Weird Graph Behaviour - Analyzing a Root in Domain 2-4

In summary, the conversation discusses a weird graph and a function with a root at y=0 for the domain x=[2,4]. There is a discussion about how to determine this root and it is revealed that the root occurs when the expression under the radical is the square of (x2-6x+8) and is simply the absolute value of (x2-6x+8). The conversation ends with appreciation for this new trick.
  • #1
Mentallic
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I came across this weird graph from another thread:

[tex]y=\left(x^2-6x+8+\sqrt{x^4-12x^3+52x^2-96x+64}\right)^2[/tex]

It seems that there is a root y=0 for the domain x=[2,4].
I've never seen such weird behaviour on a graph before. How could analysis of this function (or not?) determine that there is a root for all values [itex]2\leq x\leq 4[/itex]?
 
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  • #2
That does seem weird, but I have figured out what is going on.

The expression under the radical is the square of (x2-6x+8). So that expression (including the square root) is simply the absolute value of (x2-6x+8).

They cancel whenever (x2-6x+8) is negative, which happens for 2<x<4.
 
  • #3
Nice one Redbelly :smile:
I like this new little trick, it seems so nifty hehehe

Thanks.
 

1. What is a root in domain 2-4?

A root in domain 2-4 refers to a point on a graph where the function's output, or y-value, is equal to zero. This means that the input, or x-value, of the function at this point is a solution to the equation f(x)=0. In other words, the root in domain 2-4 is where the graph crosses the x-axis between x=2 and x=4.

2. Why is it important to analyze weird graph behavior?

Analyzing weird graph behavior can provide insight into the underlying function or equation being graphed. It can also help identify errors or inconsistencies in data or calculations. By understanding the causes of weird graph behavior, scientists can make more accurate and meaningful conclusions from their data.

3. What are some common causes of weird graph behavior?

Some common causes of weird graph behavior include mathematical errors, measurement errors, outliers in data, and incorrect assumptions about the behavior of the underlying function. Other factors such as experimental conditions, equipment limitations, and human error can also contribute to weird graph behavior.

4. How can I determine if a root in domain 2-4 is a valid solution?

To determine if a root in domain 2-4 is a valid solution, you can plug in the x-value of the root into the original function and see if the output is equal to zero. If it is, then the root is a valid solution. Additionally, you can use a graphing calculator or software to plot the function and visually confirm the root.

5. Can weird graph behavior be fixed or corrected?

In some cases, weird graph behavior can be fixed or corrected by identifying and addressing the underlying cause. For example, if the weird behavior is due to measurement errors, repeating the experiment or using more precise equipment may result in a more accurate graph. However, in other cases, the weird graph behavior may be inherent to the data or function and cannot be changed.

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