Chapter 8 3 Rational Functions and Their Graphs

Chapter 8 -3 Rational Functions and Their Graphs St. Augustine Preparatory School October 27, 2015

Transformations Guide

One last thing from 8. 2 The graph of a function is a translation of the graph of y=2/x. What is the equation of the function?

Answer We can use the asymptotes to find an equation. We have a vertical asymptote (h) equal to -3 and a horizontal asymptote (k) equal to 4.

Another example The graph is of a function is a translation of the graph y = 4/x. What is the equation for the function?

Answer Vertical Asymptote at x = 2 Horizontal asymptote at y = 3

Types of Graphs 1) Continuous graphs – no breaking points Graph 1 2) Discontinuous graphs – have a hole or asymptotes Graph 2 and 3

Important Terms Point of Discontinuity – a point where the graph is not continuous, ie. has a hole in it. Example: At x = -2, the denominator is going to be equal to 0, meaning the function is undefined at x = -2

Important Terms Removable Discontinuity – a break in the graph that can be filled by simply redefining the function at the problematic value of x. Non-removable Discontinuity – A break in the graph that cannot be filled as we cannot redefine the equation at the problematic value of x.

Examples a) Removable discontinuity b) Non-removable discontinuity c) Once factored, you will have: x+3 (x-3)(x-1) So non-removable discontinuity

Finding Points of Discontinuity What are the points of discontinuity and are they removable or non-removable? Step 1: Factor the denominator: Step 2: Find points of discontinuity: x – 3 = 0 so x = 3 and x – 1 = 0, so x = 1 Step 3: Removable or non-removable? Since the denominator is not found in the numerator, the points are both non-removable

For the following: 1. Find the points of discontinuity 2. Determine if the points are removable or nonremovable 3. Redefine the function if a point is removable. 1. 2. 3. 4.

Answers 1) x = -5 and x = -4. The point at x = -5 is removable, where the point at x = -4 is not 2) x = 9 and x = -2. The point at x = -2 is removable. The point at x = 9 is not 3) x = -1. Non-removable point. 4) x = 1/3 and x = 2. Point at x = 2 is removable and the point at x = 1/3 is non-removable