Determine visually whether the graph is symmetric with respect to the \( x \)-axis, the \( y \)-axis, or the origin. Choose the correct answer below. A. \( x \)-axis B. \( y \)-axis C. origin D. both \( x \)-axis and \( y \)-axis

I'm unable to view the graph you're referring to, but I can help you determine symmetry based on the general principles. Here's how you can check for different types of symmetry in a graph: 1. **Symmetry with respect to the \( y \)-axis**: - **Test**: Replace \( x \) with \( -x \) in the equation of the graph. - **If the equation remains unchanged**, the graph is symmetric about the \( y \)-axis. - **Example**: The equation \( y = x^2 \) is symmetric about the \( y \)-axis. 2. **Symmetry with respect to the \( x \)-axis**: - **Test**: Replace \( y \) with \( -y \) in the equation of the graph. - **If the equation remains unchanged**, the graph is symmetric about the \( x \)-axis. - **Note**: Most functions do not have \( x \)-axis symmetry because this would imply they are not functions (they would fail the vertical line test). 3. **Symmetry with respect to the origin**: - **Test**: Replace both \( x \) with \( -x \) and \( y \) with \( -y \) in the equation of the graph. - **If the equation remains unchanged**, the graph is symmetric about the origin. - **Example**: The equation \( y = x^3 \) is symmetric about the origin. 4. **Symmetry with respect to both the \( x \)-axis and the \( y \)-axis**: - **Condition**: The graph must satisfy both the \( x \)-axis and \( y \)-axis symmetry conditions individually. - **Example**: The equation \( y^2 = x^2 \) (which represents two lines, \( y = x \) and \( y = -x \)) is symmetric about both axes. **How to Choose the Correct Answer:** - **A. \( x \)-axis**: If only the \( x \)-axis symmetry condition is satisfied. - **B. \( y \)-axis**: If only the \( y \)-axis symmetry condition is satisfied. - **C. Origin**: If only the origin symmetry condition is satisfied. - **D. Both \( x \)-axis and \( y \)-axis**: If both the \( x \)-axis and \( y \)-axis symmetries are satisfied. **Example Application:** Suppose you're given the graph of \( y = x^2 \). - Replace \( x \) with \( -x \): \( y = (-x)^2 = x^2 \) (unchanged) → Symmetric about the \( y \)-axis. - Replace \( y \) with \( -y \): \( -y = x^2 \) → Not the same as the original equation → Not symmetric about the \( x \)-axis. - Replace both \( x \) with \( -x \) and \( y \) with \( -y \): \( -y = (-x)^2 = x^2 \) → Not the same as the original equation → Not symmetric about the origin. **Conclusion for the Example:** The graph is symmetric with respect to the \( y \)-axis. Therefore, the correct answer would be **B. \( y \)-axis**. If you can provide more details about the graph or its equation, I'd be happy to help you determine its symmetry more specifically!