Let a and b be positive constants. The axis of symmetry of the graph of the quadratic function \(\mathrm{f(x) =...

1. TRANSLATE the quadratic function into standard form

• Given: \(\mathrm{f(x) = 144x^2 - 12(a + 2b)x + 2ab}\)
• Standard form is \(\mathrm{f(x) = Ax^2 + Bx + C}\), so:
  • \(\mathrm{A = 144}\)
  • \(\mathrm{B = -12(a + 2b)}\)
  • \(\mathrm{C = 2ab}\) (we won't need this for axis of symmetry)

2. INFER the strategy needed

• We need the axis of symmetry, which has the formula \(\mathrm{x = \frac{-B}{2A}}\)
• The problem tells us this axis can be written as \(\mathrm{x = k(a + 2b)}\)
• This means we'll set our formula result equal to \(\mathrm{k(a + 2b)}\) and solve for \(\mathrm{k}\)

3. Apply the axis of symmetry formula

• Using \(\mathrm{x = \frac{-B}{2A}}\):
  \(\mathrm{x = \frac{-(-12(a + 2b))}{2 \times 144}}\)
  \(\mathrm{x = \frac{12(a + 2b)}{288}}\)

4. SIMPLIFY the fraction

• \(\mathrm{\frac{12(a + 2b)}{288} = (a + 2b) \times \frac{12}{288}}\)
• Reduce the fraction: \(\mathrm{\frac{12}{288} = \frac{1}{24}}\)
• So \(\mathrm{x = \frac{(a + 2b)}{24}}\)

5. INFER the final step to find k

• We have: axis of symmetry = \(\mathrm{\frac{(a + 2b)}{24}}\)
• Problem states: axis of symmetry = \(\mathrm{k(a + 2b)}\)
• Therefore: \(\mathrm{k(a + 2b) = \frac{(a + 2b)}{24}}\)
• Dividing both sides by \(\mathrm{(a + 2b)}\): \(\mathrm{k = \frac{1}{24}}\)

Answer: 1/24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly identify the coefficients, especially getting confused by the negative sign in front of \(\mathrm{12(a + 2b)}\). They might think \(\mathrm{B = 12(a + 2b)}\) instead of \(\mathrm{B = -12(a + 2b)}\).

When they use the axis formula \(\mathrm{x = \frac{-B}{2A}}\) with the wrong B value, they get:
\(\mathrm{x = \frac{-12(a + 2b)}{2 \times 144}}\)
\(\mathrm{x = \frac{-12(a + 2b)}{288}}\)
\(\mathrm{x = \frac{-(a + 2b)}{24}}\)

This leads them to think \(\mathrm{k = \frac{-1}{24}}\), which would be incorrect.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the axis formula but make arithmetic errors when simplifying \(\mathrm{\frac{12(a + 2b)}{288}}\). They might incorrectly reduce this fraction, perhaps getting \(\mathrm{\frac{1}{12}}\) or \(\mathrm{\frac{1}{8}}\) instead of \(\mathrm{\frac{1}{24}}\).

This causes them to arrive at wrong values for \(\mathrm{k}\) and potentially guess or abandon the systematic approach.

The Bottom Line:

This problem tests whether students can correctly identify coefficients in a parametric quadratic and apply the axis of symmetry formula accurately. The presence of parameters (a and b) and the need to express the final answer in terms of these parameters adds complexity that can trip up students who aren't careful with their algebraic manipulations.