INSTRUCTORS Raffi Hovasapian John Zhu

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John Zhu

John Zhu

Rational Functions

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Table of Contents

Section 1: Functions
Definitions & Properties of Functions

11m 26s

Intro
0:00
Definition
0:28
Properties: Vertical Line Test
1:32
Domain
1:38
Range
1:59
Vertical Line test
2:19
Example 1
2:33
Example 2
3:10
Properties: Roots or Zeros
4:04
Finding the Root
4:16
Properties: Forms
5:12
Graphically
5:20
List
5:46
Equation
6:11
Function
6:38
Properties: Odd & Even
7:12
Even Function
7:14
Odd Function
8:25
Properties: Increasing & Decreasing
9:17
Increasing Function
9:22
Decreasing Function
10:21
Graphing

13m 58s

Intro
0:00
Manipulating
0:10
A in the Equation
0:39
B in the Equation
0:44
C & D in the Equation
0:49
Negative values
0:59
Example 1
1:17
Example 2
1:51
Example 3: Absolute Value Functions
3:43
Example 4
4:57
Example 5
6:17
Example 6
8:02
Example 7
9:10
Example 8
11:02
Example 9
11:47
Inverse Functions

6m 47s

Intro
0:00
Inverse
0:08
Definition
0:18
Example: Finding the Inverse
1:03
Example 2
2:29
Example 3
3:12
Example 4
4:41
Polynomial Functions

5m 4s

Intro
0:00
Types of Functions: Polynomials
0:07
No Domain Restrictions
0:12
No Discontinuities
0:19
Degree Test
0:31
Types of Functions: Polynomials
1:17
Leading Coefficient Test
1:33
Leading Coefficient Positive, Even Degree
1:54
Leading Coefficient Positive, Odd Degree
2:13
Leading Coefficient Negative Even Degree
2:34
Leading Coefficient Positive, Odd Degree
2:46
Examples: Types of Functions: Polynomials
3:03
Examples: Types of Functions: Polynomials
4:18
Trigonometric Functions

6m 45s

Intro
0:00
Types of Functions: Trigonometric
0:05
6 Functions To Be Familiar With
0:14
Example 1: SIN
1:38
Example 2: COS
3:22
Example 3: TAN
4:38
Inverse Trigonometric Functions

5m 58s

Intro
0:00
Types of Functions: Trigonometric- Inverse Trig Functions
0:07
Example: Inverse SIN of X
0:45
Example: Inverse Function
2:30
Example: Inverse TAN of X
4:42
Trigonometric Identities

17m 42s

Intro
0:00
Types of Functions: Trigonometric- Trig Identities
0:07
4 Identities
0:24
Pythagorean
0:28
Double Angle
1:10
Power Reducing
1:28
Sum or Difference
1:42
Couple More Identities
1:59
Negative Angle
2:04
Product to Sum
2:39
Example 1: Prove
3:00
Example 2: Simplify Expression
5:02
Example 3: Prove
5:56
Example 4: Prove
8:02
Example 5: Prove With TAN
12:43
Exponential Functions

5m 53s

Intro
0:00
Types of Functions: Exponentials
0:07
General Form
0:10
Special Exponential Function
0:17
Example 1: Using Exponential Properties
0:46
Example 2: Using Exponential Properties
1:58
Example 3: Using Trig Identities & Exponential Properties
3:16
Example 4: Using Exponential Properties
4:37
Logarithmic Functions

7m 8s

Intro
0:00
Types of Functions: Logarithmic
0:06
General Form
0:10
2 Special Logarithmic Func.
0:19
Euler's # / Natural Log
0:27
Logarithmic & Exponential Relationship
0:45
Log form
1:56
Properties
2:09
Example 1: Apply Basic Principle of Log Func.
3:05
Example 2: Use Properties
3:40
Example 3: Regular Log
5:16
Rational Functions

15m 36s

Intro
0:00
Types of Functions: Rational - Definition
0:06
Example 1: Graph Rational Func.
0:36
Example 2: Find Asymptotes of Func.
7:02
Example 3: Find Asymptotes of Func.
8:59
Example 4: Graph Rational Func.
11:08
Conic Sections

14m 58s

Intro
0:00
Types of Conic Sections
0:06
Parabolas
0:19
Circles
1:36
Ellipses
2:40
Hyperbolas
4:42
Complete the Square
6:40
Example: Conic Sections
9:08
Example 2: Conic Sections
10:59
Example 3: Graph Conic Sections
12:21
Section 2: Limits and Continuity
Limit Definition & Properties

7m 15s

Intro
0:00
Definition
0:06
Example: Limit
0:17
Properties
1:13
1st Property
1:21
2nd Property
1:34
Special Property
1:51
Limits
2:36
Explain Example
2:49
Limits Example
4:39
Limits Example
5:21
Solving Limits with Algebra

8m 1s

Intro
0:00
Solving Limits with Algebra
0:07
Example 1: Solve Algebraically
0:30
Solving Limits with Algebra, Example 2
2:28
Solving Limits with Algebra, Example 3
3:18
Solving Limits with Algebra, Example 4
4:56
Solving Limits with Algebra, Example 5
6:26
Rational Limit Rules

3m 16s

Intro
0:00
Rational Limit Rules
0:07
Review of Solving Problem Algebraically
0:08
Limit Rules
0:28
Rule 1
0:35
Rule 2
0:40
Rule 3
0:45
Rational Limit Rules
1:02
Applying 1st Rule
1:22
Rational Limit Rules
1:50
Applying 2nd Rule
2:09
Rational Limit Rules
2:26
Applying 3rd Rule
2:40
One Sided Limits

9m 57s

Intro
0:00
Types of Limits: One-Sided Limit Rules
0:06
Example
0:19
Applying Same Rule
0:34
Rule to Keep In Mind
0:52
Types of Limits: One-Sided Limit Example 1
1:12
Limit of x² From Negative Side
2:11
Types of Limits: One-Sided Limit, Example 2
2:27
Types of Limits: One-Sided Limit, Example 3
4:26
Types of Limits: One-Sided Limit, Example 4
5:47
One-Sided Limit Example: X with Even Degree Polynomial
7:00
One-Sided Limit Example: Entire Denominator Squared
8:09
Special Trigonometric Limits

8m 28s

Intro
0:00
Types of Limits: Special Trig Limits
0:07
Pre-set Rules
0:35
Special Trig Limits, Example 1
0:58
Special Trig Limits, Example 2
2:50
Special Trig Limits, Example 3
3:55
Special Trig Limits, Example 4: With More Degrees
4:57
Special Trig Limits, Example 5
6:21
Limits & Continuity

10m 14s

Intro
0:00
Definition
0:06
3 Rules: f(x) Is Continuous…
0:21
Example 1: Finding Continuity
1:06
Types of Discontinuity
2:44
Jump
2:52
Point
3:24
Essential (Asymptote)
3:47
Removable
4:17
Example 2: Continuity Examples
4:41
Example 3: Continuity Examples
6:13
Example 4: Locate & Identify Type of Discontinuities
8:00
Limits: Multiple Choice Practice

6m 16s

Intro
0:00
Problem 1
0:08
Problem 2
1:51
Problem 3
2:54
Problem 4
4:31
Section 3: Derivatives
Derivative Definition & Properties

4m 11s

Intro
0:00
Definition
0:09
Formal Definition
0:45
Difference Quotient
1:12
Basic Derivatives
1:16
Differentiability
2:54
Basic Rules of Differentiation

7m 7s

Intro
0:00
Basic Rules of Differentiation
0:09
Constant Rule
0:14
Constant Multiple Rule
1:10
Addition and Difference Rule
1:40
Example 1: Constant Rule
2:25
Example 2: Constant Multiple Rule
3:01
Example 3: Constant Multiple Rule
3:35
Example 4: Constant Rule
4:34
Example 5: Constant Multiple Rule
5:03
Example 6
5:33
Power Rule

7m 14s

Intro
0:00
Power Rule
0:07
Power Rule Definition
0:30
Example 1
1:11
Example 2
2:25
Example 3
3:05
Example 4
4:18
Example 5
5:13
Trigonometric Rules

7m 53s

Intro
0:00
Trigonometric Rules
0:07
COS X
0:38
Find Derivative
1:02
Example 1
2:46
Example 2: COS Function
3:09
Example 3: Composite Expression
3:54
Example 4: Sec Function
5:02
Example 5: CSC
5:33
Example 6L COT
6:42
Product Rule

