# One plot with two corresponding x axes in the FrameTicks

I am having difficulty plotting the following ListPlot. Below I simplified my code. I would like the top x axis to show the corresponding w values that resulted the dots in the plot. For example for the first dot, b=9 and w=1.9. How can I add the w=1.9 value corresponding to this dot on the top x-axis? Thanks

a[b_, w_] := b/(w);
k[b_, w_] := (1 - Exp[-a[b, w]])^2*Exp[-2/w];

ListPlot[{{9, k[9, 1.9]}, {19, k[19, 3]}, {29, k[29, 4]}, {39,
k[39, 5]}}, Frame -> True,
FrameTicks -> {{Automatic, None}, {Automatic, {1.9, 3, 4, 5}}},
FrameLabel -> {{k, None}, {"b", "w"}}]


\$Version

(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)

Clear["Global*"]

a[b_, w_] := b/(w);
k[b_, w_] := (1 - Exp[-a[b, w]])^2*Exp[-2/w];

wValues = {5, 4, 3, 2.5, 1.9};


EDIT 2: Your points are on the associated w curve, the value of w is not related to the x-axis (i.e., b) in any straightforward manner.

Show[ParametricPlot[{b, k[b, w]}, {b, 9, 39}, {w, 1.9, 5},
PlotStyle -> Opacity[0.15], AspectRatio -> 1, BoundaryStyle -> None],
ParametricPlot[
Evaluate[Tooltip[{b, k[b, #]}, StringForm["w = ", #]] & /@ wValues], {b,
9, 39}, AspectRatio -> 1,
PlotLegends -> LineLegend[wValues, LegendLabel -> Style[w, 14]]],
ListPlot[{
Callout[{9, k[9, 1.9]}, 1.9],
Callout[{19, k[19, 3]}, 3],
Callout[{29, k[29, 4]}, 4],
Callout[{39, k[39, 5]}, 5, Below]}],
FrameLabel -> (Style[#, 14] & /@ {b, k})]


EDIT: The curves for w asymptoticly approach

Asymptotic[k[b, w], b -> Infinity, Assumptions -> w > 0]

(* E^(-2/w) *)


or equivalently,

Limit[k[b, w], b -> Infinity, Assumptions -> w > 0]

(* E^(-2/w) *)


For the wValues

E^(-2./#) & /@ wValues

(* {0.67032, 0.606531, 0.513417, 0.449329, 0.349018} *)

• Thanks for your time. My question was about individual points in the plot (there is no line in my plot). Is it possible to use the top x axis to show the corresponding w values for each point? Commented Nov 9, 2022 at 4:08
• Your points are on the associated w curve -- that curve represents the value of w. The curves are the w` axis Commented Nov 9, 2022 at 4:35