Reciprocal of Quadratic Function

In this section, we will analyse reciprocal of quadratic function such as f(x)= 5/ x^2 - 4x +  3.

Reciprocal of quadratic functions with two zeroes have three parts, with the middle one reaching a maximum or minimum point. This point is equidistant from the two vertical asymptotes.

The behavior of the function near the asymptotes is similar to the previous post, that is reciprocal of linear functions.

We can predict the key features of the graph by analyzing the roots of the quadratic relation in the denominator.

For example:

The denominator of the above function is x^2 - 4x +  3.

First, we have to factorize the relation. It is (x - 3)(x - 1).

The denominator cannot equal to zero. Thus the restriction of x are 3 and 1. They will be the vertical asymptotoes of the function.

To find the key features such as domain, range, end behavior and asymptotes, please refer to the previous post.

There are 5 intervals in the graph.

For the interval x<1, the sign of f(x) is positive. The sign of slope is positive and the change in slope is increasing.

For the interval 1<x<2, the sign of f(x) is negative. The sign of slope is positive and the change of slope is decreasing.

For the interval x=2, the sign of f(x) is 0. The sign of slope is positive and the change of slope is constant.

For the interval 2<x<3, the sign of f(x) is negative. The sign of slope is negative and the change of slope is decreasing.

For the interval x>3, the sign of f(x) is positive. The sign of slope is negative and the change of slope is increasing.

Source: 

McGraw-Hill Ryerson Advanced Functions 12

For further details and understanding, you can watch these videos :)