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.

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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.
The graph of $$ y = \frac{4}{x} $$ is symmetric to the origin.

Shown on the right is the graph of .
Determine its symmetries (if any).
This graph is
A. symmetric to the origin.
B. symmetric to the -axis.
C. symmetric to the -axis.
D. not symmetric to the -axis, -axis, or origin.

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Shown on the right is the graph of . Determine its symmetries (if any). This graph is A. symmetric to the origin. B. symmetric to the -axis. C. symmetric to the -axis. D. not symmetric to the -axis, -axis, or origin.