Even and Odd Functions Worksheet: Practice Problems and Solutions - Free Printable

Let's analyze the six graphs shown in the image and determine what makes them special — that is, what common property or category they all belong to.

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Step 1: Examine Each Graph

We'll go through each graph one by one and look for patterns.

#### Graph 1 (Top-left):
- A smooth, symmetric curve about the y-axis.
- Looks like a "W" shape but rounded at the top.
- Symmetric: left side mirrors right side.
- This is an even function.

#### Graph 2 (Top-middle):
- A parabola opening upwards, vertex at the origin.
- Clearly symmetric about the y-axis.
- Classic example of an even function.

#### Graph 3 (Top-right):
- A sine-like wave, repeating periodically.
- It appears symmetric about the y-axis over the interval shown.
- But wait — this looks like a cosine wave (since it starts at a peak).
- Cosine is an even function.
- So even though it's periodic, within the view, it's symmetric about y-axis.

> Note: The full function may be periodic, but the symmetry about y-axis suggests evenness.

#### Graph 4 (Bottom-left):
- A piecewise function with sharp corners.
- Starts at (-3, 3), goes down to (-2, 2), stays flat, then down to (0,0), then up again.
- Notice: The left side from x = -3 to x = 0 is mirrored exactly on the right side from x = 0 to x = 3.
- For every point (x, y), there’s a point (-x, y).
- So this is also symmetric about the y-axis → even function.

#### Graph 5 (Bottom-middle):
- A triangle-shaped graph, symmetric about the y-axis.
- Peaks at the origin, goes linearly down on both sides.
- This is clearly symmetric about the y-axis → even function.

#### Graph 6 (Bottom-right):
- A U-shaped curve, but flipped upside-down? No — it opens upward.
- Minimum near the origin, symmetric about the y-axis.
- Again, symmetric: f(-x) = f(x)
- So this is an even function.

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Step 2: Identify the Common Property

All six functions are symmetric about the y-axis.

This means:
> For every point (x, y) on the graph, the point (-x, y) is also on the graph.

In mathematical terms:
> $ f(-x) = f(x) $

This is the definition of an even function.

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✔ Conclusion:

All these functions are even functions.

They are special because they are symmetric about the y-axis, meaning their graphs remain unchanged when reflected across the vertical axis.

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Final Answer:

👉 These functions are all *even functions*. They are special because they are symmetric about the y-axis — that is, $ f(-x) = f(x) $ for all $ x $.