Predicting values based on given data

Question

   num  value
    8     ?
    10    ?
    12  24.33
    16  22.5
    20  22.29
    24  22.41
    28  22.55
    32    ?
    36    ?

let's say i have the following values. What can I do to estimate the values for the numbers 8, 10, 32, 36?

How do i come up with a function that best represent these numbers?

I feel like i can use calculus but not exactly sure; havent used calculus in longest time

Answer 1

Very open-ended question, one practical way of doing this is to enter the data into Excel and add a scatter plot. Then right click on the plot and choose Add Trendline...

You get a choice of different fitting models. Check Display Equation on chart and play with different choices. Also display the $R^2$ to see the quality of fit.

In your case, as is obvious, a quadratic polynomial will fit exactly ($R^2=1$), a cubic is very close ($R^2=0.9953$) and a quadratic is slightly worse ($R^2=0.9071$),

Picturing the cubic from Excel:

You can then use the displayed equation to fit the unknown points.