The equation \(\mathrm{y = (x - 2)^2 + 1}\) represents a parabola in the xy-plane. A new parabola is formed...

1. TRANSLATE the problem information

• Given information:
  • Original parabola: \(\mathrm{y = (x - 2)^2 + 1}\)
  • Transformations: 5 units left, 3 units down
  • Find: y-intercept of new parabola
• This tells us we need to apply transformations to create a new equation, then find where it crosses the y-axis

2. INFER the vertex of the original parabola

• The equation \(\mathrm{y = (x - 2)^2 + 1}\) is in vertex form \(\mathrm{y = a(x - h)^2 + k}\)
• The vertex \(\mathrm{(h, k)}\) can be read directly as \(\mathrm{(2, 1)}\)

3. TRANSLATE the transformations into coordinate changes

• "5 units left" means subtract 5 from the x-coordinate: \(\mathrm{2 - 5 = -3}\)
• "3 units down" means subtract 3 from the y-coordinate: \(\mathrm{1 - 3 = -2}\)
• New vertex: \(\mathrm{(-3, -2)}\)

4. SIMPLIFY to write the new parabola equation

• Using vertex form with new vertex \(\mathrm{(-3, -2)}\):
  \(\mathrm{y = (x - (-3))^2 + (-2)}\)
• SIMPLIFY: \(\mathrm{y = (x + 3)^2 - 2}\)

5. INFER how to find the y-intercept

• Y-intercept occurs when the graph crosses the y-axis (when \(\mathrm{x = 0}\))
• Substitute \(\mathrm{x = 0}\) into the new equation

6. SIMPLIFY to calculate the y-intercept

• \(\mathrm{y = (0 + 3)^2 - 2}\)
• \(\mathrm{y = 3^2 - 2}\)
• \(\mathrm{y = 9 - 2 = 7}\)

Answer: C) 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing the direction of transformations, especially thinking "left" means adding to x instead of subtracting.

Students might think: "5 units left means x + 5, so new vertex is (7, 1), then 3 units down gives (7, -2)."

This leads to equation \(\mathrm{y = (x - 7)^2 - 2}\), and y-intercept = \(\mathrm{(0 - 7)^2 - 2 = 49 - 2 = 47}\), which doesn't match any answer choice. This causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when working with the transformed equation, particularly with \(\mathrm{(x + 3)^2 - 2}\).

Students might incorrectly compute \(\mathrm{(0 + 3)^2 - 2}\) as \(\mathrm{3 - 2 = 1}\) (forgetting to square the 3), or make other arithmetic mistakes that lead them to select Choice A (2) or guess among the remaining choices.

The Bottom Line:

This problem tests whether students can correctly interpret transformation language and systematically apply it to parabola equations. The key insight is that transformations affect the vertex coordinates in predictable ways, and these changes must be carefully tracked through to the final calculation.