Is This Graph a Function? Exploring with the Vertical Line Test

Q: Why do we use vertical lines instead of horizontal lines?

The definition of a function requires each input (x-value) to have exactly one output (y-value). Vertical lines represent constant x-values, so they test this one-to-one input-output relationship!

Q: What if the graph is just a horizontal line like this one?

A horizontal line always passes the vertical line test! Every x-value corresponds to the same y-value, which is perfectly fine for a function. It's called a constant function.

Q: Can a function have curves or just straight lines?

Functions can be any shape - straight lines, curves, zigzags, or even disconnected pieces! The only requirement is passing the vertical line test.

Q: What would make a graph NOT a function?

If any vertical line intersects the graph at two or more points, it's not a function. Examples include circles, sideways parabolas, or any graph that 'loops back' on itself.

Q: How do I imagine the vertical lines when taking the test?

Picture drawing infinitely many vertical lines across the entire graph, like the bars of a fence. If any 'fence bar' crosses the graph more than once, it fails the test!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the given graph a function?

00:04 The definition of a function is that for each X value there is one Y value

00:08 Let's check the values and see if the graph meets the definition

00:31 According to our table, we can see that the graph has a slope of 0

00:36 This is the expression of a function

00:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Is the given graph a function?

2

Step-by-step solution

To determine if the graph is a function, we will use the Vertical Line Test.

The Vertical Line Test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

Let's apply this test to the given graph, where a horizontal line is drawn. This line represents the function the graph should be verified against.

Step 1: Conceptualize vertical lines passing through different x-values across the domain of the graph.
Step 2: Observe if any of these vertical lines intersect the graph at more than one point.

Upon inspection of the graph, we see that every vertical line intersects the graph at exactly one point.

This indicates that for every input (x-value), there is a unique output (y-value), fulfilling the criteria for the definition of a function.

Therefore, according to the Vertical Line Test, the given graph is indeed a function.

The correct choice is: Yes

3

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Vertical Line Test: Each vertical line must intersect graph at most once
Technique: Draw imaginary vertical lines across entire domain of graph
Check: Every x-value has exactly one y-value on this horizontal line ✓

Common Mistakes

Avoid these frequent errors

Confusing horizontal and vertical line tests
Don't use horizontal lines to test if something is a function = wrong test! The horizontal line test checks if a function is one-to-one, not if a graph IS a function. Always use vertical lines to determine if a graph represents a function.

Practice Quiz

Test your knowledge with interactive questions

Determine whether the following table represents a constant function:

FAQ

Everything you need to know about this question

Why do we use vertical lines instead of horizontal lines?

+

The definition of a function requires each input (x-value) to have exactly one output (y-value). Vertical lines represent constant x-values, so they test this one-to-one input-output relationship!

What if the graph is just a horizontal line like this one?

+

A horizontal line always passes the vertical line test! Every x-value corresponds to the same y-value, which is perfectly fine for a function. It's called a constant function.

Can a function have curves or just straight lines?

+

Functions can be any shape - straight lines, curves, zigzags, or even disconnected pieces! The only requirement is passing the vertical line test.

What would make a graph NOT a function?

+

If any vertical line intersects the graph at two or more points, it's not a function. Examples include circles, sideways parabolas, or any graph that 'loops back' on itself.

How do I imagine the vertical lines when taking the test?

+

Picture drawing infinitely many vertical lines across the entire graph, like the bars of a fence. If any 'fence bar' crosses the graph more than once, it fails the test!

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