Question

How can you tell if a function is linear, quadratic, or exponential when given four points (1,67) (2,80) (3,97) (4,118)?

Answer 1

Since we're given the values for `x=1,2,3,4` which are evenly spaced, look at differences of `y` values, then differences of differences to find that we have a quadratic:

`f(x) = 2x^2+7x+58`

Explanation:

Since the `y` values are given for `x=1, 2, 3, 4` let us write the `y` values as a sequence, then form the difference of those terms, then the difference of the differences...

Notice that it takes `2` steps to get to a constant sequence, so this is a quadratic.

Next, let us construct the `y` value for `x=0` by adding in a column on the left...

We do this by writing in another `4` on the last row, then calculating `9` as `13 - 4`, then calculating `58` as `67 - 9`.

Now we have added this row for `x=0` we can use its values to find the equation of the quadratic...

`f(x) = color(red)(58) + color(red)(9) * (x)/(1!) + color(red)(4) * (x(x-1))/(2!)=58+7x+2x^2`

`=2x^2+7x+58`