11m 11s

Intro
0:00
Product Rule
0:07
Definition
0:20
Example 1
0:43
Example 2
2:11
Example 3
4:24
Example 4
5:24
Example 5
6:42
Example 6
7:51
Quotient Rule

16m 50s

Intro
0:00
Quotient Rule
0:07
Definition
0:30
Example 1
1:17
Example 2: With No X In Numerator
2:49
Example 3
4:30
Example 4: With Decimals
6:46
Example 5
8:53
Example 6: With Trig Functions
12:55
Chain Rule

19m 48s

Intro
0:00
Chain Rule
0:07
Definition
0:17
Example 1: Applying the Chain Rule
1:33
Example 2
4:25
Example 3
6:02
Example 4
9:25
Example 5
12:47
Example 6
15:27
Higher Order Derivatives

15m

Intro
0:00
Types of Derivatives: Higher Order Derivatives
0:07
1st Derivative / F Prime
0:19
2nd Derivative
0:25
3rd Derivative
0:32
Example 1
1:48
Example 2: Find 3rd Derivative
3:13
Example 3: Acceleration
4:25
Example 4
10:20
Example 5: 2nd Derivative
12:11
Derivatives of Exponential Functions

13m 14s

Intro
0:00
Types of Derivatives: Exponential Functions
0:08
Derivatives: Definition/ Formula
0:28
Example 1
1:25
Example 2
2:47
Example 3
4:13
Example 4
7:11
Example 5
9:23
Example 6
11:06
Derivatives of Logarithmic Functions

11m 30s

Intro
0:00
Types of Derivatives: Logarithmic Functions
0:06
Rule for Logarithmic Functions
0:28
Example 1
0:58
Example 2
3:10
Example 3
4:38
Example 4
7:18
Example 5
8:48
Example 6
9:38
Derivatives of Inverse Trigonometric Functions

16m 54s

Intro
0:00
Types of Derivatives: Inverse Trigonometric Functions
0:06
6 Fundamental Properties of Inverse Trigonometric Functions
0:38
Example 1
2:17
Example 2
3:41
Example 3
5:37
Example 4
7:24
Example 5
10:08
Implicit Differentiation

16m 53s

Intro
0:00
Implicit Differentiation: First Order
0:07
Example 1: Setting Up
0:45
Example 1: Solving
1:41
Implicit Differentiation: Second Order (Ex. 2)
4:55
Example 3: Implicit Differentiation
9:11
Example 4: Implicit Differentiation
9:56
Example 5: Implicit Differentiation With Double Derivative
12:46
Multiple Choice Practice: Derivatives

11m 7s

Intro
0:00
Practice Problem 1
0:09
Answer
3:24
Practice Problem 2
3:36
Answer
6:29
Practice Problem 3
6:42
Answer
8:39
Practice Problem 4
8:43
Answer
9:33
Practice Problem 5
9:41
Answer
10:40
Section 4: Applications of Derivatives
Tangent & Normal Lines

22m 36s

Intro
0:00
Tangent and Normal Lines
0:10
Definition
0:22
Example 1
0:55
Tangent and Normal Lines: Example 2
2:43
Tangent and Normal Lines
5:21
Example 3
5:35
Tangent and Normal Lines: Example 4
9:14
Tangent and Normal Lines: Example 5
12:27
Tangent and Normal Lines: Example 6
15:54
Tangent and Normal Lines: Example 7
19:05
Position Velocity & Acceleration

18m 42s

Intro
0:00
Position, Velocity, and Acceleration
0:10
Position Function
0:14
Velocity Function
0:34
Acceleration Function
1:01
Example 1
1:20
Example 2
6:31
Example Continue: Velocity When Acceleration is Zero
6:32
Example 3: Where Is Particle Changing Directions?
8:16
Example 4: Total Distance Traveled From 0 to 2 Second
11:09
Example 5: Ball Drop Problem
16:40
Related Rates

