Explain why the equation is $y = f(x - p) +q$ for moving the original $f(x)$ by some vector $\vec{v} = (p,q)$

Question

I cannot intuitively understand why moving a graph of a function $f(x)$ has the following format when expressed ($p,q \in \mathbb{R}$):

$$ y = f(x - p) + q. $$

We obtain this via multiple observations. Firstly, we note that we can "move" a graph of a function up or down the $y$ axis by a vector $\vec{v} = (0,q)$. Hence the transformation:

$$ \mathcal{T}_y: T(x,y) \mapsto T'(x,y+q) $$

$T(x,y)$ being a point in the Euclidean plane. I can understand this intuitively. By adding a real number to a function, its $y$ coordinates increase, hence the formula when expressed states:

$$ y = f(x) +q $$

for transformation via $y$ axis.

But when we want to move our graph along $x$ axis, I get lost. Again, we assume $\vec{v} = (p,0)$. The transformation can be written as:

$$ \mathcal{T}_x: T(x,y) \mapsto T'(x+p,y). $$

All well here. But when expressing this transformation, my intuition fails. Here it is:

$$ y = f(x - p). $$

Why is there a a minus instead of a plus? What is the intuition behind it all? To add to the confusion, the full transformation for a vector $\vec{v} = (p,q)$ ca be written as:

$$ \mathcal{T}_{xy}: T(x,y) \mapsto T'(x+p,y+q). $$

Why can the transformation be done with pluses in this form, but not the other? Thank you in advance.

Answer 1

The linear transformation which maps $(x,y)=(0,0)$ to $(x',y')=(p,q)$ is $x'=x+p, y'=y+q$. Substituting $\,x=x'-p, y=y'-q\,$ into $\,y=f(x)\,$ then gives: $$y'-q=f(x'-p) \quad\iff\quad y'=f(x'-p)+q$$

Renaming the dummy variables $\,x',y'\,$ back to the more familiar $\,x,y\,$ gives: $\,y=f(x-p)+q\,$.

Answer 2

The apparent asymmetry is an artifact of an asymmetry in the way you’re representing the curve with an equation. Define instead $F:(x,y)\mapsto f(x)-y$ so that the curve is described by $F(x,y)=0$. Both $x$ and $y$ now transform the same way.

An informal way to look at this is in terms of output vs. input: generally speaking, outputs of a function transform differently from its inputs. When you rewrite the equation of the curve as above, both variables become inputs.