x=c(0,0.05,0.236,0.382,0.5,0.618,0.786,0.886,1)

x

f<-function(s){5*s*(s+1)}

y=f(x);y
z=y
z[1]=1
z[3]=1.25
z[4]=2.5 
z[6]=5
z[10]=10

curve(f,0,1.1)
points(x[c(1,3,4,6,9)],z[c(1,3,4,6,9)])

# Least square fit fb(s)=a+bs+cs^2.
xx=x[c(1,3,4,6,9)]
zz=z[c(1,3,4,6,9)]
A=matrix(1,5,3)
A[,2]=xx
A[,3]=xx^2

A

b=solve(t(A)%*%A,t(A)%*%zz);
b

fb<-function(s){b[1]+b[2]*s+b[3]*s^2} # approximation 
curve(fb,0,1.1)
points(x[c(1,3,4,6,9)],z[c(1,3,4,6,9)])

print("Start X")
print(x)
print("Start Y")
print(y)
print("Start Z")
print(z)
print("Approximation")
print(fb(x))