26m 22s

Intro
0:00
Related Rates
0:06
Finding Rate of Change: Organization & Big Picture
0:23
Example 2: Area of a Circle
1:17
Example 3: Spherical Volume Expanding
4:19
Example 4: Traveling Problem
7:57
Example 5: Square Increase
12:37
Example 6: Standard Related Rates Problem
16:59
Example 7: Standard Related Rates Problem
19:49
Minimum & Maximum

12m 22s

Intro
0:00
Extrema: First Derivative Test
0:09
Example 1
0:46
Example 2: Real World Application/ Cost Function
4:05
Example 3: Minimums & Maximums
7:10
Example 4: Find Critical Points
10:52
Concavity

11m 43s

Intro
0:00
Concavity: Second Derivative Test
0:06
Definition
0:34
Example 1
0:54
Example 2
2:51
Example 3
4:08
Example 4
5:52
Rolles Theorem

8m 28s

Intro
0:00
Rolle's Theorem
0:07
Conditions
0:11
Summary
0:41
Example 1
1:09
Example 2
3:08
Example 3
4:48
Mean Value Theorem

9m 39s

Intro
0:00
Mean Value Theorem
0:06
Rolle's Theorem
0:07
Mean Value Theorem Conditions
0:24
Mean Value Theorem Definition
0:36
Example 1
0:56
Example 2
2:44
Example 3
5:28
Example 4
7:15
Differentials

12m 25s

Intro
0:00
Differentials
0:08
1st Differential Formula
0:29
2nd Differential Formula
0:57
Example 1
1:06
Example 2
3:21
Example 3
5:49
Example 4
7:19
Example 5
9:06
Applications of Derivatives: Multiple Choice Practice

13m 21s

Intro
0:00
Practice Problem 1
0:10
Answer
1:57
Practice Problem 2
2:08
Answer
5:39
Practice Problem 3
5:45
Answer
9:59
Practice Problem 4
10:12
Answer
11:49
Practice Problem 5
11:52
Answer
13:00
Applications of Derivatives: Free Response Practice

10m 22s

Intro
0:00
Practice Problem 1
0:10
Slope
1:30
Tangent Line Equation
2:17
Absolute Minimum
2:24
2 Possible X Points With Minimums
3:15
One Interest Point
4:14
Concavity
4:33
Positive Value = Positive Concavity
4:10
Minimum Point
5:34
Absolute Minimum
6:18
Point(s) of Inflection
6:31
Definition
6:49
2 Points Of Inflection
9:59
Section 5: Integrals
Definition of Integrals

1m 8s

Intro
0:00
Definition
0:09
Definition
0:16
Example
0:20
Integrals of Power Rule

8m 50s

Intro
0:00
Power Rule
0:06
Example 1
0:25
Example 2
2:02
Example 3
2:54
Example 4
3:45
Example 5
4:49
Example 6
6:47
Integrals Basic Rules of Integration

9m 43s

Intro
0:00
Basic Rules of Integration
0:09
Constant Rule
0:22
Example 1
0:40
Addition and Difference Rule
1:40
Example 2
1:58
Example 3: Subtraction/ Difference Rule
2:47
Example 4
3:55
Example 5
5:19
Example 6
7:37
Trigonometric Rules of Integrals

8m 58s

Intro
0:00
Trigonometric Rules
0:09
Integral of SIN
0:38
Example 1: Integral of SIN
1:46
Example 2: Integral of COS
2:38
Example 3: With 2 terms of X
3:06
Example 4: Integral of SEC
4:15
Example 5: Integral of CSC
5:06
Example 6
6:18
Chain Rule

13m 59s

Intro
0:00
Chain Rule
0:07
Example 1
0:37
Example 2
3:17
Example 3
5:09
Example 4
7:53
Example 5
9:40
Example 6
11:39
Integrals of Exponential Functions

12m 52s

Intro
0:00
Types of Integrals: Exponential Functions
0:09
Rule 1
0:30
Rule 2
0:49
Example 1
1:11
Example 2
2:54
Example 3
4:19
Example 4
5:19
Example 5
7:37
Example 6
9:04
Integrals of Natural Logarithmic Functions

13m

Intro
0:00
Types of Integrals: Natural Log Functions
0:09
Example 1
0:49
Example 2
2:06
Example 3
4:01
Example 4
5:37
Example 5
7:30
Example 6
9:05
Integrals of Inverse Trigonometric Functions

