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

Options

A.
B.
C.
D.

Answer

To solve the problem of identifying the representation of the curve given by the equation \( x^2 = -2y \), we can follow these steps:

### Step 1: Rewrite the equation in terms of \( y \)
We start with the equation:
\[ x^2 = -2y \]
To express \( y \) in terms of \( x \), we rearrange the equation:
\[ y = -\frac{1}{2}x^2 \]

### Step 2: Identify the type of curve
The equation \( y = -\frac{1}{2}x^2 \) is a quadratic equation in \( x \). The general form of a quadratic equation is \( y = ax^2 + bx + c \). Here, \( a = -\frac{1}{2} \), \( b = 0 \), and \( c = 0 \).

### Step 3: Analyze the parabola
Since \( a < 0 \) (specifically, \( a = -\frac{1}{2} \)), the parabola opens downwards. The vertex of this parabola is at the origin (0,0), and it is symmetric about the y-axis.

### Step 4: Sketch the graph
To sketch the graph of \( y = -\frac{1}{2}x^2 \):
- The vertex is at (0,0).
- As \( x \) increases or decreases, \( y \) becomes more negative, indicating that the parabola opens downwards.
- The parabola will intersect the y-axis at (0,0) and will be wider than the standard parabola \( y = -x^2 \) because of the factor of \( \frac{1}{2} \).

### Step 5: Choose the correct graph from options
If provided with multiple-choice options, we look for the graph that matches the characteristics described:
- A downward-opening parabola with its vertex at the origin.

### Final Answer
The correct representation of the curve \( x^2 = -2y \) is the downward-opening parabola given by \( y = -\frac{1}{2}x^2 \).

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