A little problem

[Starsailor]

Hello,

 

I wonder if any of you could explain the answer to a question that has left me rather stumped. It's a past question from the UKMT Senior Challenge and I know I should be able to do it, but I don't even understand what it wants me to do

Anyway, enough waffling...

 

"The graph of y= l f(x) l is shown ( f(x)= x^2). Given that the graph of y= f(x) is a continuous curve, how many different possibilities are there for the graph of y=f(x)?"

 

What does it want me to do?!

 

Cheers,

Michael

[Dave]

Do you mean "possibilities are there for the graph of y = |f(x)|"?

[Starsailor]

Do you mean "possibilities are there for the graph of y = |f(x)|"?

 

No; Perhaps it would be best to see the question as it is written.

[Dave]

Ah, I get it. It wants to know what f(x) could possibly look like so that the graph of |f(x)| looks the same.

 

For example, you could have f(x) look exactly like the graph. Or, you could have the first "lump" below the x-axis with the second above the x-axis. Or...

[jcarlson]

Think of it this way. Break the graph of |f(x)| into 4 segments, where the segments are divided by where the graph touches the x-axis. Now, for the graph of f(x), each of those segments can either look like it does for |f(x)|, or it can be the reflection of |f(x)| over the x-axis, because both will look the same when you plot |f(x)|. Thus f(x) has 2 possible curves for each of 4 segments which are independent of the other segments. now use probablility to find the number of possible graphs of f(x) or just count them.

[Starsailor]

Ah right. Cheers

 

Michael

[matt grime]

now use probablility

 

 

probability????

[Dave]

Perhaps combinatorics would be a slightly better choice of words