For the quadratic function f, the graph of \(\mathrm{y = f(x)}\) in the xy-plane has its vertex at the point...

1. TRANSLATE the problem information

• Given information:
  • Vertex at (2, 5)
  • Graph passes through point (4, 13)
• We need to find the equation that defines f

2. INFER the best approach

• Since we have the vertex coordinates, vertex form is perfect here
• Vertex form: \(\mathrm{f(x) = a(x - h)^2 + k}\), where \(\mathrm{(h, k)}\) is the vertex
• We know \(\mathrm{h = 2}\) and \(\mathrm{k = 5}\), but we need to find the value of 'a'
• The additional point \(\mathrm{(4, 13)}\) will help us find 'a'

3. TRANSLATE vertex information into vertex form

• With vertex (2, 5), we get: \(\mathrm{f(x) = a(x - 2)^2 + 5}\)
• Now we need to use the point \(\mathrm{(4, 13)}\) to find 'a'

4. TRANSLATE the point condition into an equation

• Since the graph passes through \(\mathrm{(4, 13)}\), we substitute:
• \(\mathrm{13 = a(4 - 2)^2 + 5}\)

5. SIMPLIFY to solve for 'a'

• \(\mathrm{13 = a(2)^2 + 5}\)
• \(\mathrm{13 = 4a + 5}\)
• \(\mathrm{8 = 4a}\)
• \(\mathrm{a = 2}\)

6. Write the final equation

• \(\mathrm{f(x) = 2(x - 2)^2 + 5}\)

Answer: (A) \(\mathrm{f(x) = 2(x - 2)^2 + 5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students might not recognize that vertex form is the most efficient approach when given vertex coordinates. Instead, they might try to expand all answer choices and test both points, which is time-consuming and error-prone. This leads to confusion and potentially guessing rather than systematic solution.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify vertex form but make sign errors when setting up \(\mathrm{f(x) = a(x - h)^2 + k}\). They might write \(\mathrm{f(x) = a(x + 2)^2 + 5}\) instead of \(\mathrm{f(x) = a(x - 2)^2 + 5}\), forgetting that the vertex form uses \(\mathrm{(x - h)}\). This error leads them to select Choice (B) \(\mathrm{f(x) = 2(x + 2)^2 + 5}\).

The Bottom Line:

This problem rewards students who recognize the power of vertex form when vertex coordinates are given. The key insight is that you only need to find one unknown parameter rather than working backwards from standard form.