Why is elasticity not constant on a straight line graph?
Theoretically, the responsiveness of quantity demanded is different at different prices. Take the equation for the elasticity of demand:
$$\epsilon_{D}=\frac{\Delta{Q}}{\Delta{P}}\frac{P}{Q}$$
The equation itself is non-constant as it depends on P and Q. Lets assume a linear demand curve with the simple form: $P(Q)=6-Q$. If we take an equal change in price, say $\Delta{P}=1$, and since the demand curve is linear in this case, we see it has a slope = $-1$ and thus the $\Delta{Q}=1$. If we have two situations where $P=5$ and $P=4$, then, to keep in accordance with our slope, lets say $Q=1$ and $Q=2$, respectively.
The two elasticities of demand for these two prices would be $\epsilon_{D}=5$ and $\epsilon_{D}=2$ for the higher and lower price respectively.

In short, the elasticity of demand is a function of the responsiveness of quantity demanded to changes in prices that depends on the current price/quantity combination.
At a higher point up the demand curve, the elasticity is higher, or demand more elastic, than at lower points.