The function f is defined by \(\mathrm{f(x) = 3(x + 1)(x - 3)}\), where the graph of \(\mathrm{y = f(x)}\)...
GMAT Advanced Math : (Adv_Math) Questions
The function f is defined by \(\mathrm{f(x) = 3(x + 1)(x - 3)}\), where the graph of \(\mathrm{y = f(x)}\) in the xy-plane is a parabola. The functions g and h are equivalent to function f, where m and k are constants. Which of the following equations explicitly displays the x-coordinate of the vertex of the graph of \(\mathrm{y = f(x)}\) as a constant or coefficient?
- \(\mathrm{g(x) = 3x^2 + mx - 9}\)
- \(\mathrm{h(x) = 3(x - 1)^2 + k}\)
- I only
- II only
- I and II
- Neither I nor II
1. SIMPLIFY the original function to standard form
- Given: \(\mathrm{f(x) = 3(x + 1)(x - 3)}\)
- Expand: \(\mathrm{f(x) = 3(x² - 2x - 3) = 3x² - 6x - 9}\)
- INFER what we need: Find the vertex x-coordinate to compare with the given forms
2. SIMPLIFY to find the vertex
- For standard form \(\mathrm{ax² + bx + c}\), vertex x-coordinate = \(\mathrm{-b/(2a)}\)
- Here: \(\mathrm{a = 3, b = -6}\)
- Vertex x-coordinate: \(\mathrm{x = -(-6)/(2·3) = 1}\)
3. INFER what "explicitly displays" means
- Key insight: "Explicitly displays" means the value appears directly in the equation, not that it can be calculated from the equation
- We need to find which form shows \(\mathrm{x = 1}\) as a visible constant
4. INFER the analysis of each form
- Form I: \(\mathrm{g(x) = 3x² + mx - 9}\)
- For equivalence: \(\mathrm{m = -6}\) (matching our expanded form)
- Vertex calculation: \(\mathrm{x = -(-6)/(2·3) = 1}\)
- But the value 1 is NOT visible in the equation - it requires calculation
- Form II: \(\mathrm{h(x) = 3(x - 1)² + k}\)
- This is vertex form: \(\mathrm{a(x - h)² + k}\) with vertex at \(\mathrm{(h, k)}\)
- The x-coordinate \(\mathrm{h = 1}\) appears directly as the constant in \(\mathrm{(x - 1)}\)
- No calculation needed - the value 1 is explicitly shown
Answer: B (II only)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER reasoning about "explicit display": Students recognize that they can calculate the vertex from Form I using the formula \(\mathrm{x = -b/(2a)}\), and conclude this means the vertex is "explicitly displayed."
They miss the distinction between "can be determined from" versus "directly visible as a constant." Since they can find \(\mathrm{x = 1}\) from \(\mathrm{g(x) = 3x² - 6x - 9}\) using the vertex formula, they think this counts as explicit display.
This may lead them to select Choice C (I and II).
The Bottom Line:
The key challenge is understanding that mathematical "explicit display" means the value appears directly in the equation as a written constant, not that it can be calculated from the equation's coefficients. Vertex form shows the vertex coordinates directly, while standard form requires applying a formula to find them.