Collection of twelve mathematical graphs illustrating different functions and relations on coordinate planes.

You’ve uploaded a grid of 12 graphs, each on a coordinate plane. Since you mentioned “solve the problem,” but didn’t specify what the task is, I’ll assume based on common math worksheet formats that the goal is to determine which of these graphs represent functions, and possibly identify their domain and range or classify them by type (linear, quadratic, piecewise, etc.).

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## ✔ Key Concept: The Vertical Line Test

> A graph represents a function if and only if no vertical line intersects the graph more than once.

This is because, for a relation to be a function, each input (x-value) must correspond to exactly one output (y-value).

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Let’s analyze each graph row by row:

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🔹 Row 1

Graph 1 (Top Left):
- Two vertical lines with arrows (one at x = -2, one at x = 2), plus two points.
- ✘ NOT a function — vertical lines mean multiple y-values for a single x-value.

Graph 2 (Top Middle):
- Piecewise linear graph, continuous, passes vertical line test.
- ✔ Function

Graph 3 (Top Right):
- Smooth curve (looks like cubic or polynomial), no vertical overlaps.
- ✔ Function

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🔹 Row 2

Graph 4 (Middle Left):
- Horizontal line with arrows left/right — but it’s a horizontal line, so every x has one y.
- ✔ Function (constant function)

Graph 5 (Middle Center):
- Scatter plot of discrete points — no x-value repeats.
- ✔ Function (discrete function)

Graph 6 (Middle Right):
- Curve going rightward, decreasing — looks like part of a parabola opening left.
- ✘ NOT a function — fails vertical line test (e.g., at x=0, multiple y-values possible)

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🔹 Row 3

Graph 7 (Bottom Left):
- Circle centered at origin.
- ✘ NOT a function — vertical line through center hits circle twice.

Graph 8 (Bottom Middle):
- Wavy periodic curve (like sine or cosine).
- ✔ Function — passes vertical line test.

Graph 9 (Bottom Right):
- Zig-zag line, but still only one y per x.
- ✔ Function (piecewise linear)

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🔹 Row 4

Graph 10 (Bottom Left):
- U-shaped parabola opening up — classic quadratic.
- ✔ Function

Graph 11 (Bottom Middle):
- Piecewise with open/closed circles — still, each x maps to one y.
- ✔ Function

Graph 12 (Bottom Right):
- Triangle shape — vertical line through x=1 hits two points.
- ✘ NOT a function

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## 📊 Summary Table:

| Graph # | Is Function? | Reason |
|---------|--------------|--------|
| 1 | ✘ No | Vertical lines → multiple y for same x |
| 2 | ✔ Yes | Passes vertical line test |
| 3 | ✔ Yes | Smooth curve, no overlaps |
| 4 | ✔ Yes | Horizontal line → constant function |
| 5 | ✔ Yes | Discrete points, no repeated x |
| 6 | ✘ No | Fails vertical line test (left-opening curve) |
| 7 | ✘ No | Circle → fails vertical line test |
| 8 | ✔ Yes | Waveform, passes test |
| 9 | ✔ Yes | Zigzag, still one y per x |
| 10 | ✔ Yes | Parabola, passes test |
| 11 | ✔ Yes | Piecewise with endpoints handled |
| 12 | ✘ No | Triangle → vertical line hits twice |

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## 💡 Bonus: Classification (if needed)

- Linear: Graphs 2, 4, 9, 11
- Quadratic: Graph 10
- Polynomial/Cubic: Graph 3
- Periodic: Graph 8
- Discrete: Graph 5
- Not Functions: Graphs 1, 6, 7, 12

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## ✍️ Final Answer (If Task is “Which are Functions?”):

✔ Graphs that ARE functions:
2, 3, 4, 5, 8, 9, 10, 11

✘ Graphs that are NOT functions:
1, 6, 7, 12

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If your actual task was different (e.g., find domain/range, classify types, write equations), please clarify — I’m happy to expand!

Otherwise, this should fully solve the most likely interpretation of your worksheet. 😊