The graph of `y^4-3x^2+7=0` is symmetric with respect to which of the following?
the x-axis
the y-axis
the origin

Options

A. only I
B. only II
C. only III
D. I, II, and III

Explanation

To determine the symmetry of the graph of the equation \( y^4 - 3x^2 + 7 = 0 \) with respect to the x-axis, y-axis, and the origin, we will follow these steps:

### Step 1: Check for symmetry with respect to the x-axis
To check for symmetry with respect to the x-axis, we replace \( y \) with \( -y \) in the equation.

Starting with the original equation:
\[ y^4 - 3x^2 + 7 = 0 \]

Substituting \( -y \) for \( y \):
\[ (-y)^4 - 3x^2 + 7 = 0 \]

Since \( (-y)^4 = y^4 \), we have:
\[ y^4 - 3x^2 + 7 = 0 \]

This is the same as the original equation, indicating that the graph is symmetric with respect to the x-axis.

### Step 2: Check for symmetry with respect to the y-axis
To check for symmetry with respect to the y-axis, we replace \( x \) with \( -x \) in the equation.

Starting with the original equation:
\[ y^4 - 3x^2 + 7 = 0 \]

Substituting \( -x \) for \( x \):
\[ y^4 - 3(-x)^2 + 7 = 0 \]

Since \( (-x)^2 = x^2 \), we have:
\[ y^4 - 3x^2 + 7 = 0 \]

This is also the same as the original equation, indicating that the graph is symmetric with respect to the y-axis.

### Step 3: Check for symmetry with respect to the origin
To check for symmetry with respect to the origin, we replace \( x \) with \( -x \) and \( y \) with \( -y \) in the equation.

Starting with the original equation:
\[ y^4 - 3x^2 + 7 = 0 \]

Substituting \( -x \) for \( x \) and \( -y \) for \( y \):
\[ (-y)^4 - 3(-x)^2 + 7 = 0 \]

Since \( (-y)^4 = y^4 \) and \( (-x)^2 = x^2 \), we have:
\[ y^4 - 3x^2 + 7 = 0 \]

This is the same as the original equation, indicating that the graph is symmetric with respect to the origin.

### Conclusion
The graph of the equation \( y^4 - 3x^2 + 7 = 0 \) is symmetric with respect to the x-axis, y-axis, and the origin.

### Final Answer
The graph is symmetric with respect to:
- **x-axis**
- **y-axis**
- **origin**