I am not yet sure how to paste graph here in this site. so am just using equations to explain the question I have .
The graph $f(t) = 2t+1$ has a positive slope , and the initial value starts from constant $1$. In one of text book example it shows, if $t$ is subtracted with value $-2$, then the graph moves towards right. That is , $f(t-2) = 2t-3$.
Question:
When I plot this graph I get the graph moves down not right as explained below. Can you explain me if my understanding is correct ?
I have explained as below.
If plot graph for value $t=0,1,2,$ to function $f(t-2) = 2t-3$, then the function value will be
$f(-2) = -3$
$f(-1)= -1$
$f(0)= 1$.
Suppose $y=f(x)=ax+b$, as we have here. Then the graph of $y=f(x-k)$ is $$ y = f(x-k) = a(x-k)+b = ax + b-ak = f(x)-ak. $$ So in fact translating the graph right by $k$ looks the same as translating it down by $ak$. (This is only the case for $f$ of this form: $y=(x-k)^2$, for example, looks different from $y=x^2-\alpha k$ for any $\alpha$)