A quadratic function f is defined by \(\mathrm{f(x) = 2(x - 5)^2 + k}\), where k is a constant. The...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = 2(x - 5)^2 + k}\) (where k is unknown)
  • The graph passes through point (1, 40)
  • Need to find: \(\mathrm{f(0)}\)
• What this tells us: Since point (1, 40) is on the graph, we know \(\mathrm{f(1) = 40}\)

2. INFER the solution strategy

• We have an unknown constant k in our function
• To find \(\mathrm{f(0)}\), we first need to determine what k equals
• We can use the fact that \(\mathrm{f(1) = 40}\) to solve for k

3. SIMPLIFY to find the value of k

• Substitute \(\mathrm{x = 1}\) into \(\mathrm{f(x) = 2(x - 5)^2 + k}\):
  \(\mathrm{f(1) = 2(1 - 5)^2 + k = 40}\)

• Work inside parentheses first: \(\mathrm{1 - 5 = -4}\)
  \(\mathrm{f(1) = 2(-4)^2 + k = 40}\)

• Evaluate the exponent: \(\mathrm{(-4)^2 = 16}\)
  \(\mathrm{f(1) = 2(16) + k = 40}\)

• Multiply: \(\mathrm{2(16) = 32}\)
  \(\mathrm{32 + k = 40}\)

• Solve for k: \(\mathrm{k = 40 - 32 = 8}\)

4. INFER the complete function and evaluate f(0)

• Now we know the complete function: \(\mathrm{f(x) = 2(x - 5)^2 + 8}\)

• SIMPLIFY to find \(\mathrm{f(0)}\):
  \(\mathrm{f(0) = 2(0 - 5)^2 + 8}\)
  \(\mathrm{f(0) = 2(-5)^2 + 8}\)
  \(\mathrm{f(0) = 2(25) + 8}\)
  \(\mathrm{f(0) = 50 + 8}\)
  \(\mathrm{f(0) = 58}\)

Answer: D) 58

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't understand that 'the graph passes through point (1, 40)' means \(\mathrm{f(1) = 40}\). They might try to work with the function without using this crucial information, or they might confuse which coordinate represents the input and which represents the output.

Without properly translating this constraint, they can't determine k and get stuck trying to evaluate \(\mathrm{f(0)}\) with an unknown parameter. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when evaluating negative numbers squared, particularly confusing \(\mathrm{(-4)^2 = 16}\) with \(\mathrm{-(4)^2 = -16}\), or \(\mathrm{(-5)^2 = 25}\) with \(\mathrm{-(5)^2 = -25}\).

If they calculate \(\mathrm{(-4)^2}\) as -16, they get \(\mathrm{2(-16) + k = 40}\), leading to \(\mathrm{-32 + k = 40}\), so \(\mathrm{k = 72}\). Then \(\mathrm{f(0) = 2(25) + 72 = 122}\), which isn't among the choices. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem requires students to use given point information strategically - it's not just about plugging in numbers, but understanding the logical sequence of finding the unknown parameter first, then using the complete function to answer the question.