Name: where Nx ( ) and Dx ( ) are the numerator and

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Name: where Nx ( ) and Dx ( ) are the numerator and"

Transcription

1 Oblique and Non-linear Asymptote Activity Name: Prior Learning Reminder: Rational Functions In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form m n n1 n1 m1 m1 anx a x a0 m b x b x b 0 where an 0, b 0 and m and n are nonnegative integers. To make the notation a m little simpler, let s rename our rational function Nx ( ) f ( x) where Nx ( ) and are the numerator and denominator polynomials that define the rational function f ( x). The following chart provides an overview of how we determine vertical and horizontal asymptotes when dealing with rational functions. Nx ( ) Let f ( x) be a rational function, where Nx ( ) and have no common factors other than constants. The vertical asymptotes of f ( x) can be determined as follows: Finding vertical asymptotes Procedure Rational Function example Asymptotes of example Vertical asymptotes occur at each value of x which makes the denominator equal to 0. 15x 10 f ( x) 4 to 15x 10 f ( x) 4 simplifies 5(3x ) (3x)(3x) 5 3x Finding horizontal asymptotes Solving 3x 0yields x as a vertical 3 Note: 3x 0 does not yield a vertical asymptote, but does result in a hole in the graph at x. 3 Procedure Rational Function example Asymptote of example Whenever the degree of the numerator Nx ( ) is less than the degree of the denominator, the x-axis, which is the line 0, is a horizontal Whenever the degree of the numerator is equal to the degree of the denominator, the line a, where a and b are the b leading coefficients of the numerator and denominator, respectively, is the horizontal Whenever the degree of the numerator is greater than the degree of the denominator, there is no horizontal 15x 10 f ( x) has 1 as the 4 degree of the numerator and as the degree of the denominator. f ( x) has as the degree of the numerator and denominator. f ( x) 3 has as the degree of the numerator and 1 as the degree of the denominator. Since the degree of the numerator is less than the degree of the denominator, 0 is the horizontal Since the degrees of the numerator and denominator are the same and the leading coefficients of the numerator and denominator are 14 and 14 9, respectively, is the 9 horizontal Since the degree of the numerator is greater than the degree of the denominator, the function has no horizontal

2 Investigating Other Types of Asymptotes of Rational Functions Connection to Prior Learning: Rational Numbers Given r, use long division to find the quotient q and remainder r. Is equal to q 8 8 d that the divisor is the denominator. In this case, d = 8). Verify your answer., where d is the divisor? (Recall Extending to Rational Functions 1. Let s investigate a rational function where the degree of the numerator is one greater than the degree of the 3 denominator. For example, we can examine the rational function. a) Use either long division or synthetic division in the space provided to determine the quotient and remainder. (Hint: The remainder should be 45. If you don t get 45, raise your hand.) Quotient = Qx () Remainder = Rx ( ) 45 Numerator & Denominator of Rational Function Rational functions of the form Nx () Nx ( ) f ( x) can be written as Dx () ( ) ( ) ( ) N x R x y f x Q( x) D( x) D( x) where Qx ( ) and Rx ( ) are the quotient polynomial and the remainder polynomial, respectively, when we divide Nx ( ) by. Rewrite 3 ( ) in the form Q( x) Rx.

3 3 b) Let s examine the y values of the rational function and the y values of the quotient function y 16. To do this put the rational function in and the quotient function in. Next, set up your table by accessing and set the Indpnt variable to Ask so that you can input your x values into the table. Finally, select and enter the x values from the tables below, recording the corresponding y values. x x What can you say about the values in relative to the values in as x? 3 c) Graph the rational function and the quotient function y16 using your graphing calculator with the window settings below. Using two different colors, sketch the graphs of both in the box provided. x:[-5,5] scale:5 y:[-150,50] scale:5 d) Where do the graphs of the rational function and the quotient function look alike? Where do they look different?