8m 29s

Intro
0:00
Types of Integrals: Inverse Trig Functions
0:09
One Property
0:40
Example 1
1:19
Example 2
3:44
Example 3
4:53
Example 4
5:53
Integrals: Multiple Choice Practice

15m 37s

Intro
0:00
Problem 1
0:09
Answer
4:09
Problem 2
4:33
Answer
5:54
Problem 3
5:59
Answer
8:02
Problem 4
8:06
Answer
10:27
Problem 5
10:43
Answer
14:46
Section 6: Applications of Integrals
Fundamental Theorem of Calculus

15m 55s

Intro
0:00
Fundamental Theorem of Calculus: Properties
0:10
Definition of Integral
0:49
Example 1
1:14
Fundamental Theorem of Calculus: Properties
2:40
Rule 1
2:50
Rule 2
3:14
Rule 3
3:33
Rule 4
3:52
Example 2
4:07
Example 3
6:17
Example 4
9:31
Example 5
10:52
Example 6
13:34
Area Under A Curve

18m 34s

Intro
0:00
Area Under Curve
0:07
Definition of Integral
0:09
Left Endpoint
1:17
Right Endpoint
1:47
Midpoints
2:09
Example 1
2:40
Example 2
4:59
Example 3
8:48
Example 4
10:23
Example 5
12:30
Example 6
15:32
Reimann Sums

10m 35s

Intro
0:00
Reimann Sums
0:08
Definition
1:07
Example 1
2:48
Example 2
5:38
Example 3
7:21
Example 4
9:14
Trapezoid Rule

12m 46s

Intro
0:00
The Trapezoid Rule
0:09
Definition: Area Of A Trapezoid
0:26
Terms of Formula
1:35
Example 1
2:11
Example 2
4:29
Example 3
7:22
Example 4
10:01
Mean Value Theorem

11m 22s

Intro
0:00
Mean Value Theorem of Integration
0:06
Example 1
0:53
Example 2
2:29
Example 3
3:48
Example 4
6:02
Second Fundamental Theorem of Calculus

4m 44s

Intro
0:00
Second Fundamental Theorem of Calculus
0:07
Definition
0:39
Example 1
1:08
Example 2
2:07
Example 3
2:48
Example 4
3:23
Area Between Curves

16m 39s

Intro
0:00
Example 1
0:10
Example 2
3:00
Example 3
4:46
Example 4
8:22
Example 5
11:04
Example 6
13:09
Revolving Solids Washer Disk Methods

21m 9s

Intro
0:00
Revolving Solids Washer Disk Methods
0:11
Explanation
0:33
Formula
3:12
Example 1
3:42
Example 2
6:54
Example 3
9:29
Example 4
12:16
Example 5
15:35
Revolving Solids Cylindrical Shells Method

26m 46s

Intro
0:00
Revolving Solids: Cylindrical Shells Method
0:09
Volume Of A Solid
0:25
Formula
2:51
Example 1
2:56
Example 2
7:28
Example 3
11:39
Example 4
17:36
Example 5
21:45
Revolving Solids Known Cross Sections

27m 41s

Intro
0:00
Revolving Solids Known Cross Sections
0:08
Example 1
0:35
Example 2
6:01
Example 3
11:03
Example 4
17:29
Example 5
22:19
Differential Equations Eulers Method

17m 54s

Intro
0:00
Differential Equations
0:08
Example 1
0:30
Differential Equations: Euler's Method
2:33
Rules
2:39
Example 2
3:00
Example 3
5:42
Example 4
9:44
Example 5
14:14
Differential Equations Slope Fields

16m 30s

Intro
0:00
Slope Fields
0:08
What Are Slope Fields
0:21
Example 1
0:42
Example 2
6:30
Example 3
11:17
Application of Integrals: Multiple Choice Practice

14m 19s

Intro
0:00
Practice Problem 1
0:10
Answer
3:46
Practice Problem 2
3:49
Answer
6:20
Practice Problem 3
6:26
Answer
8:02
Practice Problem 4
8:07
Answer
10:58
Practice Problem 5
11:05
Answer
14:06
Application of Integrals: Free Response Practice

