This week in precalc 11, we learned many new concepts and learned a completely different vocabulary. We learned something called reciprocal functions. For example, if we have $f(x) = x + 3$ then the reciprocal would be $f(x) = \frac{1}{x + 3}$

Some new vocabulary that we learned are invariant points, hyperbola, and asymptotes

Invariant point : 1 and -1 are invariant points because once reciprocated will stay the same unlike other numbers.

Hyperbola : this is the graph of a reciprocal function

Asymptotes : the vertical and horizontal asymptotes are the boundaries where the hyperbola can not touch or overpass

Hyperbolas are quite interesting and the way to tell if they are indeed hyperbolas is to see if there are two parts to the graph. Here is an example of a hyperbola (linear) : as you can see there are two parts to the graph

For this graph, we can see that the linear graph is going through a point of -3 on the x intercept. This indicates the vertical asymptotes of the hyperbola. As for the horizontal asymptote (y), in our grade the y will always equal to 0. So for our asymptotes for this graph, it would be $x = -3$ and $y=0$. To find the invariant points, on the y axis we find 1 and -1 and from there, move horizontally until we hit a point of the linear graph. Those will be our invariant points, in this case our invariant points are $(-2,1)$ and $(-4,-1)$.

What I just showed were hyperbolas for linear graphs but there are also hyperbolas for quadratics. They are both pretty similar to each other in regards to the steps.