How to sketch the graph of $\varphi(X)=d(X,A)+d(X,B)$ when $A$ and $B$ are not given? 
$A,B$ are points in an axis, disposed in this order. Sketch the graph
  of the following function:
$$\varphi(X)=d(X,A)+d(X,B)$$

$d(A,B)$ is the distance from point $A$ to point $B$.
I'm confused with this question: The values of $A$ and $B$ are not given, I had some guesses but I feel they are not functional, for example: Presuming any arbitrary values for $A$  and $B$ such that $A<B$, for example $A=2$ and $B=3$, then I would have:
$$\varphi(1)=d(1,2)+d(1,3)$$
$$\varphi(2)=d(2,2)+d(2,3)$$
$$...$$
And then presuming new arbitrary values under the given condition:
$$\varphi(1)=d(1,3)+d(1,4)$$
$$\varphi(2)=d(2,3)+d(2,4)$$
$$...$$
But this seems to be really awkward, if I were to find values for $A$ and $B$ in $\mathbb{R}$ for this graph, I would be screwed (I guess).
I feel this isn't the purpose of the exercise, what should I do?
 A: Mark points $A$ and $B$ on the $x$-axis, with $A<B$. Let $y_0=d(A,B)$. Clearly $\varphi(A)=\varphi(B)=y_0$, so you now have two points on your graph: $\langle A,y_0\rangle$, and $\langle B,y_0\rangle$. Now try to answer the following questions in terms of $y_0$ and $X$.


*

*What is $\varphi(X)$ if $A<X<B$?  

*What is $\varphi(X)$ if $X<A$?  

*What is $\varphi(X)$ if $X>B$?


If you answer these correctly, you can sketch in the graph of $\varphi$. It has the same basic shape no matter what $A$ and $B$ are.
Added: Now that you’ve worked out most of it, here’s a rough sketch of the graph:
         \                                    /  
          \                                  /  
           \                                /  
            \                              /  
             \____________________________/  









_________________________________________________________________  
             A                            B

The line at the bottom is the $x$-axis. The horizontal segment of the graph is at height $y_0=B-A$. The slopes of the other two parts of the graph are $-2$ and $2$.
