Rational Functions
Rational Function: a type of function containing two polynomial functions
Finding Zeros, Asymptotes, and End Behavior of a Rational Function
Step 1: Look at the zeros of the Denominator
-zeros are the vertical asymptote(s) of the function
Step 2: Look at the zeros of the numerator
Step 3:Look at the end behavior (look at the leading terms of the numerator and denominator)
-if there is a higher degree in the denominator--the function is getting closer to zero
Step 4: Determine the Horizontal asymptote (from end behavior or dividing the numerator by the denominator after graphing the function)
-zeros are the vertical asymptote(s) of the function
Step 2: Look at the zeros of the numerator
Step 3:Look at the end behavior (look at the leading terms of the numerator and denominator)
-if there is a higher degree in the denominator--the function is getting closer to zero
Step 4: Determine the Horizontal asymptote (from end behavior or dividing the numerator by the denominator after graphing the function)
Finding Horizontal Asymptotes by long divison:
f(x)= x^5+1/x^2
To find the asymptote:
Divide the numerator by the denominator
-use long division
need help with long division? Go to polynomial functions
To find the asymptote:
Divide the numerator by the denominator
-use long division
need help with long division? Go to polynomial functions
Answer of f(x)= x^3-x + x+1/x^2+1
The Asymptote is X^3-x
The Asymptote is X^3-x
Example:
f(x)=(x-3)(x-5)(x-4)/(x-1)(x)(x-2)(x+1)
Step 1: The zeros of the denominator are: 1, 2, 0, -1---vertical asymptotes
Domain: all real numbers except: 1, 2, 0, and -1
Step 2: The zeros of the numerator are: 3, 5, and 4
Step 3: Look at the end behavior (looked on graphing calculator or leading terms of the numerator and denominator)
x^3/x^4 ---there is a higher degree in the denominator; lim x-->infinity f(x)-->0 lim x-->negative infinity f(x)-->0
Step 4: Horizontal asymptote is at y=0, because the function gets closer and closer to 0; it never reaches it.
Another Example:
f(x): 2/(x^2)-1
Step 1: Vertical Asymptotes at 1 and -1
Domain: all real numbers, except 1 and -1
Step 3: Limit as x--> infinity f(x)=0
Limit as x--> negative infinity f(x)=0
Now you must do the end behavior based on the asymptotes:
Limit as x-->Positive 1 from the positive side f(x)=infinity
Limit as x--> Positive 1 from the negative side f(x)=negative infinity
Limit as x--> negative 1 from the positive side f(x)= negative infinity
Limit as x--> negative 1 from the negative side f(x)=positive infinity
Step 4: Horizontal asymptote at 0--> function gets closer and closer to 0, but never reaches it
Domain: all real numbers, except 1 and -1
Step 3: Limit as x--> infinity f(x)=0
Limit as x--> negative infinity f(x)=0
Now you must do the end behavior based on the asymptotes:
Limit as x-->Positive 1 from the positive side f(x)=infinity
Limit as x--> Positive 1 from the negative side f(x)=negative infinity
Limit as x--> negative 1 from the positive side f(x)= negative infinity
Limit as x--> negative 1 from the negative side f(x)=positive infinity
Step 4: Horizontal asymptote at 0--> function gets closer and closer to 0, but never reaches it
Questions:
#1:What is the vertical asymptote(s) of f(x)=(x-3)/((x^2)+3x)?
#2:Find the horizontal asymptote of (2(x^2)+2x-3)/(x+3).
#3: Graph the function: (x+2)(x-1)(x+3)/(x+4)(x+1) and determine zeros, asymptotes, and end behavior.
ANSWERS:
Question #1
#2:Find the horizontal asymptote of (2(x^2)+2x-3)/(x+3).
#3: Graph the function: (x+2)(x-1)(x+3)/(x+4)(x+1) and determine zeros, asymptotes, and end behavior.
ANSWERS:
Question #1
Answer above.
Question #2
Question #2
Answer above.
Question #3
Question #3
Step 1: Look at the zeros of the Denominator
-zeros are the vertical asymptote(s) of the function
Step 2: Look at the zeros of the numerator
Step 3:Look at the end behavior based on the asymptotes
-zeros are the vertical asymptote(s) of the function
Step 2: Look at the zeros of the numerator
Step 3:Look at the end behavior based on the asymptotes
Common Core:
f(x)=(2x-3)(3x+2)(x+7)(x-7)/(x^2-4)(x-3)(x)
---> Graph
--->State the zeros of the numerator, end behavior, and asymptote(s) of this function
Answers:
---> Graph
--->State the zeros of the numerator, end behavior, and asymptote(s) of this function
Answers:
Step 1: Find the zeros of the denominator (Make y=0 for each of them and solve for x)
Step 2: Asymptotes @ 3, 0, 2, and -2
Step 2: Asymptotes @ 3, 0, 2, and -2
Step 3: Find the zeros of the numerator (make y=0 for each of them and solve for x)
Step 4: zeros of the numerator are -7, 7, -2/3, and 3/2
Graph:
Step 4: zeros of the numerator are -7, 7, -2/3, and 3/2
Graph:
Step 5: Look at the end behavior (Simplify the leading terms---6 (means that as the function continues to negative or positive infinity, it will get closer and closer to 6))----Horizontal asymptote at 6
Answer above.
Answer above.