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

### Step 1: Standard Form of a Parabola
Since the vertex of the parabola is at the origin, and it is tangent to a line, we can assume the general form of a vertical parabola as:
\[
y = ax^2
\]
where \(a\) is a constant that we need to find.

### Step 2: Equation of the Tangent Line
The equation of the line given is \( x + y + 1 = 0 \), or equivalently:
\[
y = -x - 1
\]
This line is tangent to the parabola at the origin (vertex).

### Step 3: Condition for Tangency
For tangency at the vertex, the slope of the parabola at \(x = 0\) must match the slope of the line at the same point.

- The slope of the line \(y = -x - 1\) is \(-1\).
- The derivative of the parabola \(y = ax^2\) is \( \frac{dy}{dx} = 2ax \). At \(x = 0\), this gives:
\[
\frac{dy}{dx} = 2a(0) = 0
\]
Thus, the parabola needs to be horizontal at the vertex, which suggests that the parabola opens vertically and does not need to match the slope of the line since tangency occurs at the vertex, not necessarily elsewhere.

### Step 4: Finding the Equation
The line is tangent to the parabola at \(x = 0\) and \(y = -1\). Therefore, we require that the parabola passes through the point \((0, -1)\).

For \( x = 0 \), we substitute into the parabola's equation \( y = ax^2 \):
\[
-1 = a(0)^2 = a(0)
\]
Thus, \( a = -1 \).

### Final Equation
The equation of the parabola is:
\[
y = -x^2
\]

This is the required equation for the parabola.

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### Related Questions:
1. What would the equation look like if the parabola opened horizontally instead of vertically?
2. How would the solution change if the vertex of the parabola wasn't at the origin?
3. What other types of curves could be used if the tangency condition was changed?
4. Can this parabola be reflected over the x-axis or y-axis?
5. How can we determine if a given line is tangent to a parabola?

### Tip:
To confirm if a line is tangent to a curve, you can check that the line and the curve have the same slope at the point of intersection.

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.

Learn how to find the equation of a parabola with its vertex at the origin, tangent to the line x + y + 1 = 0 at the vertex. This problem covers key concepts in conic sections, including parabolas and tangency conditions, suitable for students in Grades 10-12.

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