Multiple choice question asking which of the four diagrams is not the graph of a function.

To determine which diagram is not the graph of a function, we use the Vertical Line Test.

🔍 Vertical Line Test:

A graph represents a function if no vertical line intersects the graph more than once.
If a vertical line crosses the graph at two or more points, then the graph does not represent a function.

Let’s analyze each option:

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A. Ellipse

- This is a horizontal ellipse centered at the origin.
- If you draw a vertical line through the center (e.g., at $ x = 0 $), it will intersect the ellipse at two points (top and bottom).
- So, one input (x-value) gives two outputs (y-values) → Not a function.

✔ So, A fails the vertical line test.

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B. Hyperbola (with two branches)

- This looks like a hyperbola opening left and right.
- For values of $ x $ near zero (but not zero), a vertical line would intersect both the upper and lower parts?
- Wait — actually, this seems to be a hyperbola that opens up and down, but with an asymptote at $ x = 0 $.
- Let's clarify: The graph has two curves: one in the second quadrant (upper-left), one in the fourth quadrant (lower-right), and a vertical asymptote at $ x = 0 $.
- But for any $ x \neq 0 $, a vertical line intersects the graph only once.
- Even though it's discontinuous, each $ x $ maps to at most one $ y $.
- So, this passes the vertical line test → It IS a function.

Wait — let's double-check: Is this a relation where each $ x $ has only one $ y $? Yes, because for each $ x $, there is only one point on the curve.

So B is a function.

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C. Cubic-like curve

- This is a smooth S-shaped curve passing through the origin.
- It looks like $ y = x^3 $, which is a function.
- Any vertical line intersects it once.
- ✔ Passes the vertical line test → This IS a function.

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D. V-shaped graph

- This is like $ y = |x| $, symmetric about the y-axis.
- For every $ x $, there is exactly one $ y $.
- A vertical line hits only one point.
- ✔ Passes the vertical line test → This IS a function.

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✔ Final Answer:

Only A fails the vertical line test — because for many $ x $-values, there are two corresponding $ y $-values.

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🟩 Correct answer: A

> Answer: A is not the graph of a function.