x = y^2 - 1x = 3When the equations above are graphed in the xy-plane, what are the coordinates \(\mathrm{(x,...

1. INFER the solution approach

• Given information:
  • First equation: \(\mathrm{x = y^2 - 1}\) (a parabola)
  • Second equation: \(\mathrm{x = 3}\) (a vertical line)
  • Need: coordinates where these graphs intersect

• Key insight: Intersection points occur where both equations are satisfied simultaneously, so we can substitute one equation into the other.

2. INFER the substitution strategy

• Since both equations equal x, we can substitute \(\mathrm{x = 3}\) into the first equation
• This gives us: \(\mathrm{3 = y^2 - 1}\)

3. SIMPLIFY the equation

• Starting with: \(\mathrm{3 = y^2 - 1}\)
• Add 1 to both sides: \(\mathrm{4 = y^2}\)
• Take the square root: \(\mathrm{y = ±2}\)

4. CONSIDER ALL CASES for the y-values

• Since \(\mathrm{y^2 = 4}\), we have two solutions:
  • \(\mathrm{y = +2}\) (positive case)
  • \(\mathrm{y = -2}\) (negative case)

5. Form the coordinate pairs

• With \(\mathrm{x = 3}\) and \(\mathrm{y = ±2}\), our intersection points are:
  • \(\mathrm{(3, -2)}\) and \(\mathrm{(3, 2)}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Students make arithmetic errors when manipulating the equation \(\mathrm{3 = y^2 - 1}\), such as forgetting to add 1 to both sides or making sign errors.

For example, they might incorrectly get \(\mathrm{y^2 = 2}\) instead of \(\mathrm{y^2 = 4}\), leading to \(\mathrm{y = ±\sqrt{2} ≈ ±1.4}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Weak CONSIDER ALL CASES reasoning: Students remember that \(\mathrm{y^2 = 4}\) means \(\mathrm{y = 2}\), but forget about the negative solution \(\mathrm{y = -2}\).

However, since all answer choices include both positive and negative y-coordinates, this incomplete reasoning would still lead them to realize something is missing, prompting them to reconsider or guess.

The Bottom Line:

This problem tests whether students can systematically work through substitution and handle both solutions when taking square roots. The key is staying organized through the algebraic steps and remembering that quadratic equations typically have two solutions.