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

00:02 Let's find out if this graph is a function.

00:06 A function has only one Y value for each X value. Let's see if this graph fits that rule.

00:13 We'll check the values. Look at the table to see if every X has just one Y.

00:33 Our table shows the graph's slope is zero. This means it's a function!

00:38 And that's how we know the graph is a function.

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

Is the given graph a 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!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Functions questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime