# Understand MeshFunctions with Intercepts

Bug introduced in 7.0.1 or earlier and persisting through 11.0.1 or later

While I've been thinking about this question, I tried to visualize the Intercepts. I use three snippets, actually snippet 1, 2 and 3. The visualization works as expected.

p[x_] := x^4 - 4 x^2 - 2 x + 1

solutions = NSolve[p[x] == 0, x]

snippet1 =
Plot[p[x], {x, -3, 3}, PlotStyle -> Green,
MeshFunctions -> {p[x] /. x -> # &}, Mesh -> {{0}},
MeshStyle -> Directive[PointSize[Large], Red],
PlotLabel -> "Snippet 1"];

snippet2 =
Show[Plot[p[x], {x, -3, 3}, PlotStyle -> Brown],
Graphics[{Red, PointSize[Large], Point[{x, p[x]} /. solutions]}],
PlotLabel -> "Snippet 2"];

snippet3 =
Plot[p[x], {x, -3, 3}, PlotStyle -> Blue,
Epilog -> {PointSize[Large], Red, Point[{x, 0} /. NSolve[p[x]]]},
PlotLabel -> "Snippet 3"];


When applying snippet 1 with a larger xmin-value a fifth red spot appears. I do not observe this behavior on snippet 2 and 3. What's wrong with snippet 1, or with my knowledge/interpretation?

snippet1 =
Plot[p[x], {x, -3, 4}, PlotStyle -> Green,
MeshFunctions -> {p[x] /. x -> # &}, Mesh -> {{0}},
MeshStyle -> Directive[PointSize[Large], Red],
PlotLabel -> "Snippet 1"]


I'm on 10.0 for Mac OS X x86 (64-bit) (September 10, 2014).

• This problem looks familiar, but perhaps it wasn't on this site that I've seen it. At least I can't find it. The problem goes back at least to V8.0.4 (Mac OS). It doesn't happen if you use PlotRange -> All. Oct 15, 2014 at 3:28
• @MichaelE2 Should this be tagged as a bug? Oct 19, 2014 at 9:13
• @Mr.Wizard Yes, I think it ought to be considered a bug. I've edited the question, hopefully in line with our current "bugs" style. Oct 19, 2014 at 12:58
• This still happens to 10.3. Jan 17, 2016 at 12:55

A very good question. You should not rely on MeshFunctions.

Follow the old path step for step:

fun = x^4 - 4 x^2 - 2 x + 1;

xvals = NSolve[fun == 0, x] // Values // Flatten


{-1.48119, -1., 0.311108, 2.17009}

yvals = fun /. xvals // Chop


{0, 0, 0, 0}

zeroes = Transpose[{xvals, yvals}]


{{-1.48119, 0}, {-1., 0}, {0.311108, 0}, {2.17009, 0}}

snippet1 =
Plot[fun, {x, -3, 5},
Epilog -> {Red, PointSize[0.02], Point /@ zeroes},
PlotStyle -> Green,
PlotLabel -> "Snippet 1"]


It looks like a bug. However there is a simple workaround: change the sign of your mesh function. In addition, p[x] /. x -> # & is equivalent to #2 &.

Plot[p[x], {x, -3, 4}, PlotStyle -> Green, MeshFunctions -> {-#2 &},
Mesh -> {{0}}, MeshStyle -> Directive[PointSize[Large], Red]]


• Yes, it's a bug, and your "workaround" profits from this bug. +1 anyway :)
– eldo
Oct 14, 2014 at 20:39