4 3 e) Notice that as x the graph of approaches the graph of the quotient function y16. Since the quotient function s degree is one, its graph is a line. When the graph of a rational function approaches a non-horizontal line as x, that line is called an oblique asymptote of the rational function. For our rational 3 function, y = 16 is an equation of the oblique Knowing that can be expressed as 45 3 y 16, explain why the graph of must approach the line y16 as x. 45 (Hint: Think about what happens to as x.) f) In part e above you explained why the graphs are close to each other as x. Give one possible reason why the graphs are not close to each other near x 3? 3 g) In part e above you learned that the rational function has an oblique asymptote that was obtained by dividing the numerator of the rational function by the denominator of the rational function. Also, you learned that an oblique asymptote of a rational function is a non-horizontal line, so it is defined by a degree 1 polynomial. What must the relationship between the degrees of the numerator and the denominator of a rational function be in order for the rational function to have an oblique asymptote? Explain.

5 3 x 7x 13x 18. Now let s consider the rational function where the degree of the numerator is greater than x 1 the degree of the denominator. a) Use long division or synthetic division to determine the quotient polynomial Qx () and the remainder polynomial Rx (). Then write the function in the form of ( ) Q( x) Rx. Qx () Rx ( ) Rx () Q() x Dx () How does the degree of this quotient polynomial differ from the degree of the quotient polynomial in problem 1 a)? What could be the reason for this difference? b) As in 1b) examine the y values of the rational function and the quotient function Q( x ). Remember to let represent the rational function and the quotient function. x x What can you say about the values of the rational function relative to the values of the quotient function as x?

6 c) Graph the rational function and the quotient function using your graphing calculator with the settings below. Using two different colors, sketch the graphs of both in the box provided. x:[-10,10] scale:1 y:[-40,100] scale:0 3 x 7x 13x 18 d) Notice that as x the graph of approaches the graph of the quotient function x 1 Q( x ). This quotient function is a non-linear asymptote of our rational function. This asymptote is non-linear 3 x 7x 13x 18 because the quotient polynomial s degree is not 1. Knowing that can be expressed as x 1 3 x 7x 13x 18 Q( x), explain why the graph of must approach the graph of Q( x) as x 1 x 1 x. (Hint: If needed, refer to the hint in 1e.) 3. a) Given the rational function 5 3 x 5x x x1 find the following: Qx () R( x) 10x 40 Rx () Q() x Dx () The degree of the numerator is how much greater than the degree of the denominator? What is the degree of the quotient polynomial? Your answers to the previous two questions should be the same. Why should they be the same?

7 b) Graph the rational function and the quotient function using your graphing calculator with the settings below. Sketch the graphs of both in the box provided. x:[-8,8] scale:1 y:[-500,500] scale:100 c) What do the graphs in part b) suggest to you about the rational function and its corresponding quotient function? Explain. 4. Given the rational function 5 3 x 58x 00x x x 56x 56x 144. x1 x1 a) Use the above to determine the asymptote of the graph of (Note that the division has already been done for you) 5 3 x 58x 00x x 1 as x. b) Verify your answer by graphing the rational function and your asymptote in the box provided. x:[-6,6] scale:1 y:[-00,400] scale:50 5. Find the equation of the asymptote as x for each of the following rational functions. Graph each rational function and asymptote in a suitable window to verify your results. a) 4 3 x x x x x b) x x x c) 3 x x x 4 9 x 3

Exploring Rational Functions

Exploring Rational Functions Name Date Period Exploring Rational Functions Part I - The numerator is a constant and the denominator is a linear factor. 1. The parent function for rational functions is the reciprocal function: Graph

More information

Rational Functions and Their Graphs

Rational Functions and Their Graphs Objectives Rational Functions and Their Graphs Section.6 Find domain of rational functions. Use transformations to graph rational functions. Use arrow notation. Identify vertical asymptotes. Identify horizontal

More information

Precalculus A 2016 Graphs of Rational Functions

Precalculus A 2016 Graphs of Rational Functions 3-7 Precalculus A 2016 Graphs of Rational Functions Determine the equations of the vertical and horizontal asymptotes, if any, of each function. Graph each function with the asymptotes labeled. 1. ƒ(x)

More information

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

More information

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

More information

2.6 Rational Functions

2.6 Rational Functions 2.6 Rational Functions A rational function f(x) is a function which is the ratio of two polynomials, that is, f(x) = n(x) where n(x) and are polynomials. For example, f(x) = 3x2 x 4 x 2 is a rational function.

