Call out two specific points on a curve drawn with gnuplot - gnuplot

Question

I am plotting a curve in gnuplot. I would like to call out a couple of specific points on the curve.
This is what the curve looks like:
Shown below is what I'm hoping to get (but I had to draw in the points and lines to the axes with GIMP). I'd like to call out two specific points on the curve. The first, where Vgs (x-axis) equals -0.5. The second, where Id (y-axis) equals 2.5. If possible, I'd like to also have a dashed line over to the axis to aid in reading the values.
I found a reference that said to try plotting a circle using set object circle and the coordinates, but I must not have it right, because it complains about extra parameters. Browsing the manual is so far unsuccessful. I'm not even sure what term to search for.
Is there an easy way to call out a couple of points on the curve that I've drawn and have it look similar to the screenshot below?
Here are the commands I used to plot the curve:
set xlabel "Vgs (Volts)"
set ylabel "Id (mA)"
set grid
Idss=5
Vgs_off=-1
Id(Vgs) = Idss * (1 - (Vgs / Vgs_off) ) ** 2
plot [Vgs=Vgs_off:0][0:Idss + 1] Id(Vgs)
It's the characteristic curve for a Fairchild J112 JFET if anyone is curious.
Edited to replace the -1 in the denominator with Vgs_off to avoid confusion. This is a value from the JFET's datasheet and just happened to be -1 in this particular case.
ADDENDUM:
After incorporating the pieces from the answer given by #Eldrad, I came up with this much improved representation of the JFET characteristic curve:
Here are the commands used to create it:
# JFET parameters from data sheet
Vgs_off=-1
Idss=5
# JFET characteristic curve
Id(Vgs) = Idss * (1 - (Vgs / Vgs_off) ) ** 2
# Graph properties
set title "I_D vs V_{GS}"
set xlabel "V_{GS} (Volts)"
set ylabel "I_D (mA)"
set grid
set key off
set monochrome
# Plot the characteristic curve
plot [Vgs=Vgs_off:0][0:Idss] Id(Vgs)
# Plot interesting points
set object circle center 0.5 * Vgs_off, Id(0.5 * Vgs_off) radius char 0.33 fillstyle solid fillcolor rgb 'black'
set object circle center 0.293 * Vgs_off, Id(0.293 * Vgs_off) radius char 0.33 fillstyle solid fillcolor rgb 'black'
# Mark interesting points with dashed lines to where they intersect the x and y axes.
set arrow from first 0.5 * Vgs_off, graph 0 to first 0.5 * Vgs_off, graph 1 dashtype "-" nohead
set arrow from graph 0, first Id(0.5 * Vgs_off) to graph 1, first Id(0.5 * Vgs_off) dashtype "-" nohead
set arrow from 0.293 * Vgs_off, graph 0 to 0.293 * Vgs_off, graph 1 dashtype "_" nohead
set arrow from graph 0, first Id(0.293 * Vgs_off) to graph 1, first Id(0.293 * Vgs_off) dashtype "_" nohead
# Label the lines
set label "I_{DSS}" at graph 1.01, graph 1
set label "I_{DSS} / 2" at graph 1.01, graph 0.50
set label "I_{DSS} / 4" at graph 1.01, graph 0.25
set rmargin at screen 0.9
# Update the graph
replot

Answer 1

Getting specific values of a function is straightforward if the function y=f(x) and its inverse x=f(y) have an analytical expression, like it is in your case.
A couple of remarks about your function definition: You should use x as the variable, because this is the default gnuplot is looking for, it also makes the plot command shorter. Also, division by -1 is the same as putting a minus in front, so an easier function definition, including the inverse would be:
Idss = 5.0
Id (x) = Idss * (1 + x)**2
Id_inv (x) = sqrt(x/Idss) -1
Now the dashed lines can be drawn as arrows (check set arrow):
set arrow 11 from first -0.5, graph 0 to first -0.5, first Id(-0.5) lc "red" lw 2 dt 2 nohead
set arrow 12 from first -0.5, Id(-0.5) to graph 0, first Id(-0.5) lc "red" lw 2 dt 2 nohead
set arrow 21 from first Id_inv(2.5), graph 0 to Id_inv(2.5), 2.5 lc "green" lw 2 dt 2 nohead
set arrow 22 from first Id_inv(2.5), 2.5 to graph 0, first 2.5 lc "green" lw 2 dt 2 nohead
I would recommend drawing the horizontal lines to the axis where the tics are actually printed (i.e. to the left side). The reference points are the x- or y-values of the function and the border of the graphs, see the manual about coordinates for a detailed explanation.
You could add specific tic marks at those points:
set xtics add (-0.5, Id_inv(2.5))
set ytics add (Id(-0.5), 2.5)
You also asked about points in your comment and got the correct approach. The size of the circle can be chosen in any coordinate system (I think character is the width of a letter – x? – in the current font, but I'm not 100% sure)
set object 1 circle center -0.5,Id(-0.5) radius first 0.01 fs solid noborder fc "red"
set object 2 circle center Id_inv(2.5),2.5 radius first 0.01 fs solid noborder fc "green"
Now the remaining decoration can be added:
set title "{/:Italic I}_d vs {/:Italic V}_{gs}"
set xlabel "{/:Italic V}_{gs} /V"
set ylabel "{/:Italic I}_d /mA"
set xrange [-1:0]
plot Id(x)