In the xy-plane, what is the y-coordinate of the point of intersection of the graphs of \(\mathrm{y = (x -...

1. INFER what "intersection" means

• At the intersection point, both equations must give the same y-value for the same x-value
• This means we can set the right sides of both equations equal to each other:
  \(\mathrm{(x - 1)^2 = 2x - 3}\)

2. SIMPLIFY by expanding and collecting terms

• Expand the perfect square: \(\mathrm{(x - 1)^2 = x^2 - 2x + 1}\)
• Our equation becomes: \(\mathrm{x^2 - 2x + 1 = 2x - 3}\)
• Move everything to one side: \(\mathrm{x^2 - 2x + 1 - 2x + 3 = 0}\)
• Combine like terms: \(\mathrm{x^2 - 4x + 4 = 0}\)

3. SIMPLIFY by factoring the quadratic

• Recognize this as a perfect square trinomial: \(\mathrm{x^2 - 4x + 4 = (x - 2)^2}\)
• So our equation is: \(\mathrm{(x - 2)^2 = 0}\)
• Therefore: \(\mathrm{x = 2}\)

4. TRANSLATE back to find the y-coordinate

• Substitute \(\mathrm{x = 2}\) into either original equation
• Using \(\mathrm{y = 2x - 3}\): \(\mathrm{y = 2(2) - 3 = 1}\)
• (You can verify: \(\mathrm{y = (2-1)^2 = 1^2 = 1}\) ✓)

Answer: 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that intersection means "both equations give the same y-value" and instead try to solve each equation separately or graph both functions without setting them equal. This leads to confusion about what exactly they're supposed to find, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebra errors when expanding \(\mathrm{(x-1)^2}\) or when collecting terms, leading to an incorrect quadratic equation. For example, they might get \(\mathrm{x^2 - 2x + 4 = 0}\) instead of \(\mathrm{x^2 - 4x + 4 = 0}\), which would give a completely different x-value and therefore wrong y-coordinate. This leads to selecting an incorrect numerical answer.

The Bottom Line:

This problem tests whether students understand that intersection points represent solutions to systems of equations, not just individual equations. The algebra itself is straightforward once you know to set the equations equal to each other.