More information

Math Rational Functions

Math Rational Functions Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

More information

5.1 The Remainder and Factor Theorems; Synthetic Division

5.1 The Remainder and Factor Theorems; Synthetic Division 5.1 The Remainder and Factor Theorems; Synthetic Division In this section you will learn to: understand the definition of a zero of a polynomial function use long and synthetic division to divide polynomials

More information

Chapter 5: Rational Functions, Zeros of Polynomials, Inverse Functions, and Exponential Functions Quiz 5 Exam 4

Chapter 5: Rational Functions, Zeros of Polynomials, Inverse Functions, and Exponential Functions Quiz 5 Exam 4 Chapter 5: Rational Functions, Zeros of Polynomials, Inverse Functions, and QUIZ AND TEST INFORMATION: The material in this chapter is on Quiz 5 and Exam 4. You should complete at least one attempt of

More information

Graphing Rational Functions

Graphing Rational Functions Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function

More information

Solutions to Self-Test for Chapter 4 c4sts - p1

Solutions to Self-Test for Chapter 4 c4sts - p1 Solutions to Self-Test for Chapter 4 c4sts - p1 1. Graph a polynomial function. Label all intercepts and describe the end behavior. a. P(x) = x 4 2x 3 15x 2. (1) Domain = R, of course (since this is a

More information

Rational Functions 5.2 & 5.3

Rational Functions 5.2 & 5.3 Math Precalculus Algebra Name Date Rational Function Rational Functions 5. & 5.3 g( ) A function is a rational function if f ( ), where g( ) and h( ) are polynomials. h( ) Vertical asymptotes occur at

More information

Quadratic Equations and Inequalities

Quadratic Equations and Inequalities MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

More information

Chapter 2 Test Review

Chapter 2 Test Review Name Chapter 2 Test Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. If the following is a polynomial function, then state its degree and leading

More information

Rational Polynomial Functions

Rational Polynomial Functions Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of

More information

CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.

CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column. SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real

More information

Rational functions are defined for all values of x except those for which the denominator hx ( ) is equal to zero. 1 Function 5 Function

Rational functions are defined for all values of x except those for which the denominator hx ( ) is equal to zero. 1 Function 5 Function Section 4.6 Rational Functions and Their Graphs Definition Rational Function A rational function is a function of the form that h 0. f g h where g and h are polynomial functions such Objective : Finding

More information

Polynomial & Rational Functions

Polynomial & Rational Functions 4 Polynomial & Rational Functions 45 Rational Functions A function f is a rational function if there exist polynomial functions p and q, with q not the zero function, such that p(x) q(x) for all x for

More information

Official Math 112 Catalog Description

Official Math 112 Catalog Description Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A

More information

Sect 3.2 Synthetic Division

Sect 3.2 Synthetic Division 94 Objective 1: Sect 3.2 Synthetic Division Division Algorithm Recall that when dividing two numbers, we can check our answer by the whole number (quotient) times the divisor plus the remainder. This should

More information

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

More information

2.6 Graphs of Rational Functions

2.6 Graphs of Rational Functions 8 CHAPTER Polynomial, Power, and Rational Functions.6 Graphs of Rational Functions What you ll learn about Rational Functions Transformations of the Reciprocal Function Limits and Asymptotes Analyzing

More information

Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

More information

A.4 Polynomial Division; Synthetic Division

A.4 Polynomial Division; Synthetic Division SECTION A.4 Polynomial Division; Synthetic Division 977 A.4 Polynomial Division; Synthetic Division OBJECTIVES 1 Divide Polynomials Using Long Division 2 Divide Polynomials Using Synthetic Division 1 Divide

More information

Section 3.7 Rational Functions

Section 3.7 Rational Functions Section 3.7 Rational Functions A rational function is a function of the form where P and Q are polynomials. r(x) = P(x) Q(x) Rational Functions and Asymptotes The domain of a rational function consists

