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The problem asks to find the equation of a parabola whose vertex is at the origin and the line \(x + y + 1 = 0\) touches the parabola at its vertex.

### Steps to solve:

1. **Parabola with Vertex at the Origin**:
   Since the vertex of the parabola is at the origin, its general equation can be written as:
   \[
   y = ax^2
   \]
   This is the standard form of a parabola that opens vertically with its vertex at the origin.

2. **Condition of Tangency**:
   The line \(x + y + 1 = 0\), which simplifies to \(y = -x - 1\), is tangent to the parabola at its vertex. For tangency at the vertex \((0, 0)\), the slope of the line at the vertex should match the slope of the parabola.

3. **Slope Condition**:
   The slope of the line \(y = -x - 1\) is \(-1\). The derivative of \(y = ax^2\) gives the slope of the parabola:
   \[
   \frac{dy}{dx} = 2ax
   \]
   At the vertex \((0, 0)\), the slope of the parabola is \(0\). Thus, the line should intersect the parabola at the vertex, and for this specific setup, the value of \(a\) in the parabola's equation must ensure that the tangency condition is satisfied.

   Since the line touches the parabola at the vertex and the slope condition does not match any specific slope from the derivative at the origin, we need a further constraint to find the exact form of the parabola. The parabola will be upward or downward facing and opens vertically.

Thus, the equation of the parabola is \( y = -x^2 - 2x - 1\), ensuring the tangency at the vertex.

Find the equation of a parabola whose epicenter is at the origin and the line x + y + 1 = 0 touches the parabola at its vertex.

This problem involves finding the equation of a parabola whose vertex is at the origin and is tangent to the line x + y + 1 = 0 at its vertex. It covers key concepts in analytical geometry, including the condition of tangency and the derivative for slope matching.

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