Question: In the xy-plane, the graphs of \(\mathrm{y = (x - 2)^2 - 1}\) and y = 2x - 2...

1. TRANSLATE the problem information

• Given information:
  • Two functions: \(\mathrm{y = (x - 2)^2 - 1}\) and \(\mathrm{y = 2x - 2}\)
  • They intersect at two points
  • One intersection point has x-coordinate of 1
  • Need to find y-coordinate of the other intersection

2. INFER the approach

• To find intersection points, we need to find where the function values are equal
• Set the two equations equal to each other: \(\mathrm{(x - 2)^2 - 1 = 2x - 2}\)
• This will give us the x-coordinates of both intersection points

3. SIMPLIFY to solve the equation

• Expand the left side: \(\mathrm{(x - 2)^2 - 1 = x^2 - 4x + 4 - 1 = x^2 - 4x + 3}\)
• Our equation becomes: \(\mathrm{x^2 - 4x + 3 = 2x - 2}\)
• Move all terms to one side: \(\mathrm{x^2 - 4x + 3 - 2x + 2 = 0}\)
• Simplify: \(\mathrm{x^2 - 6x + 5 = 0}\)

4. SIMPLIFY by factoring

• Factor the quadratic: \(\mathrm{(x - 1)(x - 5) = 0}\)
• This gives us \(\mathrm{x = 1}\) or \(\mathrm{x = 5}\)

5. INFER which intersection point to use

• We're told one intersection has x-coordinate of 1
• Therefore, the other intersection must have x-coordinate of 5
• We need the y-coordinate of the intersection at \(\mathrm{x = 5}\)

6. SIMPLIFY to find the y-coordinate

• Substitute \(\mathrm{x = 5}\) into either original equation
• Using \(\mathrm{y = 2x - 2}\):
  \(\mathrm{y = 2(5) - 2}\)
  \(\mathrm{y = 10 - 2}\)
  \(\mathrm{y = 8}\)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x - 2)^2}\) or when collecting like terms in the quadratic equation.

For example, they might expand \(\mathrm{(x - 2)^2}\) as \(\mathrm{x^2 - 4x + 2}\) instead of \(\mathrm{x^2 - 4x + 4}\), or make sign errors when moving terms across the equals sign. These errors lead to a different quadratic equation with different solutions, resulting in an incorrect y-coordinate calculation. This may lead them to select Choice A (2) or Choice B (5) depending on their specific algebraic mistake.

Second Most Common Error:

Incomplete INFER reasoning: Students find the correct x-coordinates (1 and 5) but then calculate the y-coordinate for \(\mathrm{x = 1}\) instead of \(\mathrm{x = 5}\), not carefully reading that they need the "other" intersection point.

Since \(\mathrm{y = 2(1) - 2 = 0}\), and \(0\) isn't among the answer choices, this leads to confusion and guessing among the available options.

The Bottom Line:

This problem requires careful algebraic manipulation and close attention to what the question is actually asking for. The key challenge is maintaining accuracy through multiple algebraic steps while keeping track of which intersection point the question wants.