More information

2-5 Rational Functions

2-5 Rational Functions -5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4

More information

Situation: Dividing Linear Expressions

Situation: Dividing Linear Expressions Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product

More information

Limits at Infinity Limits at Infinity for Polynomials Limits at Infinity for the Exponential Function Function Dominance More on Asymptotes

Limits at Infinity Limits at Infinity for Polynomials Limits at Infinity for the Exponential Function Function Dominance More on Asymptotes Lecture 5 Limits at Infinity and Asymptotes Limits at Infinity Horizontal Asymptotes Limits at Infinity for Polynomials Limit of a Reciprocal Power The End Behavior of a Polynomial Evaluating the Limit

More information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Math 110 Review for Final Examination 2012 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the equation to the correct graph. 1) y = -

More information

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4. _.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

More information

3.3. GRAPHS OF RATIONAL FUNCTIONS. Some of those sketching aids include: New sketching aids include:

3.3. GRAPHS OF RATIONAL FUNCTIONS. Some of those sketching aids include: New sketching aids include: 3.3. GRAPHS OF RATIONAL FUNCTIONS In a previous lesson you learned to sketch graphs by understanding what controls their behavior. Some of those sketching aids include: y-intercept (if any) x-intercept(s)

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions

Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions A) Vertical Asymptotes A rational function, in lowest terms, will have vertical asymptotes at the real zeros of the denominator

More information

PROBLEMS AND SOLUTIONS - RATIONAL FUNCTIONS

PROBLEMS AND SOLUTIONS - RATIONAL FUNCTIONS PROBLEMS AND SOLUTIONS - RATIONAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE THAT YOU CANNOT

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

Review for Calculus Rational Functions, Logarithms & Exponentials

Review for Calculus Rational Functions, Logarithms & Exponentials Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

More information

MINI LESSON. Lesson 5b Solving Quadratic Equations

MINI LESSON. Lesson 5b Solving Quadratic Equations MINI LESSON Lesson 5b Solving Quadratic Equations Lesson Objectives By the end of this lesson, you should be able to: 1. Determine the number and type of solutions to a QUADRATIC EQUATION by graphing 2.

More information

Dividing Polynomials

Dividing Polynomials 4.3 Dividing Polynomials Essential Question Essential Question How can you use the factors of a cubic polynomial to solve a division problem involving the polynomial? Dividing Polynomials Work with a partner.

More information

Sec36NotesDone.notebook April 19, Sec. 3.6 Rational Functions and their Graphs. A rational function is of the form:

Sec36NotesDone.notebook April 19, Sec. 3.6 Rational Functions and their Graphs. A rational function is of the form: Sec. 3.6 Rational Functions and their Graphs A rational function is of the form: where P(x) and Q(x) are Polynomials The Domain of r(x) is all values of x where Q(x) is not equal to zero. The simplest

More information

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

2.4 Real Zeros of Polynomial Functions

2.4 Real Zeros of Polynomial Functions SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower

More information

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

More information

Rational Functions. In this chapter, you ll learn what a rational function is, and you ll learn how to sketch the graph of a rational function.

Rational Functions. In this chapter, you ll learn what a rational function is, and you ll learn how to sketch the graph of a rational function. Rational Functions n this chapter, you ll learn what a rational function is, and you ll learn how to sketch the graph of a rational function. Rational functions A rational function is a fraction of polynomials.

More information

3.2 The Factor Theorem and The Remainder Theorem

3.2 The Factor Theorem and The Remainder Theorem 3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

More information

Chapter 4 Fractions and Mixed Numbers

Chapter 4 Fractions and Mixed Numbers Chapter 4 Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.

