Two lines are parallel as long as their slopes are the same but with different y-intercept. By evaluating the derivatives, we can obtain the slopes of the functions using the formula:
h0f(x2) – f(x1)x2 – x1or f(x+h) – f(x)h
The slope of a function can be used to infer whether or not two lines are behaving the very same way, a.k.a. whether or not they are parallel to each other. If their derivatives are the same, it means that their rates of change are similar; therefore, they are behaving identically and thus, parallel to each other at every single value of x. However, their slope cannot have the same y-intercept because if so, they will overlap each other (and be the same equations).
For instance, we can have two hypothetical parabolas given by the functions of y = x2 and p = x2 + 18. The derivatives of these two equations are both 2x, which means they have similar slopes. As aforementioned, they will behave identically and parallel to each other. We can verify this using the Desmos tool below:
The red parabola of the function y = x2 and the blue parabola of the function y = x2+ 18 have the same derivatives (2x). We can visually observe that they are parallel to each other. We also acknowledge that the two parabolas are vertically “everywhere equidistant,” since shifting the red parabola of the function y = x2up to 18 units will make its y-intercept overlapping the one of the blue parabola of the function y = x2+ 18, thus y equals p.
In conclusion, by acknowledging two random parabolic function having the forms of y = ax2+ m and y’ = ax2 + n, we can always obtain two parallel parabola as their derivative is always 2a (as the dmdxand dndx is always 0), thus behaving similarly.
