The function f is defined by \(\mathrm{f(x) = x^2}\). In the xy-plane, the graph of \(\mathrm{y = f(x)}\) is a...

1. TRANSLATE the problem information

• Given information:
  • Original function: \(\mathrm{f(x) = x^2}\) with vertex at origin \(\mathrm{(0, 0)}\)
  • Transformed function: \(\mathrm{h(x)}\) with vertex at \(\mathrm{(3, -5)}\)
  • Need to find the equation for \(\mathrm{h(x)}\)

2. INFER the mathematical approach

• Since we know both the original and new vertex locations, we should use vertex form
• Vertex form: \(\mathrm{y = a(x - h)^2 + k}\), where \(\mathrm{(h, k)}\) is the vertex
• The shape doesn't change, so \(\mathrm{a = 1}\) (same as the original parabola)

3. SIMPLIFY by substituting the vertex coordinates

• From the vertex \(\mathrm{(3, -5)}\): \(\mathrm{h = 3}\) and \(\mathrm{k = -5}\)
• Substitute into vertex form: \(\mathrm{h(x) = 1(x - 3)^2 + (-5)}\)
• Simplify: \(\mathrm{h(x) = (x - 3)^2 - 5}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the direction of horizontal shifts in vertex form. They think moving right 3 units means using \(\mathrm{(x + 3)}\) instead of \(\mathrm{(x - 3)}\).

Their reasoning: "The vertex moved right 3, so I add 3 to x."
This leads them to \(\mathrm{h(x) = (x + 3)^2 - 5}\), causing them to select Choice B.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify vertex form but make sign errors when handling the vertical shift. They write \(\mathrm{h(x) = (x - 3)^2 + 5}\) instead of \(\mathrm{(x - 3)^2 - 5}\).

Their error: Forgetting that \(\mathrm{k = -5}\) means adding (-5), not adding 5.
This may lead them to select Choice C.

The Bottom Line:

Vertex form sign conventions are counterintuitive - rightward shifts use \(\mathrm{(x - h)}\) and downward shifts use \(\mathrm{+k}\) where k is negative. Success requires careful attention to these sign relationships.