y = x^2 - 14x + 22 The given equation relates the variables x and y. For what value of...

1. TRANSLATE the problem information

• Given: \(\mathrm{y = x^2 - 14x + 22}\)
• Find: Value of x where y reaches its minimum

2. INFER the mathematical approach

• This is a quadratic function in standard form \(\mathrm{y = ax^2 + bx + c}\)
• Since we need the minimum value, we need to find the vertex of the parabola
• The x-coordinate of the vertex gives us where the minimum occurs

3. TRANSLATE the coefficients

• Comparing \(\mathrm{y = x^2 - 14x + 22}\) to \(\mathrm{y = ax^2 + bx + c}\):
  • \(\mathrm{a = 1}\)
  • \(\mathrm{b = -14}\) (careful with the sign!)
  • \(\mathrm{c = 22}\)

4. INFER the vertex location strategy

• Since \(\mathrm{a = 1 \gt 0}\), the parabola opens upward
• This means the vertex represents the minimum point
• Use vertex formula: \(\mathrm{x = \frac{-b}{2a}}\)

5. SIMPLIFY using the vertex formula

• \(\mathrm{x = \frac{-b}{2a}}\)
  \(\mathrm{= \frac{-(-14)}{2 \times 1}}\)
  \(\mathrm{= \frac{14}{2}}\)
  \(\mathrm{= 7}\)

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mishandle the negative coefficient \(\mathrm{b = -14}\) when applying the vertex formula.

They might calculate \(\mathrm{x = \frac{-b}{2a}}\) as \(\mathrm{x = \frac{-(-14)}{2 \times 1} = \frac{-14}{2} = -7}\), forgetting that \(\mathrm{-(-14) = +14}\). This sign error leads to the wrong x-coordinate and potentially selecting an incorrect answer if -7 were among the choices.

Second Most Common Error:

Missing INFER connection: Students don't recognize this as a vertex-finding problem.

They might attempt to set \(\mathrm{y = 0}\) and solve the quadratic equation, thinking they need to find x-intercepts rather than the minimum point. This leads to unnecessary work with the quadratic formula and confusion about what the question is actually asking.

The Bottom Line:

This problem tests whether students can connect the concept of "minimum value" to the vertex of a parabola and correctly apply the vertex formula with attention to signs.