Plot Lines instead of points in scatterplot

Question

I want to replace the points in my graph with a line like in the first picture, the second picture is what I have.

but its not quite what im looking for, I want a smooth line without the points

I think I have to use predict for the 1/x curve but I am not sure how, 

Answer 1

Assuming f(1/x) fits the data well. One can use the lm() function to fix the desired function y= a/x + b and then use the predict() function to estimate the desired points.
If a more complicated nonlinear function is required to fit the data then the nls() maybe required

x<- c(176.01685819061, 21.6704613594849, 19.007554742708, 50.1865574864131, 17.6174002411188, 40.2758022496774, 11.0963214407251, 1249.94375253114, 694.894678288085, 339.786950220117, 42.1452961176151, 220.352895161601, 19.6303352674776, 9.10350287678884, 10.6222946396451, 44.1984352318898, 21.8069112975004, 42.1237630342764, 22.7551891190248, 12.9587850506626, 12.0207189111152, 20.2704921282476, 13.3441156357956, 9.13092569988769, 1781.08346869568, 71.2690023512206, 80.2376892286713, 344.114362037227, 208.830841645638, 91.1778810401913, 2220.0120768657, 41.4820962277111, 16.5730025748281, 32.30173229022, 108.703930214512, 51.6770035143256, 709.071405759588, 87.9618878732223, 10.4198968123037, 34.4951840238729, 57.8603720445067, 72.3289197551429, 30.2366643066749, 23.8696161364716, 270.014690419247, 13.8170113452005, 39.5159584479013, 27.764841260433, 18.0311836472615, 40.5709477295999, 33.1888820958952, 9.03112843931787, 4.63738971549635, 12.7591169313099, 4.7998894219979, 8.93458248803248, 7.33904760386628, 12.0940344070925, 7.17364602165948, 6.514191844409, 9.69911157978057, 6.57874454980745, 7.90556524435596)
y<- c(0.02840637, 0.230728821, 0.2630533, 0.099628272, 0.28381032, 0.12414402, 0.45059978, 0.00400018, 0.00719533500000001, 0.014715103086687, 0.118637201789886, 0.022690875, 0.254707825, 0.54923913, 0.470708088, 0.113126176837872, 0.22928510745, 0.118697847481752, 0.219730100850697, 0.38583864, 0.4159485, 0.24666396693114, 0.374696992776912, 0.547589605297248, 0.00280728, 0.070156727820596, 0.062314855376136, 0.01453005323695, 0.02394282358199, 0.0548378613646, 0.00225224, 0.120533928, 0.301695482, 0.15479046, 0.045996497, 0.096754836, 0.00705147600000001, 0.0568428, 0.47985120103071, 0.14494777, 0.08641493, 0.069128642, 0.165362156, 0.20947132, 0.018517511, 0.36187275779699, 0.126531158458224, 0.180083867690804, 0.277297380904852, 0.1232408972382, 0.15065285976048, 0.55364067, 1.07819275643191, 0.39187665, 1.04169066418176, 0.55962324, 0.68128731, 0.41342697, 0.69699564, 0.76755492, 0.515511133042674, 0.760023430328564, 0.632465844687028)


#data frame for prediction
df <- data.frame(x=sort(x))

# fit model y= a/x + b
model <-lm( y ~ I(1/x))
#summary(model)

#plot model
plot(df$x, predict(model, df), type="l", col="blue")
#optional
points(x, y)

Update - response to comments

1. x is sorted in the data frame, so that points are plotted in order. If not the line could go from x=1 to x=100, back to x=10 etc. thus making a mess. Try removing the sort and see what happens.
2. The I(1/x) term is to signal lm to perform the inverse transform on x first and then perform the least squares regression.
3. The predict() function is on the axis since that is the variable used in the plot function. To change this just assign the output from the predict function to a better variable name and plot that. Or use the "ylab= " option.