Worksheet for identifying functions from graphs.

Let’s go step by step to figure out which graphs represent functions.

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What is a function?

A function is a special kind of relationship between two things (usually x and y). For every input (x-value), there must be only one output (y-value).

In graph terms, we use the Vertical Line Test:
> If you can draw a vertical line anywhere on the graph and it touches the graph in more than one place, then it’s NOT a function.

If every vertical line hits the graph at most once → it IS a function.

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## Part A: State whether each graph represents a function.

We’ll go one by one:

Graph 1:

It’s a curve that goes up and to the right — looks like part of a sideways parabola or exponential. Try drawing a vertical line — does it ever hit more than one point?
→ No! Every vertical line hits only once.
✔ Function

Graph 2:

This is a wave-like curve (like sine or cosine). Vertical lines will always hit just one point.
✔ Function

Graph 3:

This is a circle. Draw a vertical line through the middle — it hits top and bottom! Two points for same x.
✘ Not a function

Graph 4:

Straight diagonal line. Any vertical line hits exactly once.
✔ Function

Graph 5:

Oval shape (ellipse). Like the circle — vertical line through center hits two points.
✘ Not a function

Graph 6:

Looks like a sideways “S” or hyperbola branches. Pick an x in the middle — vertical line hits two places (top and bottom branch).
✘ Not a function

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## Part B: Which of the following graphs represents a function?

Now we look at three new graphs labeled a, b, c.

Graph a:

Curvy line going from bottom left to top right — smooth, no loops. Vertical line test: passes everywhere.
✔ Function

Graph b:

Vertical line itself! That means for one x-value, there are infinite y-values. Definitely fails vertical line test.
✘ Not a function

Graph c:

Wave-like curve again — similar to graph 2 in part A. Each x has only one y.
✔ Function

Wait — the question says: “Which of the following graphs represents a function?” — implying maybe pick one? But actually, both a and c are functions. Let me check again.

Looking closely at graph b: It’s a straight vertical line — so yes, not a function.

Graphs a and c both pass the vertical line test.

But sometimes questions expect you to choose all that apply. Since it doesn’t specify “choose one”, and based on standard math problems, if multiple are correct, we list them.

However, let’s double-check graph a — is there any place where a vertical line would hit twice? From what I see, no — it’s strictly increasing or decreasing without looping back.

Same with c — it wiggles but never doubles back vertically.

So both a and c are functions.

But wait — looking again at the original image description (even though we’re not describing it), in many such worksheets, sometimes they trick you. Let me think...

Actually, in some versions, graph a might have a loop — but according to our analysis above, assuming standard shapes:

- Graph a: function
- Graph b: not
- Graph c: function

But perhaps the question expects only one answer? Hmm.

Wait — re-reading: “Which of the following graphs represents a function?” — singular “a function”, but that doesn’t mean only one exists. In math, we usually select all that qualify.

To be safe, let’s assume we should report all correct ones.

But let’s verify once more with strict vertical line test:

- a: Imagine sliding a ruler vertically across — never hits two points → ✔
- b: Ruler placed on the line — infinite hits → ✘
- c: Wavy but single path per x → ✔

So answers for Part B: a and c

But maybe the worksheet intends for us to pick one? Or perhaps I misread.

Alternatively, maybe graph a is actually failing? Let me visualize again.

Suppose graph a is like a cubic curve — still a function. Unless it turns around horizontally — but even then, as long as no two y’s for same x, it’s fine.

I think we’re good.

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Final check:

Part A:
1) Yes
2) Yes
3) No
4) Yes
5) No
6) No

Part B:
a) Yes
b) No
c) Yes

So for Part B, since the question says “which... represents a function”, and multiple do, we should include both.

But in some contexts, especially multiple choice with single answer expected, they might want only one — however, here it's open-ended.

Given that, I'll proceed with accurate mathematical answer.

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Final Answer:

Part A:
1) Function
2) Function
3) Not a function
4) Function
5) Not a function
6) Not a function

Part B:
Graphs a and c represent functions.