Chapter 2. Polynomial and Rational Functions. 2.6 Rational Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Size: px
Start display at page:

Download "Chapter 2. Polynomial and Rational Functions. 2.6 Rational Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc."

Transcription

1 Chapter Polynomial and Rational Functions.6 Rational Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1

2 Objectives: Find the domains of rational functions. Use arrow notation. Identify vertical asymptotes. Identify horizontal asymptotes. Use transformations to graph rational functions. Graph rational functions. Identify slant asymptotes. Solve applied problems involving rational functions. Copyright 014, 010, 007 Pearson Education, Inc.

3 Rational Functions Rational functions are quotients of polynomial functions. This means that rational functions can be expressed as px ( ) f( x) qx ( ) where p and q are polynomial functions and qx ( ) 0. The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero. Copyright 014, 010, 007 Pearson Education, Inc. 3

4 Example: Finding the Domain of a Rational Function Find the domain of the rational function: x 5 f( x). x 5 Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0. x 5 0 x 5 The domain of f consists of all real numbers except 5. Copyright 014, 010, 007 Pearson Education, Inc. 4

5 Example: Finding the Domain of a Rational Function (continued) Find the domain of the rational function: x 5 f( x). x 5 The domain of f consists of all real numbers except 5. We can express the domain in set-builder or interval notation. Domain of f = xx 5 Domain of f = (,5) (5, ) Copyright 014, 010, 007 Pearson Education, Inc. 5

6 Example: Finding the Domain of a Rational Function Find the domain of the rational function: x gx ( ). x 5 Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0. x 5 0 x 5 x 5 The domain of g consists of all real numbers except 5 or 5. Copyright 014, 010, 007 Pearson Education, Inc. 6

7 Example: Finding the Domain of a Rational Function (continued) Find the domain of the rational function: x gx ( ). x 5 The domain of g consists of all real numbers except 5 or 5. We can express the domain in set-builder or interval notation. Domain of g = xx 5, x 5 Domain of g = (, 5) ( 5,5) (5, ) Copyright 014, 010, 007 Pearson Education, Inc. 7

8 Example: Finding the Domain of a Rational Function Find the domain of the rational function: x 5 hx ( ). x 5 Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0. x 5 0 x 5 x 5i No real numbers cause the denominator of h(x) to equal 0. The domain of h consists of all real numbers. Copyright 014, 010, 007 Pearson Education, Inc. 8

9 Example: Finding the Domain of a Rational Function (continued) Find the domain of the rational function: x 5 hx ( ). x 5 The domain of h consists of all real numbers. We can write the domain in interval notation: Domain of h = (, ) Copyright 014, 010, 007 Pearson Education, Inc. 9

10 Arrow Notation We use arrow notation to describe the behavior of some functions. Copyright 014, 010, 007 Pearson Education, Inc. 10

11 Definition of a Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a. Copyright 014, 010, 007 Pearson Education, Inc. 11

12 Definition of a Vertical Asymptote (continued) The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a. Copyright 014, 010, 007 Pearson Education, Inc. 1

13 Locating Vertical Asymptotes px ( ) If f( x) is a rational function in which p(x) qx ( ) and q(x) have no common factors and a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f. Copyright 014, 010, 007 Pearson Education, Inc. 13

14 Example: Finding the Vertical Asymptotes of a Rational Function Find the vertical asymptotes, if any, of the graph of the rational function: x f( x). x 1 There are no common factors in the numerator and the denominator. x 1 0 x 1 x 1 The zeros of the denominator are 1 and 1. Thus, the lines x = 1 and x = 1 are the vertical asymptotes for the graph of f. Copyright 014, 010, 007 Pearson Education, Inc. 14

15 Example: Finding the Vertical Asymptotes of a Rational Function (continued) Find the vertical asymptotes, if any, of the graph of the rational function: x f( x). The zeros of the x 1 denominator are 1 and 1. Thus, x 1 the lines x = 1 and x = 1 are the vertical x 1 asymptotes for the graph of f. Copyright 014, 010, 007 Pearson Education, Inc. 15

16 Example: Finding the Vertical Asymptotes of a Rational Function Find the vertical asymptotes, if any, of the graph of the rational function: x 1 hx ( ). x 1 We cannot factor the denominator of h(x) over the real numbers. The denominator has no real zeros. Thus, the graph of h has no vertical asymptotes. Copyright 014, 010, 007 Pearson Education, Inc. 16

17 Definition of a Horizontal Asymptote Copyright 014, 010, 007 Pearson Education, Inc. 17

18 Locating Horizontal Asymptotes Copyright 014, 010, 007 Pearson Education, Inc. 18

19 Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function: 9x f( x). 3x 1 The degree of the numerator,, is equal to the degree of the denominator,. The leading coefficients of the numerator and the denominator, 9 and 3, are used to obtain the equation of the horizontal asymptote. 9 The equation of the horizontal asymptote is y 3 or y = 3. Copyright 014, 010, 007 Pearson Education, Inc. 19

