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#1 2006-05-02 05:31:13

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Curve Sketching

I am having a little trouble with scaling curves, for example:

y = -2x^3

I know that;
y = af(x) is scaled in the y direction by a
y = f(ax) is scaled in the x direction by 1/a

So how can I tell if y = -2x^3 is y = -2f(x) or f(-2x)???


Aloha Nui means Goodbye.

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#2 2006-05-02 08:57:39

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Curve Sketching

y = -2(x)^3 or y = (-2x)^3?

y = -2(x)^3:

Draw the sketch x^3, then flip it over the y axis.  Now stretch it upwards by a factor of two.  Plotting some points also helps.   For example, plot x = -2, 1, 0, 1, 2, then connect the dots keeping in mind it should look something like x^3.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2006-05-02 11:14:51

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Curve Sketching

I know that the curve is exactly the same but if you are asked to describe the transformation, should I say that it's scaled in the y-direction by a factor -2 or in the x-direction by factor -1/2?


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#4 2006-05-02 11:41:23

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Curve Sketching

Separate the - from the 2.  You flip because of the negative, you strech/shirk because of the 2.

But remember that streching the y by a factor of 2 is exactly the same as shrinking the x by a factor of 2, which is exactly the same as streching the x by a factor of 1/2.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#5 2006-05-03 00:12:18

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Curve Sketching

So I wont have any questions in the format y = -2x^3 asking in which axis I need to scale?

FYI.  The correct term for streching or shrinking the curve is to "scale" it by a "factor" smile


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#6 2006-05-03 00:23:12

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Curve Sketching

Right, but "scale" it by a "factor" may not make sense to someone who doesn't know that's the correct term for streching/shrinking wink

And you probably will have questions like this, but there are just multiple correct answers.  Just like if I asked you to give me two numbers when added together that give me 9.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#7 2006-05-03 00:43:29

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Curve Sketching

Ah good point!!! I'll write both answers smile


Aloha Nui means Goodbye.

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#8 2006-05-03 02:58:54

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: Curve Sketching

Compare y=f(ax[sub]2[/sub]) and y=f(x)
when the two y equals, usually the value in the bracket equals.
ax[sub]2[/sub]=x
thus x[sub]2[/sub]=x/a


X'(y-Xβ)=0

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#9 2006-05-03 04:14:07

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Curve Sketching

-2x² - 7x + 15
-2[x² + 7/2x] + 15
-2[(x + 7/4)² - 49/16] + 15
-2(x + 7/4)² + 18 + 1/16

Sketch the curve of -2x² - 7x + 15:
2(x + 7/4)² = 18 + 1/16
(x + 7/4)² = 9 + 1/32
x + 7/4 = ±√9 + 1/32
x = - 7/4 ±√9 + 1/32

So the curve would cross the x-axis at -4.76 and +1.26 which, as you already know, is wrong!.


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#10 2006-05-04 15:52:02

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: Curve Sketching

Yes, but a trick will solve the controversy
devide the original curve or function into such pieces  that in each piece x and y are one by one, use my procedure seperately and then stick the pieces by original sequence.


X'(y-Xβ)=0

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