More information

3. Power of a Product: Separate letters, distribute to the exponents and the bases

3. Power of a Product: Separate letters, distribute to the exponents and the bases Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same

More information

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005 Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions. (x 1)(x 2 + 1) Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

More information

CHAPTER 4. Test Bank Exercises in. Exercise Set 4.1

CHAPTER 4. Test Bank Exercises in. Exercise Set 4.1 Test Bank Exercises in CHAPTER 4 Exercise Set 4.1 1. Graph the quadratic function f(x) = x 2 2x 3. Indicate the vertex, axis of symmetry, minimum 2. Graph the quadratic function f(x) = x 2 2x. Indicate

More information

An Insight into Division Algorithm, Remainder and Factor Theorem

An Insight into Division Algorithm, Remainder and Factor Theorem An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the

More information

1. Use Properties of Exponents

1. Use Properties of Exponents A. Polynomials Polynomials are one of the most fundamental types of functions used in mathematics. They are very simple to use, primarily because they are formed entirely by multiplication (exponents are

More information

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if 1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

More information

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

1 Algebra - Partial Fractions

1 Algebra - Partial Fractions 1 Algebra - Partial Fractions Definition 1.1 A polynomial P (x) in x is a function that may be written in the form P (x) a n x n + a n 1 x n 1 +... + a 1 x + a 0. a n 0 and n is the degree of the polynomial,

More information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

More information

Dividing Polynomials VOCABULARY

Dividing Polynomials VOCABULARY - Dividing Polynomials TEKS FOCUS TEKS ()(C) Determine the quotient of a polynomial of degree three and degree four when divided by a polynomial of degree one and of degree two. TEKS ()(A) Apply mathematics

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

Solving Quadratic Equations by Completing the Square

Solving Quadratic Equations by Completing the Square 9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application

More information

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s).

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s). 1. Simplify each expression, and eliminate any negative exponent(s). a. b. c. 2. Simplify the expression. Assume that a and b denote any real numbers. (Assume that a denotes a positive number.) 3. Find

More information

+ k, and follows all the same rules for determining. y x 4. = + c. ( ) 2

+ k, and follows all the same rules for determining. y x 4. = + c. ( ) 2 The Quadratic Function The quadratic function is another parent function. The equation for the quadratic function is and its graph is a bowl-shaped curve called a parabola. The point ( 0,0) is called the

More information

Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships

Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships These materials are for nonprofit educational purposes

More information

Determinants can be used to solve a linear system of equations using Cramer s Rule.

Determinants can be used to solve a linear system of equations using Cramer s Rule. 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

More information

Main page. Given f ( x, y) = c we differentiate with respect to x so that

Main page. Given f ( x, y) = c we differentiate with respect to x so that Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching - asymptotes Curve sketching the

More information

Mathematics Book One. Polynomial, Radical, and Rational Functions Transformations and Operations Exponential and Logarithmic Functions

Mathematics Book One. Polynomial, Radical, and Rational Functions Transformations and Operations Exponential and Logarithmic Functions Mathematics 30- Book One Polynomial, Radical, and Rational Functions Transformations and Operations Exponential and Logarithmic Functions A workbook and animated series by Barry Mabillard Copyright 204

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

In this lesson you will learn to find zeros of polynomial functions that are not factorable.

In this lesson you will learn to find zeros of polynomial functions that are not factorable. 2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has

More information

Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if

Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if Section 2.1: Vertical and Horizontal Asymptotes Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if, lim x a f(x) =, lim x a x a x a f(x) =, or. + + Definition.

More information

PreCalculus Generic Notes by Scott Surgent. We are reacquainted with quadratic equations, this time as a function. A quadratic 2

PreCalculus Generic Notes by Scott Surgent. We are reacquainted with quadratic equations, this time as a function. A quadratic 2 Quadratic Functions We are reacquainted with quadratic equations, this time as a function. A quadratic function is of the form f ( x) = ax + bx + c, which is also called the general form. We know already

More information

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x). .7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

More information

3.4 Limits at Infinity - Asymptotes

3.4 Limits at Infinity - Asymptotes 3.4 Limits at Infinity - Asymptotes Definition 3.3. If f is a function defined on some interval (a, ), then f(x) = L means that values of f(x) are very close to L (keep getting closer to L) as x. The line

More information

If a product or quotient has an even number of negative factors, then its value is positive.

