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Alive

Member

Registered: 2010-08-14

Posts: 92

Transformations with Matrices

I have a test in two days, so please help me out with these matrice questions. Not much time left so please write up the way to the questions..

I've done everything up until 3... Please complete a few from question 3...

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All things can be forgiven if we can progress.
Yusuf Islam (formerly known as Cat Stevens)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Transformations with Matrices

hi Alive,

Hopefully you said this reflects the x coordinates in the y axis and doubles the y coordinates.  Is that what you call a dilation?  I would call it a stretch in the y direction, but it sounds like the same thing.

So what does that do to the curve

Well this curve is an 'even' function, meaning the y axis is a line of symmetry anyway so the -1 in the matrix has no effect.

The 2 doubles the y coordinates so the graph is stretched away from the x axis with a scale factor of x 2.

See diagrams below.

The curve is reflected in the y axis and stretched away from the x axis to double the distance.  Note: That means the crossing point of the curve on the y axis will change from (0,-5) to (0,-10) and the curve will go up more steeply.

Last edited by Bob (2011-03-13 05:01:27)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Alive

Member

Registered: 2010-08-14

Posts: 92

Re: Transformations with Matrices

These are the answers... I still don't know how to do 3)...

These are examples from the textbook

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All things can be forgiven if we can progress.
Yusuf Islam (formerly known as Cat Stevens)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Transformations with Matrices

hi Alive,

All your answers for 1 and 2 are correct, well done! 

All the matrices in these questions have top row second element and bottom row first element = zero.

That helps a lot because you can say what happens to x coordinates and y coordinates separately as you have done.

So for question 3 I did the following steps:

(i) What does the graph look like before it is transformed?

(ii)  How is it effected by the two independant changes to x and y coordinates?

(iii) Check with some numbers.

(i)  This graph is symmetrical in the y axis.
It zooms off to + infinity as x approaches zero from either side.
As x tends to infinity it drops lower and lower towards the x axis.
(Both axes are asymptotes.)

(ii) The -1 reflects the x coordinates in the y axis.  But the y axis is already a line of symmetry so the graph will look the same after this transformation.
The 2 doubles the y coordinates so for any point on the original graph just stretch the y coordinate to twice its original value.

(iii)  (1,1) becomes (-1, 2);    (-1,1) becomes (1, 2).  These points have swapped over x coordinates  but you wouldn't see this when you look at the finished result.

(10, 0.01) becomes (-10, 0.02) etc.

(i) This is a standard cubic shape but moved down 5 units so it goes through (0,-5)

(ii) The -1 reflects the graph so it goes from the second quadrant to the fourth.
The 2 stretches it from the horizontal axis.  So it goes up more steeply. 
The point (0,-5) is stretched to (0,-10)

(iii) (1,-4) becomes (-1,-8)  (2, 3) becomes (-2, 6) and so on.

The book examples do this algebraically but I prefer to think about the graph shape and check with numbers.  It works so if this makes it easier, do the same.

Hope that helps,

Bob

Last edited by Bob (2011-03-13 21:59:59)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Alive

Member

Registered: 2010-08-14

Posts: 92

Re: Transformations with Matrices

Could you do it algebraically please.

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All things can be forgiven if we can progress.
Yusuf Islam (formerly known as Cat Stevens)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Transformations with Matrices

hi Alive

I'll have a go.

so

Now that is true whatever curve you transform so if

and for

Bob

Last edited by Bob (2011-03-14 01:31:18)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob