In the xy-plane, the line y = -3 and the parabola \(\mathrm{y = (x + 4)^2 - 3}\) intersect at...

1. TRANSLATE the problem information

• Given information:
  • Line: \(\mathrm{y = -3}\) (horizontal line)
  • Parabola: \(\mathrm{y = (x + 4)^2 - 3}\) (vertex form)
  • They intersect at exactly one point
• What this tells us: We need to find where both equations give the same y-value

2. INFER the approach

• To find intersection points, set the y-values of both equations equal
• This gives us one equation with one unknown (x)
• Solve for x to get the x-coordinate of intersection

3. SIMPLIFY by setting up the equation

Set the right sides equal since both equal y:

\(\mathrm{-3 = (x + 4)^2 - 3}\)

4. SIMPLIFY through algebraic manipulation

Add 3 to both sides:

\(\mathrm{-3 + 3 = (x + 4)^2 - 3 + 3}\)

\(\mathrm{0 = (x + 4)^2}\)

5. SIMPLIFY to solve for x

Take the square root of both sides:

\(\mathrm{\sqrt{0} = \sqrt{(x + 4)^2}}\)

\(\mathrm{0 = x + 4}\)

Subtract 4 from both sides:

\(\mathrm{x = -4}\)

Answer: C (-4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when moving terms across the equals sign or forget to add 3 to both sides correctly.

Instead of getting \(\mathrm{0 = (x + 4)^2}\), they might get \(\mathrm{-6 = (x + 4)^2}\) or \(\mathrm{0 = (x + 4)^2 - 6}\). This leads to incorrect x-values like \(\mathrm{x = -4 \pm \sqrt{6}}\), which don't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might not recognize that intersection means setting the equations equal, and instead try to solve each equation separately or substitute answer choices without systematic approach.

This causes them to get stuck early in the problem and resort to random answer selection.

The Bottom Line:

This problem tests whether students can systematically set up and solve the intersection condition. The key insight is that "intersect at exactly one point" translates directly to a single solution when the equations are set equal.