If a product or quotient has an even number of negative factors, then its value is positive. MATH 0 SOLVING NONLINEAR INEQUALITIES KSU Important Properties: If a product or quotient has an even number of negative factors, then its value is positive. If a product or quotient has an odd number of

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

Notes on Curve Sketching. B. Intercepts: Find the y-intercept (f(0)) and any x-intercepts. Skip finding x-intercepts if f(x) is very complicated.

Notes on Curve Sketching. B. Intercepts: Find the y-intercept (f(0)) and any x-intercepts. Skip finding x-intercepts if f(x) is very complicated. Notes on Curve Sketching The following checklist is a guide to sketching the curve y = f(). A. Domain: Find the domain of f. B. Intercepts: Find the y-intercept (f(0)) and any -intercepts. Skip finding

More information

Rational Functions ( )

Rational Functions ( ) Rational Functions A rational function is a function of the form r P Q where P and Q are polynomials. We assume that P() and Q() have no factors in common, and Q() is not the zero polynomial. The domain

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

8 Polynomials Worksheet

8 Polynomials Worksheet 8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems Factor each polynomial completely using the given factor and long division. 1. x 3 + 2x 2 23x 60; x + 4 So, x 3 + 2x 2 23x 60 = (x + 4)(x 2 2x 15). Factoring the quadratic expression yields x 3 + 2x 2

More information

Lesson 2.3 Exercises, pages 114 121

Lesson 2.3 Exercises, pages 114 121 Lesson.3 Eercises, pages 11 11 A. For the graph of each rational function below: i) Write the equations of an asmptotes. ii) State the domain. a) b) 0 6 8 8 0 8 16 i) There is no vertical asmptote. The

More information

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

More information

Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:

Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes

More information

Chapter R - Basic Algebra Operations (69 topics, due on 05/01/12)

Chapter R - Basic Algebra Operations (69 topics, due on 05/01/12) Course Name: College Algebra 001 Course Code: R3RK6-CTKHJ ALEKS Course: College Algebra with Trigonometry Instructor: Prof. Bozyk Course Dates: Begin: 01/17/2012 End: 05/04/2012 Course Content: 288 topics

More information

6.4 The Remainder Theorem

6.4 The Remainder Theorem 6.4. THE REMAINDER THEOREM 6.3.2 Check the two divisions we performed in Problem 6.12 by multiplying the quotient by the divisor, then adding the remainder. 6.3.3 Find the quotient and remainder when x

More information

SYNTHETIC DIVISION AND THE FACTOR THEOREM

SYNTHETIC DIVISION AND THE FACTOR THEOREM 628 (11 48) Chapter 11 Functions In this section Synthetic Division The Factor Theorem Solving Polynomial Equations 11.6 SYNTHETIC DIVISION AND THE FACTOR THEOREM In this section we study functions defined

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

Text: A Graphical Approach to College Algebra (Hornsby, Lial, Rockswold)

Text: A Graphical Approach to College Algebra (Hornsby, Lial, Rockswold) Students will take Self Tests covering the topics found in Chapter R (Reference: Basic Algebraic Concepts) and Chapter 1 (Linear Functions, Equations, and Inequalities). If any deficiencies are revealed,

More information

Sometimes it is easier to leave a number written as an exponent. For example, it is much easier to write

Sometimes it is easier to leave a number written as an exponent. For example, it is much easier to write 4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall

More information

This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0).

This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

Section 3.2 Polynomial Functions and Their Graphs

Section 3.2 Polynomial Functions and Their Graphs Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P(x) = 3, Q(x) = 4x 7, R(x) = x 2 +x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 +2x+4 (b)

More information

Analyzing Polynomial and Rational Functions

Analyzing Polynomial and Rational Functions Analyzing Polynomial and Rational Functions Raja Almukahhal, (RajaA) Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as

More information

Solutions for 2015 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama

Solutions for 2015 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama Solutions for 0 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama. For f(x = 6x 4(x +, find f(. Solution: The notation f( tells us to evaluate the

More information

Procedure for Graphing Polynomial Functions

Procedure for Graphing Polynomial Functions Procedure for Graphing Polynomial Functions P(x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information