About This Chapter
Rational and Radical Functions - Chapter Summary
Sorting out the differences between radical and rational functions could be easier than you think. Let our experienced instructors show you what a rational function is and provide you with strategies for finding its domain and range. They can also help you tell the differences between the vertical, horizontal and slant asymptotes found in their graphs and provide you with tips for finding zeros of these functions, determining their end behavior and locating their y-intercepts. The uses of inverse variation and the steps required to graph rational expressions are covered as well.
The second half of this chapter helps you perform some of the same tasks, but with radical functions. In addition to learning how to find their domain and range, you can practice graphing radical functions and solving radical equations. Lessons also show you how to graph square and cube root functions. Use this chapter to examine the following:
- Vertical asymptotes
- Horizontal asymptotes
- Slant asymptotes
- End behavior
- Inverse variation
Our chapter on rational and radical functions features entertaining video lessons lasting between five and ten minutes each. You can view them in any order and as many times as you like. The same thing goes for the matching transcripts, which are supplemented with links to text lessons that define key mathematics terminology. If you'd like to check your understanding of the material, each lesson also includes a self-assessment quiz that links back to video content.
1. Rational Function: Definition, Equation & Examples
Rational functions are an extremely useful type of function found in mathematics. They are functions that are fractions whose numerator and denominator are both polynomials. Learn all about them in this lesson!
2. Vertical Asymptotes: Definition & Rules
Vertical asymptotes are interesting mathematical phenomena that occur with certain functions. You will learn where you will see them, what they look like, and how to find them in this lesson.
3. How to Find Horizontal Asymptotes
In this lesson, we will learn rules and definitions that will enable us to find horizontal asymptotes. Through applying these rules and through examples, we will deepen our understanding of finding horizontal asymptotes.
4. Finding Slant Asymptotes of Rational Functions
Watch this video lesson to learn how you can find the slant asymptote to certain types of rational functions. Learn what you need to look for and how polynomial long division comes into play.
5. Finding Rational Zeros Using the Rational Zeros Theorem & Synthetic Division
After completing this lesson, you will know what the rational zeros theorem says. You will also know how to apply this theorem to find zeros of polynomial functions.
6. Inverse Variation: Definition, Equation & Examples
Have you heard of an inverse variation? Don't worry, most people haven't, but it's really quite simple. If you've ever ridden a bike, you've seen inverse variations in action. This lesson will fill you in - read on for details.
7. Graphing & Analyzing Rational Functions
After watching this video lesson, you will be able to graph a rational function by analyzing its horizontal and vertical asymptotes. Learn to find how the graph behaves around the asymptotes.
8. Analyzing the Graph of a Rational Function: Asymptotes, Domain, and Range
A rational function arises from the ratio of two polynomial expressions. The graphs of rational functions often have distinct characteristics. In this lesson, we look at how to analyze some of those characteristics.
9. Solving Radical Equations: Steps and Examples
Solving radical equations is not any more difficult than solving other algebraic equations. This lesson will show you how to solve equations containing a square root and give some real-world examples.
10. Graphing Square Root & Cube Root Functions
This lesson will briefly review the most basic square root and cube root functions. Then we will look at transformations of these basic functions and their corresponding algebraic operations that we can use to graph more involved square root and cube root functions.
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Other chapters within the TExES Mathematics 7-12 (235): Practice & Study Guide course
- About the TExES Math 7-12 Exam
- Real Numbers
- Mathematical Models
- Complex Numbers & the Complex Plane
- Number Theory
- Number Patterns
- Functions and Graphs
- Linear Functions
- Quadratic Functions & Polynomials
- Evaluating Piecewise & Composite Functions
- Inequalities and Absolute Values
- Exponentials & Logs
- The Unit Circle
- Trigonometric Functions
- Using a Scientific Calculator for Calculus
- Understanding Limits in Math
- Understanding Rate of Change
- Calculating Derivatives of Functions
- Derivatives and Graphs
- Optimization in Calculus
- Definite Integrals and Sums
- Integration Applications in Calculus
- Working with Measurement
- Finding Volume, Area & Perimeter
- Introduction to Proofs and Constructions
- Congruence and Similarity
- Real World Shapes
- Coordinate Geometry
- Understanding Transformations in Math
- Conic Sections
- Understanding Vectors
- Measuring & Displaying Data
- Data Distribution Overview
- Sampling in Statistics
- Distribution & Inference in Statistics
- Inference About a Mean
- Regression and Correlation
- Finding Probability
- Probability Distributions and Statistical Inference
- Experiments and Surveys
- Mathematical Process & Perspectives
- Teaching Strategies & Activities for the Math Classroom
- Differentiated Instructional Strategies for the Math Classroom
- Using Student Assessments in the Math Classroom
- TExES Mathematics 7-12 Flashcards