One way to describe the pattern of covariation for a linear function is:

Question

One way to describe the pattern of covariation for a linear function is: As input value increases by 1, the output value changes by a constant (fixed) amount k where k is some real number. Explain why this is the case for linear functions and not for quadratic functions.

Answer 1

Take $y = ax + b$. If $x' = (x + d)$, then $y' = a(x') + b = a(x + d) + b = (ax + b) + ad = y + ad$. If $d$ is very small, you can see that this is the proportion at which $y$ increases if you add $d$ to $x'$.

But you cannot do this with quadratics:

Let $y = a(x-b)^2 + c$.

$x' = x + d \implies y' = a(x + d - b)^2 + c = a((x - b) + d)^2 +c = (a(x - b)^2 + c) + 2ad(x -b) + ad^2 = y + 2ad(x - b) + ad^2.$

You can see that the additional terms $(2ad(x-b) + ad^2)$ have a dependency on $x$, which means that if you add $d$ at different points in the real axis, the variance will differ. For example, if you have a quadratic evaluated at $x = 1$, then add $d$, this difference will not be the same as the difference if you had started with $x = 2$. Thus, the covariation is not constant for a quadratic.