Pfthb

I am basically trying to make this figure in Tikz only better, but I can't figure out how to make the vertical lines from the x-axis to the graphs M(x). The function is M(x)=−1/2·q·x^2+1/2·q·L·x, where the values of q and L doesn't matter in the first place.

(The figure is made in Maple 2018. It hints at the mistakes made in the vertical lines.)

This is a very 'hacky' solution, and I'm sure some people will be able to give you a more elegant answer, but this definitely works!

I also like it because it's been a very broadly applicable technique for me (especially using intersections).

MWE:

documentclassarticle
usepackageamsmath
usepackagetikz
usetikzlibrarycalc,intersections

begindocument
 begintikzpicture
 draw[<->] (0,6) node[above]M--(0,0)--(9,0)node[right]L;
 draw[name path = C] (0,0) .. controls (3,6) and (5,6) .. (8,0) node[pos = 0.5, above] $M_max = dfrac18 cdot q cdot L^2$;

 foreach i in 0,0.5,...,8
 draw[draw opacity = 0, name path = L] (i,0)--(i,6);
 draw [name intersections =of = L and C] let p1 = (intersection-1) in (x1,y1)--(x1,0);
 
 endtikzpicture
enddocument

Output:

PS: If you need the curve to look differently, just edit the code for draw[name path = C].... For instance, by changing it to:

draw[name path = C] (0,0) .. controls (2,4) and (6,4) .. (8,0) node[pos = 0.5, above] $M_max = dfrac18 cdot q cdot L^2$;

you get this (which is closer to your original image). I illustrated this just in case you wanted to see a sample of how to draw curves. :)

As I mention in a comment, a ycomb plot can be used for this. Here are two examples, the first a modified version of marmot's code, the second a more verbose (and probably more complicated than it needs to be) version using pgfplots.

documentclass[border=5mm]standalone
usepackagepgfplots
pgfplotssetcompat=1.3
begindocument
begintikzpicture[
 declare function=
 M(x) = 4-(x-4)*(x-4)/4;
 
]
draw[latex-latex] (0,6) node[left] $mathbfM$ |- (8.5,0) node[below]
$mathbfL$;
 draw[thick] plot[variable=x,domain=0:8,smooth] (x,M(x));
 draw[thick] plot[variable=x,domain=0:8,ycomb] (x,M(x));
endtikzpicture

begintikzpicture
beginaxis[
 declare function=
 t = 2;
 mid = 5;
 d=4.5;
 M(x) = -(x-mid)^2*(t/d^2) + t;
 ,
 axis lines=middle,
 xtick=empty, ytick=empty,
 ylabel=$M$, xlabel=$L$,
 enlarge x limits,
 enlarge y limits=value=0.5,upper,
 domain=mid-d:mid+d
]

addplot [thick] M(x) node[midway, above] $M_max = frac18 qL$;
addplot [ycomb, samples=15] M(x);
endaxis
endtikzpicture

enddocument

Here is one more possibility: use a pattern. And if you use clip, as suggested by @nidhin, I'd use a grid rather than a foreach loop. In fact, if you use foreach, since the function is known, you do not need clip.

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarypatterns
usepackageamsmath
begindocument
begintikzpicture
draw[latex-latex] (0,6) node[left] $boldsymbolM$ |- (8.5,0) node[below]
$boldsymbolL$;
draw[thick,pattern=vertical lines] plot[variable=x,domain=0:8,smooth] (x,4-(x-4)*(x-4)/4);
endtikzpicture

begintikzpicture
draw[latex-latex] (0,6) node[left] $boldsymbolM$ |- (8.5,0) node[below]
$boldsymbolL$;
draw[thick] plot[variable=x,domain=0:8,smooth] (x,4-(x-4)*(x-4)/4);
clip plot[variable=x,domain=0:8,smooth] (x,4-(x-4)*(x-4)/4);
draw (0,0) grid[xstep=1cm,ystep=6cm] (8,5);
endtikzpicture
enddocument

Using clip

documentclassstandalone
usepackageamsmath
usepackagetikz
begindocument
 begintikzpicture[>=latex]
 draw(4,5) node $M_max = dfrac18 cdot q cdot L^2$;
 draw[<->] (0,6) node[above]M--(0,0)--(9,0)node[right]L;
 draw[clip] (0,0) .. controls (3,6) and (5,6) .. (8,0) --cycle;
 foreach i in 0,0.5,...,8
 draw (i,0) -- ++ (0,10);
 endtikzpicture
enddocument