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I SERIOUSLY don't understand increasing/decreasing functions!?

So for instance, in a quadratic equation, if I'm looking for the x values in an increasing function why should I include the ones below 0??
For Eg. Why should include x<-4 if I'm looking for the x values in an increasing function? Isn't -4 below zero?? and also how do I know which direction to shade the graphs for both increasing and decreasing function?

(edited 8 years ago)

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Original post by FluffyCherry
So for instance, in a quadratic equation, if I'm looking for the x values in an increasing function why should I include the ones below 0??
For Eg. Why should include x<-2 if I'm looking for the x values in an increasing function? Isn't -2 below zero?? and also how do I know which direction to shade the graphs for both increasing and decreasing function?


Could you please post an example of a question that you're finding difficult? :smile:
Original post by SeanFM
Could you please post an example of a question that you're finding difficult? :smile:


Okay, so we're told that this function f(x)=x^2+x-6 is an increasing function and we had to find the x values but the highlighted part is decreasing, and x < -3 is below 0 so why is that value included?

and also, how am I supposed to now which side to shade?
(edited 8 years ago)
hello.PNG41.6KB
Original post by FluffyCherry
Okay, so we're told that this function f(x)=x^2+x-6 is an increasing function and we had to find the x values but the highlighted part is decreasing, and x < -3 is below 0 so why is that value included?

x


Thank you.

I would argue that the function is decreasing until the minimum point, after which it is increasing. Using that increasing means if x>y, then f(x)>f(y) which is clearly not true between x = -2 and x = -3 for example.

Are you sure that those values are included?
Original post by SeanFM
Thank you.

I would argue that the function is decreasing until the minimum point, after which it is increasing. Using that increasing means if x>y, then f(x)>f(y) which is clearly not true between x = -2 and x = -3 for example.

Are you sure that those values are included?


I got this from the internet just to demonstrate you my problem with 'increasing function' so I wasn't aware if it was actually wrong or right,:K: so here's one



Why do I need to include x<-4 ? increasing function is when x >0 ?!
hh.PNG76.2KB
Original post by FluffyCherry
I got this from the internet just to demonstrate you my problem with 'increasing function' so I wasn't aware if it was actually wrong or right,:K: so here's one



Why do I need to include x<-4 ? increasing function is when x >0 ?!


Ohh. This is rather different to just a graph of f(x).

Remember f'(x) is the gradient. When the gradient is positive the function is increasing.

So that graph is the graph of the gradient, and so when it is >0 then those values of x are when it's an increasing function.
Original post by SeanFM
Ohh. This is rather different to just a graph of f(x).

Remember f'(x) is the gradient. When the gradient is positive the function is increasing.

So that graph is the graph of the gradient, and so when it is >0 then those values of x are when it's an increasing function.


Exactly. The gradient of the graph is increasing (positive) when x < -4 and x > 2/3. Looking at the graph is proof of this because the graph is above 0 for those values of x onwards.

The graph would be decreasing when -4<x<2/3 since the graph is below 0 at that point.

The fact that -4 is a negative number has nothing to do with the fact that the function is increasing. You need to look at the gradient as that decides whether a function is increasing or decreasing. The x values just tell you at what point on the graph the gradient is increasing/decreasing.
Original post by SeanFM
Ohh. This is rather different to just a graph of f(x).

Remember f'(x) is the gradient. When the gradient is positive the function is increasing.

So that graph is the graph of the gradient, and so when it is >0 then those values of x are when it's an increasing function.


but why don't we include the values between -4 and 2/3 ? are the values >0?
Original post by FluffyCherry
but why don't we include the values between -4 and 2/3 ? are the values >0?


When dy/dx is positive the function is increasing. When negative, decreasing. Can you see why? :smile:

Then look at the graph of dy/dx and answer your own question :biggrin:
Original post by SeanFM
When dy/dx is positive the function is increasing. When negative, decreasing. Can you see why? :smile:

Then look at the graph of dy/dx and answer your own question :biggrin:


Original post by mollyxrose
Exactly. The gradient of the graph is increasing (positive) when x < -4 and x > 2/3. Looking at the graph is proof of this because the graph is above 0 for those values of x onwards.

The graph would be decreasing when -4<x<2/3 since the graph is below 0 at that point.

The fact that -4 is a negative number has nothing to do with the fact that the function is increasing. You need to look at the gradient as that decides whether a function is increasing or decreasing. The x values just tell you at what point on the graph the gradient is increasing/decreasing.


So is this graph a decreasing function

and this is an increasing function
Both functions have increasing and decreasing sections. When it slopes upwards (positive gradient) it is increasing, when it slopes downwards (negative gradient) it is decreasing.

Your confusion is because the graphs you have shown are the graphs of the derivative - i.e the gradient. To make this clearer you should draw the graph of the original equation, to the same scale, and compare it to that of the gradient. You'll find that where the original graph slopes upwards, the gradient graph will be positive, and where the original graph slopes downwards, the gradient graph will be negative.

In the example you posted the original graph is: y = x^3 + 5x^2 - 8x +1, and the gradient graph is: y = 3x^2 + 10x -8
(edited 8 years ago)
Original post by FluffyCherry
So is this graph a decreasing function

and this is an increasing function


If that's a graph of y then when for the first one half of it is increasing, (left half) then rest decreasing. the second one, the left half is decreasing (by half I mean where there are actually lines)
Original post by FluffyCherry
So is this graph a decreasing function

and this is an increasing function


No, they are simply negative and positive graphs. They are both increasing and decreasing at certain points.
Looking at them, I would say the first graph is increasing (sloping upwards) until the point (2,4) and decreasing (sloping downwards) after that.
The second graph is decreasing until (2,-4) and increases after that point.

However, in an exam the only way you could prove that is by calculating the gradient function (derivative) and plotting the graph of that.

I'll send a photo of an example I've done to try and help you understand :smile:
Original post by mollyxrose
No, they are simply negative and positive graphs. They are both increasing and decreasing at certain points.
Looking at them, I would say the first graph is increasing (sloping upwards) until the point (2,4) and decreasing (sloping downwards) after that.
The second graph is decreasing until (2,-4) and increases after that point.

However, in an exam the only way you could prove that is by calculating the gradient function (derivative) and plotting the graph of that.

I'll send a photo of an example I've done to try and help you understand :smile:


Original post by SeanFM
If that's a graph of y then when for the first one half of it is increasing, (left half) then rest decreasing. the second one, the left half is decreasing (by half I mean where there are actually lines)


so I did these, are they correct now?
olo.png14.3KB
Original post by FluffyCherry
so I did these, are they correct now?


If those are the graphs of dy/dx then yes. If graphs of y, then no.
Original post by SeanFM
If those are the graphs of dy/dx then yes. If graphs of y, then no.


So increasing/decreasing functions are only applied to graphs of dy/dx
functions1.jpg
Attachment not found


This might help you understand further :smile:
functions2.jpg496.1KB
Original post by FluffyCherry
So increasing/decreasing functions are only applied to graphs of dy/dx


No, but in terms of what you are asking and what you are shading, it's only correct for graphs of dy/dx. Did you try/read what was said in my previous post...?
Original post by FluffyCherry
So increasing/decreasing functions are only applied to graphs of dy/dx


I'll leave to let Molly explain it so that you don't have two people saying things. :smile:
Original post by mollyxrose
functions1.jpg
Attachment not found


This might help you understand further :smile:
Original post by Speckle
No, but in terms of what you are asking and what you are shading, it's only correct for graphs of dy/dx. Did you try/read what was said in my previous post...?


AAh, sorry I missed it!

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