9m 14s

Intro
0:00
Problem 1
0:10
Part A
0:24
Part A: Solution
2:04
Part B
2:10
Problem 1, Part B Continue
2:23
Part B: Solution
6:15
Problem 1, Part C
6:58
Part C: Solution
12:40
Problem 2
12:52
Part A
13:02
Part A: Solution
15:34
Part B
16:03
Part B: Solution
18:48
Section 7: Sample AP Test
AP Calculus AB Practice test: Section 1: Multiple Choice Part 1

17m 50s

Intro
0:00
Problem 1
0:20
Problem 2
1:24
Problem 3
2:53
Problem 4
3:56
Problem 5
8:18
Problem 6
9:06
Problem 7
10:14
Problem 8
12:16
Problem 9
14:13
AP Calculus AB Practice test: Section 1: Multiple Choice Part 2

17m 32s

Intro
0:00
Problem 10
0:18
Problem 11
2:26
Problem 12
6:11
Problem 13
7:04
Problem 14
8:06
Problem 15
10:32
Problem 16
11:40
Problem 17
13:00
Problem 18
14:43
AP Calculus AB Practice test: Section 1: Multiple Choice Part 3

22m 14s

Intro
0:00
Problem 19
0:21
Problem 20
2:33
Problem 21
7:23
Problem 22
10:24
Problem 23
12:18
Problem 24
13:13
Problem 25
15:52
Problem 26
17:03
Problem 27
19:44
AP Calculus AB Practice test: Section 1: Multiple Choice Part 4

19m 35s

Intro
0:00
Problem 28
0:23
Problem 29
3:50
Problem 30
5:31
Problem 31
9:02
Problem 32
10:07
Problem 33
11:27
Problem 34
13:47
Problem 35
15:21
Problem 36
16:53
AP Calculus AB Practice test: Section 1: Multiple Choice Part 5

25m 43s

Intro
0:00
Problem 37
0:22
Problem 38
2:27
Problem 39
5:36
Problem 40
7:21
Problem 41
10:08
Problem 42
11:29
Problem 43
13:07
Problem 44
18:18
Problem 45
21:08
AP Calculus AB Practice Test: Section 2: Free Response Part 1

16m 50s

Intro
0:00
Problem 1, Part A
0:20
Problem 1, Part B
3:03
Problem 1, Part C
4:11
Problem 1, Part D
5:36
Problem 2, Part A
7:37
Problem 2, Part B
9:02
Problem 2, Part C
12:31
AP Calculus AB Practice Test: Section 2: Free Response Part 2

21m 36s

Intro
0:00
Problem 3, Part A
0:18
Problem 3, Part B
5:57
Problem 4, Part A
11:26
Problem 4, Part B
12:28
Problem 4, Part C
15:35
Problem 4, Part D
18:56
AP Calculus AB Practice Test: Section 2: Free Response Part 3

13m 39s

Intro
0:00
Problem 5, Part A
0:21
Problem 5, Part B
3:07
Problem 5, Part C
6:43
Problem 6
8:24
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Lecture Comments (14)

0 answers

Post by Nolan Zhang on April 9, 2015

If the degrees are the same, what I did was to divide the coefficients instead of their degrees. It worked out for a function like (17x^2-8x)/(4X^2+9), because if you divide their degrees, you will always get y=1 which appears not to be the case with the graph of that particular function. Please correct me if I'm wrong.(im talking about ex #3)

0 answers

Post by John Michael Musaazi on August 9, 2014

for the last example why is it (x-2)(x-5) instead of (x+2)(x-5)

1 answer

Last reply by: Adina Tung
Fri Apr 27, 2018 11:20 AM

Post by Julia Qiu on February 21, 2014

The graph of the first example that you draw is different from what I have done on the calculators. I think the behavior of your final function maybe have some problem.

0 answers

Post by John Michael Musaazi on January 8, 2014

How do you factor these equations and bring them down to a small number,am sorry i have not done math for a while.

0 answers

Post by Mirza Baig on December 18, 2013

I don't get it why do you guys look at your paper

2 answers

Last reply by: John Zhu
Mon Aug 12, 2013 8:52 PM

Post by Taylor Wright on August 11, 2013

In example 3:

Wouldn't there be another vertical asymptote at -4?

