Shown on the right is the graph of \( y=\frac{4}{x} \). Determine its symmetries (if any). This graph is A. symmetric to the origin. B. symmetric to the \( x \)-axis. C. symmetric to the \( y \)-axis. D. not symmetric to the \( x \)-axis, \( y \)-axis, or origin.

To determine the symmetries of the graph \( y = \frac{4}{x} \), let's analyze each option: **A. Symmetric to the origin.** - **Check:** A graph is symmetric to the origin if replacing \( x \) with \( -x \) and \( y \) with \( -y \) results in the same equation. \[ y = \frac{4}{x} \quad \rightarrow \quad -y = \frac{4}{-x} \quad \Rightarrow \quad -y = -\frac{4}{x} \quad \Rightarrow \quad y = \frac{4}{x} \] The equation remains unchanged, confirming symmetry about the origin. **B. Symmetric to the \( x \)-axis.** - **Check:** A graph is symmetric to the \( x \)-axis if replacing \( y \) with \( -y \) results in the same equation. \[ -y = \frac{4}{x} \quad \Rightarrow \quad y = -\frac{4}{x} \] This is a different equation, so the graph is **not** symmetric to the \( x \)-axis. **C. Symmetric to the \( y \)-axis.** - **Check:** A graph is symmetric to the \( y \)-axis if replacing \( x \) with \( -x \) results in the same equation. \[ y = \frac{4}{-x} \quad \Rightarrow \quad y = -\frac{4}{x} \] This is a different equation, so the graph is **not** symmetric to the \( y \)-axis. **D. Not symmetric to the \( x \)-axis, \( y \)-axis, or origin.** - Based on the above checks, this is incorrect because the graph **is** symmetric to the origin. **Conclusion:** The graph of \( y = \frac{4}{x} \) is **symmetric to the origin** only. **Answer:** A. symmetric to the origin.