Let f be a quadratic function. The table below shows three values of x and their corresponding values of y,...

1. TRANSLATE the given information

• Given information:
  • f is a quadratic function
  • Table shows \(\mathrm{y = f(6 - x)}\) with values: \(\mathrm{(1, 42), (3, 50), (5, 42)}\)
  • Need to find \(\mathrm{f(0)}\)

• The key insight: We need to convert \(\mathrm{y = f(6 - x)}\) into points that lie directly on \(\mathrm{f(x)}\)

2. TRANSLATE the table to find points on f(x)

For each row in the table where \(\mathrm{y = f(6 - x)}\):

• When \(\mathrm{x = 1}\), \(\mathrm{y = 42}\) → This means \(\mathrm{f(6 - 1) = f(5) = 42}\)
• When \(\mathrm{x = 3}\), \(\mathrm{y = 50}\) → This means \(\mathrm{f(6 - 3) = f(3) = 50}\)
• When \(\mathrm{x = 5}\), \(\mathrm{y = 42}\) → This means \(\mathrm{f(6 - 5) = f(1) = 42}\)

So the actual points on \(\mathrm{f(x)}\) are: \(\mathrm{(1, 42), (3, 50), (5, 42)}\)

3. INFER the structure of the quadratic

• Notice that \(\mathrm{f(1) = f(5) = 42}\) (same y-value for different x-values)
• For any quadratic, when two different inputs give the same output, the axis of symmetry lies exactly halfway between those x-values
• Axis of symmetry: \(\mathrm{x = \frac{1 + 5}{2} = 3}\)

• Since \(\mathrm{(3, 50)}\) is on our list and lies on the axis of symmetry, this must be the vertex: \(\mathrm{(3, 50)}\)

4. INFER the vertex form and SIMPLIFY to find the coefficient

• Using vertex form: \(\mathrm{f(x) = a(x - 3)^2 + 50}\)
• To find coefficient a, substitute any other known point, like \(\mathrm{(1, 42)}\):

\(\mathrm{42 = a(1 - 3)^2 + 50}\)

\(\mathrm{42 = a(-2)^2 + 50}\)

\(\mathrm{42 = 4a + 50}\)

\(\mathrm{-8 = 4a}\)

\(\mathrm{a = -2}\)

• Complete function: \(\mathrm{f(x) = -2(x - 3)^2 + 50}\)

5. SIMPLIFY to find f(0)

\(\mathrm{f(0) = -2(0 - 3)^2 + 50}\)

\(\mathrm{f(0) = -2(-3)^2 + 50}\)

\(\mathrm{f(0) = -2(9) + 50}\)

\(\mathrm{f(0) = -18 + 50}\)

\(\mathrm{f(0) = 32}\)

Answer: 32

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret \(\mathrm{y = f(6 - x)}\) and try to work directly with the given table values as if they were points on \(\mathrm{f(x)}\) itself.

They might set up a system using \(\mathrm{(1, 42), (3, 50), (5, 42)}\) as direct points on \(\mathrm{f(x) = ax^2 + bx + c}\), not realizing these are actually \(\mathrm{f(5), f(3)}\), and \(\mathrm{f(1)}\) respectively. This leads to finding a completely different quadratic function, and when they calculate \(\mathrm{f(0)}\), they get an incorrect result. This leads to confusion and potentially guessing among answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify the points on \(\mathrm{f(x)}\) and even recognize the vertex form approach, but make algebraic errors when solving for coefficient a or when calculating \(\mathrm{f(0)}\).

For example, they might incorrectly compute \(\mathrm{(-3)^2 = -9}\) instead of \(\mathrm{9}\), or make sign errors when solving \(\mathrm{42 = 4a + 50}\). These calculation mistakes lead them away from the correct answer of 32.

The Bottom Line:

This problem tests whether students can "unwrap" a function composition to find the actual function, then apply their knowledge of quadratic symmetry. The translation step is absolutely critical - without it, students are solving a completely different problem.