidden behind the calculation

4. INFER the answer based on direct visibility

Only in standard form (Option B) does the y-intercept appear as the standalone constant term that requires no manipulation to identify.

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students focus on calculating the y-intercept rather than understanding what "directly visible" means.

They correctly find that all equations give the same y-intercept value, but miss that the question asks specifically about which equation shows this value without requiring any algebraic work. They might think all answers are equivalent since they all have the same y-intercept, leading to confusion and guessing among the choices.

Second Most Common Error:

Insufficient TRANSLATE reasoning: Students misinterpret "directly as a coefficient or constant term" to mean any number that appears in the equation.

They might select Option A because they see the number 4 twice, or Option D because they see the number 2, not recognizing that these numbers don't represent the y-intercept value. This may lead them to select Choice A or Choice D based on visible numbers rather than the actual y-intercept.

The Bottom Line:

This problem tests whether students understand the difference between computing a value and having that value explicitly displayed in an equation. The key insight is recognizing that standard form \(\mathrm{ax^2 + bx + c}\) automatically reveals the y-intercept as the constant term \(\mathrm{c}\).