Worksheet featuring four graphs to determine if functions are even, odd, or neither.

Let’s solve each problem by analyzing the graphical symmetry of the functions.

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🔁 Definitions:

- Even function: Symmetric about the y-axis.
→ If you fold the graph along the y-axis, both sides match.
→ Mathematically: `f(-x) = f(x)`

- Odd function: Symmetric about the origin (rotational symmetry of 180°).
→ If you rotate the graph 180° around the origin, it looks the same.
→ Mathematically: `f(-x) = -f(x)`

- Neither: Doesn’t satisfy either symmetry.

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## ✔ Problem 1:

Graph: A wavy curve that goes up and down symmetrically on both sides of the y-axis.

- Look at x = 5 and x = -5 → same height (same y-value).
- The entire graph is mirrored across the y-axis.
- This is classic even function behavior.

✔ Answer: Even

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## ✔ Problem 2:

Graph: A curve that peaks near the origin and falls steeply on both sides — but it’s not symmetric about the y-axis or origin.

- Left side: as x → -∞, y → -∞
- Right side: as x → ∞, y → -∞
- But look at x = 2 and x = -2 — they are NOT the same height → not even.
- Also, if you rotate 180°, it doesn’t match → not odd.
- It’s symmetric about the y-axis? No — the left and right aren’t mirror images.
Wait — actually, looking closely: the graph is symmetric about the y-axis! The peak is centered at x=0, and both sides fall identically.

➡️ Let me double-check: yes, this graph is symmetric about the y-axis. The shape on the left is a mirror image of the right.

✔ Answer: Even

*(Note: Some might think it's neither because of the vertical drop, but symmetry is about the overall shape — and it is symmetric about y-axis.)*

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## ✔ Problem 3:

Graph: A hyperbola-like curve in quadrants II and IV — one branch in top-left, one in bottom-right.

- This is symmetric about the origin.
- Example: Point (2, -1) would correspond to (-2, 1) — which matches the graph.
- Rotate 180° → graph maps onto itself.

✔ Answer: Odd

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## ✔ Problem 4:

Graph: Similar to #3, but flipped — branches in quadrants I and III.

- Again, symmetric about the origin.
- For example, point (1, 2) corresponds to (-1, -2) — which matches the graph.
- Rotating 180° around origin gives the same graph.

✔ Answer: Odd

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## 📝 Final Answers:

1. Even
2. Even
3. Odd
4. Odd

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Let me know if you’d like to see the algebraic reasoning too!