I've have got some vertical and horizontal distances for a projectile-like motion.
In order to work out the trajectory, why is it better to plot on the x-axis, "horizontal distance^2", and on the y axis, "vertical distance"?
The equations of motion, given the usual x and y axis choices, are $y=-0.5gt^2 +v_yt +y_0$ and $x=v_xt + x_0$. That is, the initial conditions are velocity $v_x$ or $v_y$, and position $y_0$ or $x_0$. The only difference is that gravity operates in the y direction.
Plotting y versus x gives a parabola.
Plotting y versus $x^2$ means you are plotting points of the form:
$(y,x) = (-0.5gt^2 + v_yt + y_0, v_x^2t^2 + 2v_xx_0t + x_0^2)$
For large $t$, these points will approach a straight line through the origin with slope $-0.5g/v_x^2$.
Not sure how this gives a great advantage in plotting. Maybe you are to assume that $x(0) = y(0) = 0$, which makes it a little simpler. Then you can get $v_y$ from the behavior near $(0,0)$.