1 answer

Last reply by: John Zhu
Mon Aug 12, 2013 8:48 PM

Post by Taylor Wright on August 11, 2013

In example 2:


(4-x) does not equal -(x+4)

(4-x) = -(x-4)    but    -(x+4) = -4-x

2 answers

Last reply by: Angela Patrick
Sun Sep 1, 2013 2:27 PM

Post by Taylor Wright on August 11, 2013

How could there be a Horizontal Asymptote at y=0 if the point (1,0) is on the graph???

Rational Functions

  • Factor to get terms, where c is a constant
  • Numerator: zeros (x-intercepts)
  • Denominator: Asymptotes (undefined x values)
  • Plug-in values into function to get point of function

Rational Functions

Find the roots of x2 + 5x + 6
  • x2 + 5x + 6 = (x + 3)(x + 2)
  • (x + 3)(x + 2) = 0
x = −3, −2
Find the roots of 4x2 + 8x + 2
  • 4x2 + 8x + 2 = 2(2x2 + 4x + 1) = 2(2x + 1)2
Two identical roots, x = −[1/2], −[1/2]
Find the asymptote(s) of f(x) = [1/(4x2 + 8x + 2)]
We already found the roots on the previous problems. The asymptote is where the function is undefined. Here, it's when the denominator is equal to zero. We know that the denominator is equal to zero when x = −[1/2]. The function goes to positive infinity when approaching the root (−[1/2]) from the positive direction. The function also goes to positive infinity when approaching the root from the negative direction (The degree of the polynomial in the denominator is even).
Graph y = [1/(4x2 + 8x + 2)]
  • We already determined the asymptote behavior, what about the edge behavior? As x gets very large and positve, y approaches zero. As x gets very large and negative, y also approaches zero from the positive side.
Find the asymptote(s) of f(x) = [1/x]
The function is undefined at x = 0. The function goes to positive infinity when approaching zero from the positive direction. The function goes to negative infinity when approaching zero from the negative direction (Remember, the polynomial in the denominator is of odd degree).
Find the asymptote(s) of f(x) = tanx
  • tanx = [sinx/cosx]. The function is undefined when cosx = 0. cosx = 0 when x = ..., −[(π)/2], [(π)/2], [(3π)/2], [(5π)/2], ...
There are an infinite number of asymptotes for f(x) = tanx and those asymptotes are at odd integer multiples of [(π)/2]
Graph y = [x/(x2 + 10x + 25)] + [5/(x2 + 10x + 25)]
  • y = [x/(x2 + 10x + 25)] + [5/(x2 + 10x + 25)]
  • = [(x + 5)/(x2 + 10x + 25)]
  • = [(x + 5)/((x + 5)2)]
  • = [1/(x + 5)]
  • This is similar to the graph of y = [1/x], but it is shifted 5 to the left.
Graph y = [1/(x2 − 4)]
  • Even degree polynomial in the denominator. As x gets very large in either direction, y approaches zero. There are two asymptotes here, -2 and 2. That gives us 4 different cases for asymptotic behavior. Approaching -2 from the left, Approaching -2 from the right, approaching 2 from the left, and approaching 2 from the right. Plugging in values to test for these cases.
  • f(−2.001) ≈ 250
  • f(−1.999) ≈ −250
  • f(1.999) ≈ −250
  • f(2.001) ≈ 250
Graph y = [x/(x2 −4)]
  • Asymptotes are at same locations as previous problem, so we can find the asymptote behavior in a similar fashion
  • f(−2.001) ≈ −500
  • f(−1.999) ≈ 500
  • f(1.999) ≈ −500
  • f(2.001) ≈ 500
Graph f(x) = [1/lnx]
  • Remember the properties of lnx. Domain (0, ∞), ln1 = 0, and lnx is an increasing function. As x gets larger, f(x) approaches zero and f(x) has an asymptote at x = 1.
  • f(.999) ≈ −1000
  • f(1.001) ≈ 1000

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Rational Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Types of Functions: Rational - Definition 0:06
  • Example 1: Graph Rational Func. 0:36
  • Example 2: Find Asymptotes of Func. 7:02
  • Example 3: Find Asymptotes of Func. 8:59
  • Example 4: Graph Rational Func. 11:08

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