Transformation Matrices

In summary, the conversation is about a person preparing for an AQA exam in Pure Maths MPC2 and asking for help with a problem on past papers. The problem involves finding a single geometrical transformation that maps the graph of y=6^x onto the graph of y=6^(3x) and also translating the graph of y=6^x by a matrix [1; -2] to give the graph of the curve with equation y=f(x). After some thought and confusion, it is determined that the transformation involves translating each point on the graph one unit right and two units down, and the use of transformation matrices is not necessary to solve the problem.
  • #1
22
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Hi

I'm sitting for an AQA exam tomorrow (Pure Maths MPC2) and while going through some past papers I encountered this problem which I'm not sure how to solve. I'd appreciate any help :)

Homework Statement


i) Describe a single geometrical transformation that maps the graph of y=6^x onto the graph of y=6^(3x).

ii) The graph of y=6^x is translated by the matrix [1; -2] (it is a 2x1 matrix) to give the graph of the curve with equation y=f(x) . Write down an expression for f(x).
 
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  • #2
If you are taking an exam in these, surely you must know something! What have you tried?
 
  • #3
well, if i were doing the exam right now, for i) i'd say it would compress the x-axis
however I'm pretty uncertain about (ii)
 
  • #4
while thinking about the second part I came up with two possible solutions which both seem right and ended up getting more confused

you shift every (x,y) points by the vector [1 -2] (ie move one to the right and down two)
or
use the matrix transformation equation and end up with y=x-2*6^x
 
  • #5
exis said:
ii) The graph of y=6^x is translated by the matrix [1; -2] (it is a 2x1 matrix) to give the graph of the curve with equation y=f(x) . Write down an expression for f(x).

After some thought, I think what this means is that each point on the graph of y = 6^x is translated one unit right and two units down. The description is a bit confusing in its description of the graph being translated by a matrix. Although [1 -2]^T is indeed a matrix, it might have been clearer to describe this as a translation by an amount represented by the given vector.
 
  • #6
Ok. I completely got it now. Thanks a lot for your help
 
  • #7
And BTW, this has nothing to do with transformation matrices, which you used as the title for this thread.
 
  • #8
Sorry about that. At the time I posted the problem I still thought that I needed to use the matrix transformation equation to solve it. The wording of the question confused me
 

1. What is a transformation matrix?

A transformation matrix is a mathematical representation of a transformation that is applied to a geometric shape, such as a translation, rotation, or scaling. It is a square matrix that contains the coefficients of the transformation equations.

2. How is a transformation matrix used?

A transformation matrix is used to perform a transformation on a set of points or vectors in a coordinate system. It can be multiplied by a vector or a set of coordinates to produce the transformed coordinates of the points.

3. How do you create a transformation matrix?

To create a transformation matrix, you need to know the type of transformation you want to perform (translation, rotation, or scaling) and the parameters of the transformation. These parameters are then used to construct the appropriate transformation matrix using the rules and equations for that type of transformation.

4. Can a transformation matrix be combined with other matrices?

Yes, transformation matrices can be multiplied together to combine multiple transformations. The resulting matrix will represent the combined transformation of the individual matrices.

5. What is the inverse of a transformation matrix?

The inverse of a transformation matrix is another matrix that, when multiplied by the original matrix, produces the identity matrix. It represents the opposite transformation of the original matrix, effectively "undoing" the transformation.

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