Question:The graph of a quadratic function g is a parabola in the xy-plane with vertex \((\mathrm{h},\mathrm{k})\). Given that h =...
GMAT Advanced Math : (Adv_Math) Questions
The graph of a quadratic function g is a parabola in the xy-plane with vertex \((\mathrm{h},\mathrm{k})\). Given that \(\mathrm{h = -3}\), which of the following equations, all of which define g for all values of x, shows the value of k as a constant term or coefficient?
1. TRANSLATE the problem requirements
- Given information:
- Vertex is at \((h,k)\) where \(h = -3\)
- Need to find which form shows \(k\) as a constant term or coefficient
- What this means: I need to identify which algebraic form explicitly displays the y-coordinate of the vertex (k value) without requiring any calculations.
2. INFER which form explicitly shows k
- The vertex form of a parabola is \(\mathrm{g(x)} = \mathrm{a(x-h)}^2 + \mathrm{k}\)
- With \(\mathrm{h} = -3\), this becomes \(\mathrm{g(x)} = \mathrm{a(x+3)}^2 + \mathrm{k}\)
- In vertex form, \(\mathrm{k}\) appears directly as the constant term added to the squared expression
3. INFER why other forms don't show k directly
- Standard form: \(\mathrm{g(x)} = \mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\) shows the y-intercept \((\mathrm{c})\), not the vertex y-coordinate
- Factored forms: require expanding or other calculations to find the vertex
- Only vertex form displays \(\mathrm{k}\) immediately without any work
4. Examine each choice
Looking for the pattern \(\mathrm{g(x)} = \mathrm{a(x+3)}^2 + \mathrm{k}\):
- \(\mathrm{g(x)} = -\mathrm{x}^2 - 6\mathrm{x} - 2\) → Standard form (hides k)
- \(\mathrm{g(x)} = -\mathrm{x(x+6)} - 2\) → Partially factored (hides k)
- \(\mathrm{g(x)} = -(\mathrm{x}+1)(\mathrm{x}+5) + 3\) → Factored form (hides k)
- \(\mathrm{g(x)} = -(\mathrm{x}+3)^2 + 7\) → Vertex form (shows \(\mathrm{k} = 7\) explicitly)
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students see constant terms in multiple answer choices (like -2 in choices A and B, or +3 in choice C) and incorrectly assume these represent k.
They don't understand that k specifically refers to the y-coordinate of the vertex, which only appears explicitly in vertex form. They might think "any constant term shows k" rather than recognizing that k has a specific mathematical meaning tied to the vertex location.
This may lead them to select Choice A (-2) or Choice C (+3) based on seeing obvious constant terms.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misunderstand what "shows the value of k as a constant term" means, thinking it just means "contains a constant number somewhere."
They fail to connect this requirement to the vertex form structure, not realizing that the question is asking which form makes k immediately visible without any calculation or conversion work.
This leads to confusion and guessing among the choices that contain constants.
The Bottom Line:
This problem tests whether students understand that different algebraic forms of the same quadratic function reveal different information directly. Only vertex form makes the vertex coordinates immediately visible.