Question:In the xy-plane, the equation of parabola P is \(\mathrm{y = 2(x + 5)^2 - 3}\). Parabola Q is obtained...

1. TRANSLATE the problem information

• Given information:
  • Original parabola P: \(\mathrm{y = 2(x + 5)^2 - 3}\)
  • Transform: down 3 units and left 2 units
  • Need: equation of transformed parabola Q

2. INFER the vertex of the original parabola

• The vertex form is \(\mathrm{y = a(x - h)^2 + k}\) where \(\mathrm{(h, k)}\) is the vertex
• From \(\mathrm{y = 2(x + 5)^2 - 3 = 2(x - (-5))^2 + (-3)}\)
• The vertex of P is \(\mathrm{(-5, -3)}\)

3. TRANSLATE the transformation into coordinate changes

• Left 2 units: subtract 2 from x-coordinate
• Down 3 units: subtract 3 from y-coordinate
• New vertex coordinates: \(\mathrm{(-5 - 2, -3 - 3) = (-7, -6)}\)

4. INFER the equation of the transformed parabola

• Use vertex \(\mathrm{(-7, -6)}\) with the same coefficient \(\mathrm{a = 2}\)
• Substitute into vertex form: \(\mathrm{y = 2(x - (-7))^2 + (-6)}\)
• Simplify: \(\mathrm{y = 2(x + 7)^2 - 6}\)

Answer: C. \(\mathrm{y = 2(x + 7)^2 - 6}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the direction of horizontal transformations in vertex form. They know that moving left 2 units changes the x-coordinate from -5 to -7, but they incorrectly think this means the number in the parentheses should get smaller. They might reason: "If I move left, the number should decrease, so \(\mathrm{(x + 5)}\) becomes \(\mathrm{(x + 3)}\)."

This leads them to select Choice A (\(\mathrm{y = 2(x + 3)^2 - 6}\)) after correctly handling the vertical transformation but mishandling the horizontal one.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that they need to move the vertex, but they apply the transformations in the wrong direction. They might think "left 2" means add 2 to the x-coordinate instead of subtract 2, getting vertex \(\mathrm{(-3, -6)}\) instead of \(\mathrm{(-7, -6)}\).

This may lead them to select Choice B (\(\mathrm{y = 2(x + 3)^2 + 0}\)) if they also make errors with the vertical transformation.

The Bottom Line:

The key challenge is understanding that in vertex form \(\mathrm{y = a(x - h)^2 + k}\), the relationship between the vertex coordinates and what appears in the equation can be counterintuitive, especially for horizontal transformations where moving left makes the h-value more negative but the expression \(\mathrm{(x - h)}\) shows a larger positive number.