Vertical Line Test: Determining if an Elliptical Graph Represents a Function

Vertical Line Test with Elliptical Graphs

Is the given graph a function?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Is the given graph a function?
00:05 The definition of a function is that for each X value there is one Y value
00:10 It seems that for an X value there are several Y values, therefore not a function
00:15 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?

–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777–4–4–4–3–3–3–2–2–2–1–1–1111222333000

2

Step-by-step solution

It is important to remember that a function is an equation that assigns to each element in domain X one and only one element in range Y

Let's note that in the graph:

f(0)=2,f(0)=2 f(0)=2,f(0)=-2

In other words, there are two values for the same number.

Therefore, the graph is not a function.

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic
  • Definition: Function assigns exactly one output for each input value
  • Technique: Use vertical line test - if line intersects graph twice, not function
  • Check: At x = 0, ellipse has two y-values (2 and -2) ✓

Common Mistakes

Avoid these frequent errors
  • Confusing equations with functions
    Don't assume every graph represents a function = wrong classification! Just because you can draw or write an equation doesn't mean it passes the function test. Always apply the vertical line test to determine if it's truly a function.

Practice Quiz

Test your knowledge with interactive questions

Determine whether the following table represents a constant function:

XY02468-3-3-3-3-3

FAQ

Everything you need to know about this question

What exactly is the vertical line test?

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The vertical line test is a visual way to check if a graph represents a function. Draw or imagine any vertical line through the graph. If it touches the graph at more than one point, it's not a function.

Why does this ellipse fail the vertical line test?

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An ellipse is symmetric around both axes, so most vertical lines through it will intersect two points. For example, at x = 0, we get both y = 2 and y = -2, violating the function rule.

Are circles also not functions?

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Correct! Circles fail the vertical line test just like ellipses. Any vertical line through a circle (except at the very edges) intersects the circle at two points.

Can part of an ellipse be a function?

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Yes! If you take just the top half or bottom half of an ellipse, it would pass the vertical line test and represent a function.

How is this different from a parabola?

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A vertical parabola (like y=x2 y = x^2 ) passes the vertical line test because each x-value gives only one y-value. But a horizontal parabola would fail, just like this ellipse!

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