\(\mathrm{g(x) = x^2 + 55}\). What is the minimum value of the given function?

1. TRANSLATE the problem information

• Given function: \(\mathrm{g(x) = x^2 + 55}\)
• Need to find: minimum value of this function (the smallest y-value the function produces)

2. INFER the approach

• This is a quadratic function with coefficient of \(\mathrm{x^2}\) equal to 1 (positive)
• Since the coefficient is positive, the parabola opens upward, meaning it has a minimum point
• We can find this minimum using vertex form properties

3. INFER the vertex form structure

• Rewrite \(\mathrm{g(x) = x^2 + 55}\) as \(\mathrm{g(x) = 1(x - 0)^2 + 55}\)
• This matches vertex form: \(\mathrm{g(x) = a(x - h)^2 + k}\)
• Here: \(\mathrm{a = 1, h = 0, k = 55}\)

4. INFER the minimum value

• For vertex form \(\mathrm{g(x) = a(x - h)^2 + k}\) with \(\mathrm{a \gt 0}\), the minimum value is k
• Therefore, minimum value = 55

Alternative reasoning: Since \(\mathrm{x^2 \geq 0}\) always, the smallest value of \(\mathrm{x^2 + 55}\) occurs when \(\mathrm{x^2 = 0}\), giving us \(\mathrm{0 + 55 = 55}\).

Answer: B. 55

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse what "minimum value of the function" means

Students find where the minimum occurs (\(\mathrm{x = 0}\)) instead of what the minimum value is (\(\mathrm{y = 55}\)). They think the question is asking for the x-coordinate of the vertex rather than the y-coordinate.

This leads them to select Choice A (0).

Second Most Common Error:

Missing conceptual knowledge: Students don't recognize vertex form or quadratic properties

Without understanding that \(\mathrm{x^2 + 55}\) is in vertex form or that \(\mathrm{x^2 \geq 0}\), students may attempt incorrect approaches like setting the function equal to zero or making calculation errors with the given values.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key challenge is distinguishing between the input value where the minimum occurs (\(\mathrm{x = 0}\)) versus the actual minimum output value of the function (\(\mathrm{y = 55}\)). Students must clearly understand that "minimum value of the function" refers to the range value, not the domain value.