A Line Keeps Its Rate

Take the line $g\left(x\right) = 4x-1$. From x = 0 to x = 2, the outputs are -1 and 7, so the rate is $\frac{7-(-1)}{2}$. Subtracting -1 adds 1, so the top is 8 and the rate is 4. From x = 5 to x = 10, the outputs are 19 and 39, a change of 20 over an input change of 5: the rate is 4 again.

That is no accident. Every time the input grows by 1, $4x-1$ grows by exactly 4. Over any stretch of inputs the output change is 4 times the input change, and dividing by the input change leaves 4. For a line $kx+c$, the average rate of change is always k, its slope, whichever two inputs you pick.

Watch the double sign when an output is negative. Subtracting a negative output turns into adding, so write the bracket and work it out before you divide.