Identifying Functions from Graphs Worksheets - Free Printable

To determine whether each graph represents a function, we use the Vertical Line Test:

> ✔ A graph represents a function if no vertical line intersects the graph at more than one point.
> ✘ If any vertical line crosses the graph at two or more points, it is not a function.

Let’s go through each graph one by one:

---

1)

- This graph is a curve that opens to the right, like a sideways parabola.
- It fails the vertical line test because a vertical line (e.g., at $ x = 0 $) will intersect the graph at two points.
- Answer: Not a function

---

2)

- This is a smooth wave-like curve (like a sine or cosine), but only over a limited domain.
- Every vertical line will intersect the graph at only one point.
- Answer: Function

---

3)

- This is a parabola opening upward, vertex at the bottom.
- For every $ x $, there is only one y-value.
- Passes the vertical line test.
- Answer: Function

---

4)

- This graph looks like a sideways "S" shape, possibly part of a cubic or similar.
- But notice: for some values of $ x $ (e.g., around $ x = 0 $), a vertical line would cross the graph twice.
- So, it fails the vertical line test.
- Answer: Not a function

---

5)

- This is a straight diagonal line, going from top-left to bottom-right.
- A straight line (unless vertical) passes the vertical line test.
- Each $ x $ maps to exactly one $ y $.
- Answer: Function

---

6)

- This is an ellipse, centered at the origin.
- A vertical line through the center (e.g., $ x = 0 $) intersects the ellipse at two points (top and bottom).
- So, it fails the vertical line test.
- Answer: Not a function

---

7)

- This graph has two separate parts: one on the left and one on the right.
- The left part is a downward-opening curve, and the right is an upward-opening curve.
- But look closely: both curves are independent, and for each $ x $, there is only one $ y $ value.
- Even though it's disconnected, as long as no vertical line hits more than one point, it's okay.
- Answer: Function

> ⚠️ Note: There may be a potential issue if two branches share the same $ x $, but in this case, they appear to be on different $ x $-ranges. So, no overlap → passes vertical line test.

---

8)

- This is a piecewise linear graph, with a sharp corner.
- It appears to be a single continuous path, and for each $ x $, only one $ y $ exists.
- Passes the vertical line test.
- Answer: Function

---

9)

- This graph is a vertical line at $ x = 2 $, extending from $ y = -1 $ to $ y = 2 $.
- A vertical line (in fact, *this* line) intersects itself at infinitely many points.
- So, for $ x = 2 $, there are multiple $ y $-values → violates definition of a function.
- Answer: Not a function

---

✔ Final Answers:

| Graph | Is it a function? |
|-------|-------------------|
| 1 | ✘ No |
| 2 | ✔ Yes |
| 3 | ✔ Yes |
| 4 | ✘ No |
| 5 | ✔ Yes |
| 6 | ✘ No |
| 7 | ✔ Yes |
| 8 | ✔ Yes |
| 9 | ✘ No |

---

📝 Summary:

Use the Vertical Line Test:
- If any vertical line crosses the graph more than once → not a function
- Otherwise → it is a function

This applies regardless of how complex or unusual the graph looks — just check for multiple outputs ($ y $) for a single input ($ x $).

Let me know if you'd like this explained visually or with coordinates!