2.6 Rational Functions

What is a Rational Function ? 𝑓 𝑥 = 1 𝑥 ;𝑥≠0 *Also referred to as the reciprocal function The standard form of a rational function is: f(x)= N(x) D(x) *A function is rational if ‘x’ is in the denominator, after the function has been simplified

Are the following functions rational? 𝑓 𝑥 = 1 2𝑥 𝑓 𝑥 = 𝑥 𝑥 𝑓 𝑥 = ( 𝑥 3 −2 𝑥 2 −15𝑥) (𝑥+3)

Finding Domain/Range of Rational Functions… Domain will generally be all real numbers except for the Vertical Asymptote(s) or holes of the function Range – generally all real numbers except for the Horizontal Asymptote or holes of the function

Domain continued… Ex: 1 𝑥 2 +7𝑥+12 = 1 (𝑥+4)(𝑥+3) -4 and -3 both make the denominator equal to zero, so they are both excluded from the domain What is the domain of the following functions? f(x)= 1 ( 𝑥 2 +10𝑥+24) 2. f(x)= 1 ( 𝑥 2 −5𝑥+6)

Describing the domain… Ex: f(x) = 1 𝑥 Description: As x decreases to 0 y increases without bound, as x increases to 0 y decreases without bound. Ex. 2: Describe the domain of f(x) = 1 𝑥 2 +10𝑥+24

What are Asymptotes? Asymptote- the line that the function approaches, helps determine end behaviors. *graphs will NEVER cross vertical asymptotes but they can cross horizontal or slant Holes - point of discontinuity (function is undefined at this value) Asymptotes

Finding Vertical Asymptotes and Holes… Vertical Asymptote- set the denominator equal to zero. Holes – occur when a factor in the denominator is simplified (reduced to 1) by same factor in the numerator

Identify any vertical asymptotes or holes in the following rational functions: Ex 1: (𝑥+4)(𝑥 −2) (𝑥−7)(𝑥−2) Ex. 2: 4 𝑥 2 −1 2 𝑥 2 −11𝑥−6

Horizontal Asymptotes…

Cases of Horizontal Asymptotes Case 1: N < D 𝑦= 𝑎 𝑥+ℎ +𝑘 Horizontal asymptote: y = k *most scenarios, y = 0 Ex 1. 𝑦= 3 𝑥+2 −1 Ex 2. 𝑦= −2 𝑥+3 +4 Ex 3. 𝑦= 𝑥−6 𝑥 2 +1

Cases of Horizontal Asymptotes Case 2: N = D 𝑦= 𝑎𝑥 𝑛 𝑏𝑥 𝑑 Horizontal asymptote is y = 𝑎 𝑏 *the leading coefficients Ex. 1) y= 2𝑥 𝑥−3 Ex 2.) 𝑦= 3 𝑥 2 6 𝑥 2 +2𝑥

Cases of Horizontal Asymptotes Case 3: N > D 𝑦= 𝑥 𝑛 𝑥 𝑑 Horizontal Asymptote is NONE *improper fraction! This case will have a SLANT Asymptote Ex. 1) 𝑦= 𝑥 3 −9 𝑥 2 +5

Homework Day 1: Pg 174 #’s 1, 7, 11, 13-16, 17, 21, 25, 33, 39 Have fun     

Day 2 – Warm Up Without a calculator…. Please graph(not sketch) the rational function: 𝑓 𝑥 = 1−3𝑥 1−𝑥

What is a Slant Asymptote? An oblique line that the graph approaches, it helps describe/determine the end behaviors If the degree of the numerator is exactly 1 more than the denominator it has a slant asymptote

Finding Slant Asymptotes Check the Degree Use either long or synthetic division to find asymptote Find the Slant Asymptote Ex 1.) 𝑓 𝑥 = 2 𝑥 2 +3𝑥+1 𝑥+1 Ex 2.) 𝑓 𝑥 = 𝑥 2 −𝑥−2 𝑥−1

Identify any asymptotes of the following rational functions… f(x) = 2𝑥 𝑥 3 −6 𝑥 2 +2𝑥−12 f(x) = 4 𝑥 2 𝑥 2 −9 f(x) = 3 𝑥 2 −𝑥−2 𝑥 2 −6𝑥−7 𝑓 𝑥 = 4 𝑥 3 +7𝑥−3 𝑥 2 −4

Graphing Rational Functions Find the zeros of the denominator. These will be the vertical asymptotes/holes. Draw dotted line(s). Find the horizontal asymptotes in the ways learned earlier. Draw dotted line(s). See if there will be any slant asymptotes in the graph. Draw dotted line(s). Find the zeros of the numerator. These will be the x- intercepts unless it is also a zero of the denominator. Evaluate f(0) to find y intercept of function. Plot. Create a table of values and plug in at least one point between and one point beyond each x-intercept and vertical asymptote. Connect points with smooth curves

Graphs of a Rational Function Things to keep in mind: Positive numerator, functions will be in quadrants 1 and 3 Negative numerator, functions will be in quadrants 2 and 4 Graph the following functions 𝑓 𝑥 = 𝑥 2 −𝑥−5 𝑥−1

Some more practice with graphing… 1) 𝑓 𝑥 = 1 𝑥 2 2) 𝑓 𝑥 = 𝑥 3 𝑥 2 −1 3) 𝑓 𝑥 = 4 𝑥 2 −4

Homework Day 2… Pg. 174-175 #’s 3, 9, 19, 23, 35, 45, 49, 51, 55, 69