Which graph shows a set of ordered pairs that represent a function?

On a coordinate plane, solid circles appear at the following points: (negative 2, 3), (negative 1, 2), (1, 1), (2, negative 1), (2, negative 3), (4, negative 5).

On a coordinate plane, solid circles appear at the following points: (negative 4, negative 4), (negative 2, 2), (1, 1), (2, negative 5), (4, negative 3), (4, 4).

On a coordinate plane, solid circles appear at the following points: (negative 3, 2), (negative 2, 2), (0, 1), (1, 3), (2, negative 4), (4, negative 1).

On a coordinate plane, solid circles appear at the following points: (negative 4, 2), (negative 2, 1), (negative 2, negative 1), (0, negative 2), (0, negative 4), (2, negative 5).

To determine which graph shows a set of ordered pairs that represent a function, we must check if each x-coordinate corresponds to exactly one y-coordinate. This is known as the "vertical line test": if a vertical line intersects the graph at more than one point, then the graph does not represent a function.

Let's analyze each set of ordered pairs:

1. \(( -2, 3 ), ( -1, 2 ), ( 1, 1 ), ( 2, -1 ), ( 2, -3 ), ( 4, -5 )\)
   - Here, \( x = 2 \) corresponds to both \( y = -1 \) and \( y = -3 \).
   - Therefore, this set does **not** represent a function.

2. \(( -4, -4 ), ( -2, 2 ), ( 1, 1 ), ( 2, -5 ), ( 4, -3 ), ( 4, 4 )\)
   - Here, \( x = 4 \) corresponds to both \( y = -3 \) and \( y = 4 \).
   - Therefore, this set does **not** represent a function.

3. \(( -3, 2 ), ( -2, 2 ), ( 0, 1 ), ( 1, 3 ), ( 2, -4 ), ( 4, -1 )\)
   - Each x-coordinate corresponds to exactly one y-coordinate.
   - Therefore, this set **does** represent a function.

4. \(( -4, 2 ), ( -2, 1 ), ( -2, -1 ), ( 0, -2 ), ( 0, -4 ), ( 2, -5 )\)
   - Here, \( x = -2 \) corresponds to both \( y = 1 \) and \( y = -1 \).
   - Here, \( x = 0 \) corresponds to both \( y = -2 \) and \( y = -4 \).
   - Therefore, this set does **not** represent a function.

Hence, the graph that shows a set of ordered pairs that represent a function is the third one: \(( -3, 2 ), ( -2, 2 ), ( 0, 1 ), ( 1, 3 ), ( 2, -4 ), ( 4, -1 )\).

Would you like further details or have any questions about this analysis?

Here are 8 questions related to this topic:
1. What is the vertical line test and how is it used to determine functions?
2. Can a function have two identical y-values for different x-values?
3. What is the difference between a relation and a function?
4. How do you identify the domain and range of a function from a set of ordered pairs?
5. Why is the concept of a function important in mathematics?
6. Can a function have more than one output for a given input?
7. How can you visually confirm a function on a graph without using the vertical line test?
8. How do transformations affect the graph of a function?

**Tip:** Always remember that for a graph to represent a function, each input (x-value) must map to exactly one output (y-value).

Learn how to identify functions on a coordinate plane using the vertical line test. This analysis discusses different sets of ordered pairs and applies the vertical line test to determine which sets represent functions. Suitable for high school students studying functions and graphing.

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