You need more values in your vector x. Try it with
x0 = linspace(0.002, 0.2, 100);
and alter your code to be
hyprb = @(b,x) b(1) + b(2)./(x + b(3)); % Generalised Hyperbola
NRCF = @(b) norm(y - hyprb(b,x)); % Residual Norm Cost Function
B0 = [1; 1; 1];
B = fminsearch(NRCF, B0); % Estimate Parameters
figure(1)
plot(x, y, 'pg')
hold on
plot(x0, hyprb(B,x0), '-r')
hold off
grid
text(0.7, 0.52, sprintf('y = %.4f %+.4f/(x %+.4f)', B)).
The green points have stayed the same, using your original x and y, but now the red curve has 100 points instead of six, so appears smoother.
