Which of the following graphs is symmetric about the origin ?

Options

A.
B.
C.
D.

Explanation

To determine which of the given graphs is symmetric about the origin, we will analyze each option step by step.

### Step 1: Understand Symmetry About the Origin
A graph is symmetric about the origin if rotating it 180 degrees around the origin results in the same graph. Mathematically, this means that if (x, y) is a point on the graph, then (-x, -y) must also be a point on the graph.

**Hint:** Remember that symmetry about the origin involves flipping both the x and y coordinates.

### Step 2: Analyze Each Option
- **Option 1:** A parabola along the y-axis (e.g., y = x²).
- If we rotate this graph 180 degrees, the graph will look like an inverted parabola (e.g., y = -x²). This does not match the original graph.

**Hint:** Check if flipping the graph changes its shape or position.

- **Option 2:** Another parabola along the y-axis (similar to option 1).
- Similar reasoning applies; it will also become inverted and will not match the original graph.

**Hint:** Look for characteristics of the graph that change upon rotation.

- **Option 3:** A parabola along the x-axis (e.g., x = y²).
- Rotating this graph 180 degrees will also result in an inverted graph (e.g., x = -y²), which does not match the original.

**Hint:** Consider how the orientation of the graph changes with rotation.

- **Option 4:** A rectangular hyperbola in the first and second quadrants.
- When this graph is rotated 180 degrees, it will change its shape and orientation, resulting in a different graph.

**Hint:** Visualize how the hyperbola would look after rotation.

- **Option 5:** A rectangular hyperbola in the first and third quadrants.
- When this graph is rotated 180 degrees, it remains unchanged. The points (x, y) become (-x, -y), and the shape stays the same.

**Hint:** Check if the graph looks identical after a 180-degree rotation.

### Step 3: Conclusion
After analyzing all options, we find that **Option 5** is the only graph that is symmetric about the origin.

**Final Answer:** The graph that is symmetric about the origin is **Option 5**.