The monohumpoid curve

styletext[
t_
]:=
Style[t,Bold,FontSize->14,FontFamily->"Helvetica"];
styletext["N"]

Block[{r=0.5,k=3,nd=(n|->r n (k-n)),ndmax},
ndmax=r ((k^2)/4);
Plot[nd[n],{n,0,k}
,Axes->True
,PlotStyle->{Thickness[0.02],Automatic}
(*,Frame->{{True,False},{True,False}}*)
,Ticks->{{{0,styletext@"0"},{k,styletext@"K"}},{{0,styletext@"0"},{ndmax,styletext@"(Subscript[(OverDot[N]), max])"}}}
,AspectRatio->1/2
,AxesLabel->{TraditionalForm[styletext["N"[t]]],styletext@(OverDot[N])}
,Epilog->{
Dashed,
Line[{{0,ndmax},{k/2,ndmax}}]
}
]
]

Block[{r=0.5,k=3,nd=(n|->r n (k-n)),ndmax},
ndmax=r((k^2)/4);
Plot[{
nd[n],
(n^2)
},{n,-k/2,k}
,Axes->True
,PlotStyle->{Thickness[0.02],Automatic}
,PlotRange->{Full,{-r,All}}
(*,Frame->{{True,False},{True,False}}*)
,Ticks->{{{0,styletext@"0"},{k,styletext@"K"}},{{0,styletext@"0"},{ndmax,styletext@"(Subscript[(OverDot[N]), max])"}}}
,AspectRatio->1/2
,AxesLabel->{TraditionalForm[styletext["N"[t]]],styletext@(OverDot[N])}
,PlotLegends->{"(OverDot[N])","(N^2)"}
,Epilog->{
Dashed,
Line[{{0,ndmax},{k/2,ndmax}}]
}
]
]

Block[{r=0.5,k=3,nd=(n|->r n (k-n)),ndmax},
ndmax=r((k^2)/4);
Plot[{
nd[n],
-(n^2)
},{n,-k/2,k}
,Axes->True
,PlotStyle->{Thickness[0.02],Automatic}
,PlotRange->{Full,{-r,All}}
(*,Frame->{{True,False},{True,False}}*)
,Ticks->{{{0,styletext@"0"},{k,styletext@"K"}},{{0,styletext@"0"},{ndmax,styletext@"(Subscript[(OverDot[N]), max])"}}}
,AspectRatio->1/2
,AxesLabel->{TraditionalForm[styletext["N"[t]]],styletext@(OverDot[N])}
,PlotLegends->{"(OverDot[N])","-(N^2)"}
,Epilog->{
Dashed,
Line[{{0,ndmax},{k/2,ndmax}}]
}
]
]

Block[{r=0.5,k=3,nd=(n|->r n (k-n)),ndmax,n0},
ndmax=r((k^2)/4);
n0=(Sqrt[ndmax]);
Plot[{
nd[n],
ndmax-(n^2)
},{n,-k/2,k}
,Axes->True
,PlotStyle->{Thickness[0.02],Automatic}
,PlotRange->{Full,{-r,All}}
(*,Frame->{{True,False},{True,False}}*)
,Ticks->{{{0,styletext@"0"},{k,styletext@"K"}},{{0,styletext@"0"},{ndmax,styletext@"(Subscript[(OverDot[N]), max])"}}}
,AspectRatio->1/2
,AxesLabel->{TraditionalForm[styletext["N"[t]]],styletext@(OverDot[N])}
,PlotLegends->{"(OverDot[N])","(Subscript[(OverDot[N]), max])-(N^2)"}
,Epilog->{
Dashed,
Line[{{0,ndmax},{k/2,ndmax}}],
PointSize[Large],
Point[{{-ndmax,0}}],
Text[styletext@"-(Sqrt[(Subscript[(OverDot[N]), max])])",{-ndmax,0},{-1.2,-1}]
}
]
]

Block[{r=0.5,k=3,nd=(n|->r n (k-n)),m,ndmax,n0,n1},
ndmax=r((k^2)/4);
n0=(Sqrt[ndmax]);
m=n|->ndmax-(((n-n0))^2);
n1=Max[n/.Solve[m[n]==0,n]];
Plot[{
nd[n],
m[n]
},{n,-k/2,k}
,Axes->True
,PlotStyle->{Thickness[0.02],Automatic}
,PlotRange->{Full,{-r,All}}
(*,Frame->{{True,False},{True,False}}*)
,Ticks->{{{0,styletext@"0"},{k,styletext@"K"}},{{0,styletext@"0"},{ndmax,styletext@"(Subscript[(OverDot[N]), max])"}}}
,AspectRatio->1/2
,AxesLabel->{TraditionalForm[styletext["N"[t]]],styletext@(OverDot[N])}
,PlotLegends->{"(OverDot[N])","(Subscript[(OverDot[N]), max])-(N-(Sqrt[(Subscript[(OverDot[N]), max])]))^2"}
,Epilog->{
Dashed,
Line[{{0,ndmax},{k/2,ndmax}}],
PointSize[Large],
Point[{{n1,0}}],
Text[styletext@"2(Sqrt[(Subscript[(OverDot[N]), max])])",{n1,0},{0,-1.2}]
}
]
]

Block[{r=0.5,k=3,nd=(n|->r n (k-n)),m,ndmax,n0,n1},
ndmax=r((k^2)/4);
n0=(Sqrt[ndmax]);
m=n|->ndmax-(((n-n0))^2);
n1=Max[n/.Solve[m[n]==0,n]];
m=n|->ndmax-((((n1/k) n-n0))^2);
Plot[{
nd[n],
m[n]
},{n,-k/2,k}
,Axes->True
,PlotRange->{Full,{-r,All}}
,PlotStyle->{Thickness[0.02],Automatic}
,Ticks->{{{0,styletext@"0"},{k,styletext@"K"}},{{0,styletext@"0"},{ndmax,styletext@"(Subscript[(OverDot[N]), max])"}}}
,AspectRatio->1/2
,AxesLabel->{TraditionalForm[styletext["N"[t]]],styletext@(OverDot[N])}
,PlotLegends->{"(OverDot[N])","(Subscript[(OverDot[N]), max])-(((2(Sqrt[(Subscript[(OverDot[N]), max])]))/K)N-(Sqrt[(Subscript[(OverDot[N]), max])]))^2"}
,Epilog->{
Dashed,
Line[{{0,ndmax},{k/2,ndmax}}],
PointSize[Large]
}
]
]

Solve[ndmax-(((n-n0))^2)==0/.{n0->(Sqrt[ndmax])},n]

Now, (Subscript[N, 0])=(Sqrt[(Subscript[(OverDot[N]), max])]) and (Subscript[N, 1])=2(Sqrt[(Subscript[(OverDot[N]), max])])=2(Subscript[N, 0]), giving for the eventual function:

ndmax-((((n1/k)n-n0))^2)//.{n0->(Sqrt[ndmax]),n1->2n0}//Simplify

Learning curve exponents

Log[2,0.75]

Log[2,0.95]