Understanding Vector Transformation with <1,-1> Translation - Explained

In summary, "Translation" in general means to add a vector. When applying the vector <-1,-1> to translate f(x) to h(x), we want to find a function h(x) that has the same graph as f(x) but is translated by <-1,-1>. To do this, we can write h(x) as g(x)= f(x+1) and then subtract 1 from the result to get it in the desired form of ax^3+bx^2+cx+d.
  • #1
kvzrock
3
0
what does it mean by applying vector <-1,-1> to translate f(x) to h(x)?
 
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  • #2
"Translation" in general means to add a vector. Since you haven't told us what "f(x)" and "h(x)" represent I can't be more specific but if they are vector valued functions, then I would suspect you are to add the vector <-1,-1> to f(x).
 
  • #3
f(x)=x^3-3x^2
h(x) should be in the form of ax^3+bx^2+cx+d

edit: problem solved. Thanks.
 
Last edited:
  • #4
I suspect that you are intended to find a function whose graph looks exactly line y= f(x) but is translated (one translates geometric things- points and sets of points like graphs- not functions) by <-1,-1>. In particular, that means that, since f(0)= 0, we want g(-1)= -1. First, we want x= -1, in g, to "act like" x=0 in f. That is
g(x)= f(u) for some u so that when x= -1, u= 0: okay the simplest possible thing is u= x+1. If we write g(x)= f(x+1)= (x+1)3- 3(x+1)2, we have g(-1)= f(0) but we are not quite done: g(-1)= f(0)= 0 and we want g(-1)= f(0)-1. Fine: just subtract 1 from what we just got:

g(x)= (x+1)3- 3(x+1)2- 1.

In order to get it in the form "ax3+ b2+ cx+ d", you will need to multiply it out.
 

1. What is vector transformation?

Vector transformation is the process of changing the coordinates of a vector from one coordinate system to another. This is done by multiplying the vector by a transformation matrix, which represents the rotation, scaling, and shearing of the vector in the new coordinate system.

2. Why is vector transformation important?

Vector transformation is important because it allows us to represent the same vector in different coordinate systems, making it easier to analyze and manipulate. It also helps in solving problems in fields such as physics, engineering, and computer graphics.

3. What is the difference between vector transformation and vector translation?

Vector transformation involves changing the coordinates of a vector, while vector translation involves moving a vector from one point to another without changing its direction or magnitude. In other words, vector transformation changes the vector's reference frame, while vector translation changes its position in the same reference frame.

4. How is vector transformation related to linear algebra?

Vector transformation is a fundamental concept in linear algebra. It involves using matrices to represent linear transformations, which are used to change the coordinates of vectors. This is important in solving systems of linear equations, finding eigenvalues and eigenvectors, and performing other operations in linear algebra.

5. Can vector transformation be applied to higher dimensions?

Yes, vector transformation can be applied to any number of dimensions. In fact, it is often used in three-dimensional and higher-dimensional spaces to represent rotations, translations, and other transformations. The process is the same as in two dimensions, but the transformation matrix will have more rows and columns to account for the additional dimensions.

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