The parabola `y ^(2) =x` is symmetric about

Options

A. X-axis
B. Y-axis
C. Both X-axis and Y-axis
D. The line `y=x`

Answer

To determine the axis of symmetry for the parabola given by the equation \( y^2 = x \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Standard Form of the Parabola**:
The equation \( y^2 = x \) is in the standard form of a parabola that opens to the right. This form is generally expressed as \( y^2 = 4px \), where \( p \) is the distance from the vertex to the focus.

2. **Graph the Parabola**:
To visualize the parabola, we can plot some points. For example:
- If \( y = 0 \), then \( x = 0 \) (point: (0,0)).
- If \( y = 1 \), then \( x = 1 \) (point: (1,1)).
- If \( y = -1 \), then \( x = 1 \) (point: (1,-1)).
- If \( y = 2 \), then \( x = 4 \) (point: (4,2)).
- If \( y = -2 \), then \( x = 4 \) (point: (4,-2)).

Plotting these points will show that the parabola opens to the right.

3. **Determine the Symmetry**:
To find the axis of symmetry, we need to check if the parabola is symmetric about any line. A parabola of the form \( y^2 = x \) is symmetric about the x-axis. This means that for every point \( (x, y) \) on the parabola, the point \( (x, -y) \) is also on the parabola.

4. **Conclusion**:
Since the parabola \( y^2 = x \) is symmetric about the x-axis, we conclude that the axis of symmetry is the x-axis.

### Final Answer:
The parabola \( y^2 = x \) is symmetric about the x-axis.

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