Is the parabola `y ^(2) = 4x` symmetrical about x-axis ?

Answer

To determine if the parabola given by the equation \( y^2 = 4x \) is symmetric about the x-axis, we can follow these steps:

### Step 1: Understand the concept of symmetry about the x-axis
A curve is symmetric about the x-axis if for every point \( (x, y) \) on the curve, the point \( (x, -y) \) is also on the curve. This means that if we replace \( y \) with \( -y \) in the equation of the curve, the resulting equation should be equivalent to the original equation.

### Step 2: Start with the given equation
The equation of the parabola is:
\[ y^2 = 4x \]

### Step 3: Replace \( y \) with \( -y \)
To check for symmetry about the x-axis, we substitute \( -y \) into the equation:
\[ (-y)^2 = 4x \]

### Step 4: Simplify the equation
Since \( (-y)^2 = y^2 \), we can rewrite the equation as:
\[ y^2 = 4x \]

### Step 5: Compare the equations
We see that the modified equation \( y^2 = 4x \) is identical to the original equation \( y^2 = 4x \). This means that for every point \( (x, y) \) on the parabola, the point \( (x, -y) \) is also on the parabola.

### Conclusion
Since the equation remains unchanged when \( y \) is replaced with \( -y \), we conclude that the parabola \( y^2 = 4x \) is symmetric about the x-axis.

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