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

Aug 1, 2015

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:

$f \left(x\right) = 2 {x}^{2} + 7 x + 58$

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

$= 2 {x}^{2} + 7 x + 58$