Solve the following parametric equation graphically: $$-x^2+2x=3a, a\in \mathbb{R}$$
To solve the equation graphically, we must draw the graph for each side, member of the equation. $$y_1=-x^2+2x$$ and $$y_2=3a$$
The vertex of $y_1=-x^2+2x$ is $V(1;1)$ and $G_1\cap Ox =\{O,A\}:O(0;0);A(2;0)$. Also we see $G_1 \cap Oy = O(0;0)$.
*$G_1$ is the graph of $y_1$
In my opinion, if $a=\dfrac{1}{3}$, the equation will have one root; if $a>\dfrac{1}{3}$, the equation won't have any roots; if $a<\dfrac{a}{3}$, the equation will have two roots.
Is there any algorithm to tell what to do after we analyze the first function? (in this case $y_1=-x^2+2x$)