# Adding one point to a plot

I want to add a point to a plot. The plot shows the solution to Solve for different values of a parameter (a). The point that I want to add is where the parameter is zero (a=0). I know that if a->0 then [Tau]1=0.485, but I don't know why the following doesn't show this point on the graph.

gamma4 = .4;
gamma3 = .284;
gamma2 = .265;
gamma1 = .484;

z3 = 10000;
z2 = 32000;
z1 = 75000;
zbar4 = 4800;
zbar3 = 20050;
zbar2 = 46200;
zbar1 = 138100;
h4 = .301;
h3 = .404;
h2 = .25;
h1 = .045;
p1 = 3.89;
p2 = 4.92;
p3 = 5.22;
p4 = 5.39;
g1 = 0.00001;
g2 = 0.44;
g3 = 0.88;
g4 = 1.76;
a =.;

eq1 = (\[Tau]1/(1 - \[Tau]1)) - ((1 -
g1 - (g2 a p2 + g3 a p3 + g4 a p4)) (zbar1 -
z1)/((1 - ((g1 a p1/\[Tau]1) + g2 a p2 + g3 a p3 +
g4 a p4)) zbar1 gamma1));
Print["0=", eq1];

sol[a_] := NSolve[{eq1 == 0  }, {\[Tau]1}, Reals][[1]]

Show[Plot[\[Tau]1 /. sol[a], {a, 0, 2},
AxesLabel -> (Style[#, 14, Bold] & /@ {a, \[Tau]1})],
Graphics[{PointSize[Medium], Blue, Point[{0, sol[0] }]}]]


Thanks,

Update:

gamma4 = .4; gamma3 = .284; gamma2 = .265; gamma1 = .484; z3 = 10000; \
z2 = 32000; z1 = 75000; zbar4 = 4800; zbar3 = 20050; zbar2 = 46200;
zbar1 = 138100; h4 = .301; h3 = .404; h2 = .25;
h1 = .045; p1 = 3.89; p2 = 4.92; p3 = 5.22;
p4 = 5.39; g1 = 0.77; g2 = 0.77; g3 = 0.77;
g4 = 1.54; a =.;

eq3 = (\[Tau]3/(1 - \[Tau]3)) - (((1 - g3 - g1 h1 a p1 - g2 h2 a p2 -
g3 (h3 - 1) a p3 - g4 h4 a p4) h3 (zbar3 -
z3) + ((1 - g2 - g1 h1 a p1 - g3 h3 a p3 -
g4 h4 a p4) h2 + (1 - g1 - g2 h2 a p2 - g3 h3 a p3 -
g4 h4 a p4) h1) (z2 - z3))/((1 - g1 h1 a p1 -
g2 h2 a p2 - g3 a p3 (h3 + ((1 - \[Tau]3)/\[Tau]3)) -
g4 h4 a p4) zbar3 h3 gamma3));
Print["0=", eq3];

daSol = Solve[{eq3 == 0}, {\[Tau]3}] // Flatten // Last;
sol[aa_] := daSol /. a -> aa
t3 = Show[
Plot[\[Tau]3 /. sol[a], {a, 0, 2},
AxesLabel -> (Style[#, 11] & /@ {"Abatement \n Cost",
"Second Bracket"}), PlotRange -> {0, 1}],
Graphics[{PointSize[Large], Blue, Point[{0, \[Tau]3 /. sol[0]}]}]]


The plot glitch is that your sol[0] is returning a rule (when it works) and not a scalar.

(Also for some reason your NSolve seemed a little temperamental, and since eq1 is a fairly straightforward equation, I just solve analytically. )

Just use \[Tau]1 /. sol[0] in your Graphics:

  daSol = Solve[eq1 == 0, \[Tau]1] // Flatten // Last;
sol[aa_] := daSol /. a -> aa

t3=Show[Plot[\[Tau]1 /. sol[a] // Evaluate, {a, 0, 2},
AxesLabel -> (Style[#, 14, Bold] & /@ {a, \[Tau]1}),
PlotRange -> All],  Graphics[{PointSize[Medium], Blue,
Point[{0, \[Tau]1 /. sol[0]}]}]]


Update As per your comment requesting a tick at $\tau_3$, to add tick marks at your favorite values on the "y" axis:

yTicksToAdd = {\[Tau]3 /. sol[0]};

Show[t3, bzz =
Ticks -> {First[Ticks /. AbsoluteOptions[t3]],
Join[Last[
Ticks /.
AbsoluteOptions[t3]], {#,
Style[#, {Red, 13}], {0.0375, 0.}, {Red,
AbsoluteThickness[1.5]}} & /@ (yTicksToAdd)]}]

• Thanks @John! I have updates the codes. Is there a way I can show the number too? I mean adding a tick for that point to the existing ticks on the vertical axis. This is suggested in the forum but doesn't work: Ticks->{Automatic,Join[Last[Ticks/.AbsoluteOptions[t3]],{{#,Style[#,\ Red,20],{0,-.02},Directive[Red,Thick]}&@{0,\[Tau]3/.sol[0]}}]} Commented Aug 28, 2017 at 19:00
• Some glitches in your comment code, updated my answer with something that works in V.11 for "y-axis" tick marks. Please let me know if it works for you. Commented Aug 28, 2017 at 20:33
• Yes, it worked, Thanks! Commented Aug 28, 2017 at 21:29