Deriving the "third" kinematic equation using graph

[brotherbobby]

TL;DR

The three equations describing one-dimensional kinematics are well known from high school. The third connects velocity to displacement : ${\boldsymbol{(3)}}\;\boxed{v^2(x)=v^2_0+2a_0(x-x_0)}$, where $v_0,\, x_0$ are the velocity and position of the body with respect to the origin at time $t=0$ and $a_0$ is the (constant) acceleration. (3) can be derived using both algebra and calculus. Question is - how to derive it using graph? (!)

Problem Statement : For a particle moving in one dimension along the $x$ axis with constant acceleration $a_0$ (see figure alongside), the three well known equations of high school kinematics are $(1)\; v_(t)=v_0+a_0t\quad (2)\; x(t)=x_0+v_0t+\dfrac{1}{2}a_0t^2\quad \text{and}\quad\boxed{(3)\; v^2(x) = v^2_0+2a_0(x-x_0)}$. These equations can be derived using the methods of (1) algebra and (2) calculus - and I have been able to do it [I would be happy to provide the details if asked]. Additionally, there's a third method, the method of graphs. I have been able to derive (1) and (2) using the graph depicting $v(t)-t$, its slope and the area under the straight line lead to (1) and (2). [I would be happy to provide the details if asked].

I am stuck with (3). Knowing it already (or beforehand), I can see that the graph of $v(x)\;\text{versus}\; x$ will be a parabola that "curves" with a decreasing slope $-$ an attempt of which I have made in the graph alongside. But that doesn't help. It is "how" to derive the equation using graphs where I am at a loss. More specifically, which graph to use? It cannot be the one I have shown, for it was sketched using the equation that I seek to derive. I am left with $v\;\text{versus}\; t$ graph which is a straight line, or the $x\;\text{versus}\; t$ graph which is parabolic. For reference, I put both graphs to the left.

Can either of the two graphs above be used to derive (3), namely : $v^2(x)=v^2_0+2a_0(x-x_0)$?

Many thanks.

 

[Ibix]

I suspect all the same questions from your previous similar thread will arise here too. Basically, what do you regard as a "using a graph"? Because some algebra is inevitable.

 

[brotherbobby]

Yes, am afraid I entirely forgot about that thread.
True, you'd need some algebra but that can be readily done with equations (1) and (2). With (3), none seems to be possible.