In the xy-plane, the graph of the equation \(\mathrm{y = (x - 3)^2 + 5}\) is a parabola. The parabola...

1. TRANSLATE the given equation into vertex information

• Given equation: \(\mathrm{y = (x - 3)^2 + 5}\)
• This is in vertex form \(\mathrm{y = a(x - h)^2 + k}\)
• Reading the vertex coordinates: \(\mathrm{h = 3, k = 5}\)
• Original vertex: \(\mathrm{(3, 5)}\)

2. TRANSLATE the transformation description into coordinate changes

• Translation: "4 units to the left and 2 units down"
• Left 4 units → subtract 4 from x-coordinate
• Down 2 units → subtract 2 from y-coordinate

3. INFER the new vertex location

• New x-coordinate: \(\mathrm{3 - 4 = -1}\)
• New y-coordinate: \(\mathrm{5 - 2 = 3}\)
• New vertex: \(\mathrm{(-1, 3)}\)

4. TRANSLATE the new vertex back into equation form

• Using vertex form with new coordinates \(\mathrm{(h', k') = (-1, 3)}\):
• \(\mathrm{y = (x - h')^2 + k' = (x - (-1))^2 + 3}\)
• SIMPLIFY: \(\mathrm{y = (x + 1)^2 + 3}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing the sign conventions for horizontal translations

Students often think "move 4 units left" means the x-coordinate should increase by 4, leading them to calculate the new x-coordinate as \(\mathrm{3 + 4 = 7}\). This gives a new vertex of \(\mathrm{(7, 3)}\) and equation \(\mathrm{y = (x - 7)^2 + 3}\).

This may lead them to select Choice A (\(\mathrm{y = (x - 7)^2 + 3}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Mixing up which coordinate is affected by which direction

Students might correctly subtract for "left" but apply it to the y-coordinate instead, or correctly subtract for "down" but apply it to the x-coordinate. This creates confusion about the final vertex location and leads to guessing among the remaining choices.

The Bottom Line:

This problem tests whether students truly understand the connection between geometric transformations and algebraic representations. The key insight is that translations are applied directly to vertex coordinates, with careful attention to sign conventions.