Week 13 – Pre Calculus 11

This week we learned how to graph reciprocal LINEAR functions. At first glance, this lesson may seem hard, although after doing a few question you get the hang of it pretty fast. The first thing you need to remember is that the reciprocal is the flipped version of a function $(ex. 2 -> \frac{1}{2})$ . I will start with linear functions – there are 4 steps to graphing a linear reciprocal function

Graph the original parent function – ex. $\frac{1}{x+3}$ (so you would graph x+3)
If you were to write out “y value” points in reciprocal form on a table of values all numbers will be put into a faction form expect for 1, -1. In the second step you need to find the 1, -1 points on your linear function that you graphed. These are called invariant points.
The third step is finding your vertical and horizontal asymptotes. These are defined as a line that represents undefined values to which the reciprocal graph line will never reach a finite distance (the line of your graph will never reach 0). Your horizontal asymptote will usually be y=0 and to find the vertical asymptote it will be shown as the x-intercept on your graph. If you are asked to find the vertical aymptote and are not given a graph you can do as followed. We know the horizontal one will be y=0, so if given the function $y= \frac{1}{2x}$ you can simply put 0 in place of the y and algebraically solve it (remember to solve it in it’s original function). So, $0= 2x$ -> $0/2 = 2/2$ -> $0=x$ .
The fourth and final step is to graph the function using something called a hyperbola, a hyperbola graph looks like two curvy lines each one intersecting with the invariant points, and in the middle of the two hyperbolas is our vertical and horizontal asymptotes. A key thing to remember is the hyperbola will never touch 0 on the y/x axis because 0 cannot be a reciprocated number as 0 can’t be divided by a number. It will also never touch the x-intercept of the vertical asymptote.