# Marking Specific Points on Graph

Is there any way that I can label/mark points (axes intercepts, maximum/minimum values etc.) on Mathematica? (as in without using the drawing tools)

Such as in the case of: Plot[{x^3 - 6*x^2 + 9*x + 10}, {x, 0, 4}]

Thanks

• You can use Epilog to add whatever you want, but you have to calculate those point's positions. Related: Labeling points of intersection between plots
– Kuba
Oct 15 '15 at 10:00
• Hi @Kuba Thanks for your response - I'm new to mathematica, and although this might sound a bit amateurish/childish - Can you please show me what the coding may look like in a basic graph? Such as in the aforementioned example or something of the like.
– Sven
Oct 15 '15 at 10:12
• Type in Epilog to Mathematica notebook, press F1 :)
– Kuba
Oct 15 '15 at 10:14
• Alright, I think I get it now - Thank you so much!!
– Sven
Oct 15 '15 at 10:20
• @Sven, the point is the the documentation of Epilog does contain the examples you are after, so the pointer provided by Kuba is actually enough to answer your question. Oct 15 '15 at 13:37

Here is an example of some of the things you could do using Epilog, which essentially allows you to combine any 2D graphics primitives on top of an existing plot.

I will define your function as fun[x]:

Clear[fun, realzero]
fun[x_] := x^3 - 6*x^2 + 9*x + 10


I can use Solve to find the function's zeroes, i.e. its intersections with the horizontal axis:

realzero = {x, 0} /. First@Solve[fun[x] == 0., x, Reals]

(* Out: {-0.721892, 0} *)


I then have Mathematica calculate the expression's first derivative and set up an equation to find its zeroes, i.e. the function's extrema.

Solve[D[fun[x], x] == 0, x]

(* Out: {{x -> 1}, {x -> 3}} *)


I can then combine what I found in a plot:

Plot[fun[x], {x, -1, 4},
Epilog -> {PointSize[0.03],
Red, Tooltip[#, #[]] &@Point[{1, fun}],
Blue, Tooltip[#, #[]] &@Point[{3, fun}],
Darker@Green, Tooltip[#, #[]] &@Point[realzero],
Orange, Tooltip[#, #[]] &@Point[{0, fun}]
},
AxesStyle -> Directive[Gray, Dashed], AxesOrigin -> {0, 0},
PlotRange -> {Automatic, {-4, All}},
Frame -> True,
FrameLabel -> (Style[#, Bold, FontSize -> 14] & /@ {"x", "y"})
] If you execute the code in Mathematica and hover over those points with the mouse, you will also notice that a tooltip pops up with the coordinates for those points.

I would suggest that you use this code as a starting point to explore the commands and options showcased: play around with it, modify it, dig into the documentation... Mathematica can be a lot of fun!

• Thank you very much!! This is a great starting point - Can't wait to discover more!! :D
– Sven
Oct 15 '15 at 10:28