How does one consider what a graph looks like in a mathematical proof

Question

Mostly I am wondering for example what it would be like to prove that a linear graph (negative slope) shifted right would look the same as one shifted up.

Can you consider how a graph looks when doing a proof?

Answer 1

Let $L$ be the line $y=-mx+c$, $m>0$.

The line shifted right by $a$ has the equation $$\begin{align*} y &= -m(x-a)+c\\ y &= -mx +c+ma\\ y-ma &= -mx+c \end{align*}$$

Since $m>0$ and $a>0$, $ma>0$, so the graph is equivalent to $L$ shifted up by $ma$.