Algebra Tutorials!
 
 
   
Home
Rational Expressions
Graphs of Rational Functions
Solve Two-Step Equations
Multiply, Dividing; Exponents; Square Roots; and Solving Equations
LinearEquations
Solving a Quadratic Equation
Systems of Linear Equations Introduction
Equations and Inequalities
Solving 2nd Degree Equations
Review Solving Quadratic Equations
System of Equations
Solving Equations & Inequalities
Linear Equations Functions Zeros, and Applications
Rational Expressions and Functions
Linear equations in two variables
Lesson Plan for Comparing and Ordering Rational Numbers
LinearEquations
Solving Equations
Radicals and Rational Exponents
Solving Linear Equations
Systems of Linear Equations
Solving Exponential and Logarithmic Equations
Solving Systems of Linear Equations
DISTANCE,CIRCLES,AND QUADRATIC EQUATIONS
Solving Quadratic Equations
Quadratic and Rational Inequalit
Applications of Systems of Linear Equations in Two Variables
Systems of Linear Equations
Test Description for RATIONAL EX
Exponential and Logarithmic Equations
Systems of Linear Equations: Cramer's Rule
Introduction to Systems of Linear Equations
Literal Equations & Formula
Equations and Inequalities with Absolute Value
Rational Expressions
SOLVING LINEAR AND QUADRATIC EQUATIONS
Steepest Descent for Solving Linear Equations
The Quadratic Equation
Linear equations in two variables
   
Try the Free Math Solver or Scroll down to Resources!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Graphs of Rational Functions

Graph

1. Factor, state domain, and THEN reduce.
Factor:

Domain.

Cannot be reduced.

2. Plot intercepts. For x-intercepts, state whether graph touches (multiplicity is even) or
crosses (multiplicity is odd) x-axis.

y-intercept


 

x-intercepts

Touch or cross:
x = –2 has multiplicity odd so graph crosses here.
x = –3 has multiplicity odd so graph crosses here.

3. Draw vertical asymptotes (the zeros of the denominator).

x +1= 0
x = -1
x = -1

4. Draw horizontal asymptotes:
deg num = deg denom + 1 so use long division to find oblique asymptote:

So, there is an oblique asymptote at y = x + 4

5. Plot where the graph crosses a horizontal or oblique asymptote, if it does.

Find where R crosses horizontal or oblique asymptotes.

This has no solution so the graph does
NOT cross the oblique asymptote.

6. If needed, plot a few extra points.
R(–6) = –2.4
R(3) = 7.5


7. Connect the dots.

Graph:


1. Factor, state domain, and THEN reduce.

Since the (x + 1) factors cancel, there will be a hole at x = –1.
To find the y-value of the hole, find


2. Plot intercepts. For x-intercepts, state whether graph touches (multiplicity is even) or
crosses (multiplicity is odd) x-axis.

y-intercept:



x-intercepts.

Touch or cross:
x = –1 has multiplicity odd so graph crosses here.

3. Draw vertical asymptotes (the zeros of the denominator).
x -1= 0
x = 1
x = 1

4. Draw horizontal asymptotes:
Degree of p = degree of q so

5. Plot where the graph crosses a horizontal or oblique asymptote, if it does.
Function = asymptote
R(x) = y
Original function:

But, x = –1 is not in the domain so R does not cross the horizontal asymptote.
Reduced version:

This has no solution so R does not cross the horizontal asymptote.

6. If needed, plot a few extra points.
R(2) = 3


7. Connect the dots.
Note that we can see how the right side behaves by
looking at the multiplicity of the vertical asymptote.

It is odd so the graph must go up on one side and
down on the other.

If it had been even, then both sides of the graph
would have had to go up or down together.
Think of the graph of

versus
.

  Copyrights © 2005-2024