Which of the following can represent the curve `x^(2) = -2y` ?

Options

A.
B.
C.
D.

Answer

To solve the equation \( x^2 = -2y \) and determine its representation, we can follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the equation to express \( y \) in terms of \( x \):
\[ x^2 = -2y \implies y = -\frac{1}{2}x^2 \]

### Step 2: Understanding the Form of the Equation
The equation \( y = -\frac{1}{2}x^2 \) is a quadratic equation in \( x \). It represents a parabola that opens downwards because the coefficient of \( x^2 \) is negative.

### Step 3: Identifying the Vertex
The vertex of the parabola can be found from the equation \( y = -\frac{1}{2}x^2 \). The vertex occurs at the point where \( x = 0 \):
\[ y = -\frac{1}{2}(0)^2 = 0 \]
Thus, the vertex is at the point \( (0, 0) \).

### Step 4: Analyzing the Symmetry
The parabola is symmetric about the y-axis because the equation involves \( x^2 \). This means that for every positive \( x \), there is a corresponding negative \( x \) that gives the same \( y \) value.

### Step 5: Sketching the Graph
To sketch the graph, we can plot a few points:
- For \( x = 1 \), \( y = -\frac{1}{2}(1)^2 = -\frac{1}{2} \) → Point (1, -0.5)
- For \( x = -1 \), \( y = -\frac{1}{2}(-1)^2 = -\frac{1}{2} \) → Point (-1, -0.5)
- For \( x = 2 \), \( y = -\frac{1}{2}(2)^2 = -2 \) → Point (2, -2)
- For \( x = -2 \), \( y = -\frac{1}{2}(-2)^2 = -2 \) → Point (-2, -2)

Plotting these points will show that the parabola opens downwards and is symmetric about the y-axis.

### Step 6: Conclusion
The curve represented by the equation \( x^2 = -2y \) is a downward-opening parabola with its vertex at the origin (0, 0).