Can I Use Vector Notation for Graph Translations?

[R_Sarav]

Homework Statement

Hi. Its not really a problem but just a general question. When doing graph translations, such as move the parabola x units right or y units up etc, Is it okay to use vector format. So instead of saying move this equation 4 units left, could you write it like this -> <-4,0>

Homework Equations

I know this is used for translations of shapes but my main question is can it also be used for graphs

The Attempt at a Solution

I asked my teacher but he just shouted at me for asking him concepts he didnt talk about.

 

[fresh_42]

Homework Statement

Hi. Its not really a problem but just a general question. When doing graph translations, such as move the parabola x units right or y units up etc, Is it okay to use vector format. So instead of saying move this equation 4 units left, could you write it like this -> <-4,0>

Homework Equations

I know this is used for translations of shapes but my main question is can it also be used for graphs

The Attempt at a Solution

I asked my teacher but he just shouted at me for asking him concepts he didnt talk about.

A graph $\mathcal{G}$ of usually a function $x \mapsto f(x)$ is a set of points: $\mathcal{G}=\{\,(x,f(x)\,|\,x \in \operatorname{domain}(f)\,\}$. This is a subset of $\mathbb{R}^2$ and to move it means to move it within this plane $\mathbb{R}^2$. Thus you need a direction in which it is moved and a distance by which it is moved. But this exactly defines a vector $v=(v_x,v_y)$. So the moved object will be $v+\mathcal{G}=\{\,(x+v_x,f(x)+v_y)\,|\,x \in \operatorname{domain}(f)\,\}$. So in a sense this is even better than just to say move by $"v_0"$ in direction $"x"$ or $"y"$ since you allow more than two directions.

However, whether you should argue with your teacher is a complete different question. It's not always important to be right. Sometimes it's smarter not to be.

 

[R_Sarav]

A graph $\mathcal{G}$ of usually a function $x \mapsto f(x)$ is a set of points: $\mathcal{G}=\{\,(x,f(x)\,|\,x \in \operatorname{domain}(f)\,\}$. This is a subset of $\mathbb{R}^2$ and to move it means to move it within this plane $\mathbb{R}^2$. Thus you need a direction in which it is moved and a distance by which it is moved. But this exactly defines a vector $v=(v_x,v_y)$. So the moved object will be $v+\mathcal{G}=\{\,(x+v_x,f(x)+v_y\,|\,x \in \operatorname{domain}(f)\,\}$. So in a sense this is even better than just to say move by $"v_0"$ in direction $"x"$ or $"y"$ since you allow more than two directions.

However, whether you should argue with your teacher is a complete different question. It's not always important to be right. Sometimes it's smarter not to be.

So is it okay to write like this ? If i wrote like this In a paper would it be correct ?

 

[fresh_42]

So is it okay to write like this ? If i wrote like this In a paper would it be correct ?

I'm not quite sure what you mean by this. That's why I wrote the formulas, which are correct. 4 to the left would be $v_x=-4$ and $v_y=0$. It would be correct, if it would be o.k. depends on a lot of human factors. I wouldn't argue just to be right. 4 to the left is usually as good.

 

[Ray Vickson]

So is it okay to write like this ? If i wrote like this In a paper would it be correct ?

It is correct, but I suspect that if you wrote it like fresh_42 did your teacher would really yell at you.

If I were your teacher, what I would prefer to see are either of the following (assuming you are graphing a parabola of the form $y = a + b x^2$). If you want to move the whole graph $A$ units to the right (to the left if $A < 0$) and $B$ units up (down if $B < 0$), you could either say that (i) the vertex moves from $(a,0)$ to $(a+A, 0+B)$ but the shape remains unchanged; or (ii) explain that the new graph has equation $y-B = a + b(x-A)^2 \Rightarrow y = a+B+b(x-A)^2.$ Since you are plotting graphs in two-dimensional cartesian coordinate systems, I don't think you could be yelled at for doing it using method (i). However, my personal preference would be that you show your understanding by using method (ii).