20 Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function: 9x f( x). 3x 1 y 3 The equation of the horizontal asymptote is y = 3. Copyright 014, 010, 007 Pearson Education, Inc. 0

21 Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function: 9x gx ( ). 3x 1 The degree of the numerator, 1, is less than the degree of the denominator,. Thus, the graph of g has the x-axis as a horizontal asymptote. The equation of the horizontal asymptote is y = 0. Copyright 014, 010, 007 Pearson Education, Inc. 1

22 Example: Finding the Horizontal Asymptote of a Rational Function (continued) Find the horizontal asymptote, if any, of the graph of the rational function: 9x gx ( ). 3x 1 The equation of the horizontal asymptote is y = 0. y 0 Copyright 014, 010, 007 Pearson Education, Inc.

23 Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of 3 the rational function: 9x hx ( ). 3x 1 The degree of the numerator, 3, is greater than the degree of the denominator,. Thus the graph of h has no horizontal asymptote. Copyright 014, 010, 007 Pearson Education, Inc. 3

24 Example: Finding the Horizontal Asymptote of a Rational Function (continued) Find the horizontal asymptote, if any, of the graph of 3 the rational function: 9x hx ( ). 3x 1 The graph has no horizontal asymptote. Copyright 014, 010, 007 Pearson Education, Inc. 4

25 Basic Reciprocal Functions Copyright 014, 010, 007 Pearson Education, Inc. 5

26 Example: Using Transformations to Graph a Rational Function 1 Use the graph of f( x) to graph x Begin with f( x) 1 x We ve identified two points and the asymptotes. Vertical asymptote x = 0 ( 1,1) 1 gx ( ) 1. x (1,1) Horizontal asymptote y = 0 Copyright 014, 010, 007 Pearson Education, Inc. 6

27 Example: Using Transformations to Graph a Rational Function (continued) 1 Use the graph of f( x) to graph x 1 gx ( ) 1. x The graph will shift units to the left. Subtract from each x-coordinate. The vertical asymptote is now x = Vertical asymptote x = ( 3, 1) ( 1,1) Horizontal asymptote y = 0 Copyright 014, 010, 007 Pearson Education, Inc. 7

28 Example: Using Transformations to Graph a Rational Function (continued) 1 Use the graph of f( x) to graph x 1 gx ( ) 1. x The graph will shift 1 unit down. Subtract 1 from each y-coordinate. The horizontal asymptote is now y = 1. Vertical asymptote x = ( 3, ) ( 1,0) Horizontal asymptote y = 1 Copyright 014, 010, 007 Pearson Education, Inc. 8

29 Strategy for Graphing a Rational Function The following strategy can be used to graph ( ) f( x) px. qx ( ) where p and q are polynomial functions with no common factors. 1. Determine whether the graph of f has symmetry. f ( x) f( x) y-axis symmetry f ( x) f( x) origin symmetry. Find the y-intercept (if there is one) by evaluating f(0). Copyright 014, 010, 007 Pearson Education, Inc. 9

30 Strategy for Graphing a Rational Function (continued) The following strategy can be used to graph ( ) f( x) px. qx ( ) where p and q are polynomial functions with no common factors. 3. Find the x-intercepts (if there are any) by solving the equation p(x) = Find any vertical asymptote(s) by solving the equation q(x) = 0. Copyright 014, 010, 007 Pearson Education, Inc. 30

31 Strategy for Graphing a Rational Function (continued) px ( ) The following strategy can be used to graph f( x) qx ( ) where p and q are polynomial functions with no common factors. 5. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. Copyright 014, 010, 007 Pearson Education, Inc. 31

32 Strategy for Graphing a Rational Function (continued) px ( ) The following strategy can be used to graph f( x) qx ( ) where p and q are polynomial functions with no common factors. 7. Use the information obtained previously to graph the function between and beyond the vertical asymptotes. Copyright 014, 010, 007 Pearson Education, Inc. 3

33 Example: Graphing a Rational Function Graph: 3x 3 f( x). x Step 1 Determine symmetry. 3( x) 3 3x 3 f( x) ( x) x Because f( x) does not equal either f(x) or f(x), the graph has neither y-axis symmetry nor origin symmetry. Copyright 014, 010, 007 Pearson Education, Inc. 33

34 Example: Graphing a Rational Function (continued) Graph: 3x 3 f( x). x Step Find the y-intercept. Evaluate f(0). 3(0) 3 3 f (0) 0 Step 3 Find x-intercepts. 3x 3 0 3x 3 x 1 Copyright 014, 010, 007 Pearson Education, Inc. 34

35 Example: Graphing a Rational Function (continued) Graph: 3x 3 f( x). x Step 4 Find the vertical asymptote(s). x 0 x Step 5 Find the horizontal asymptote. The numerator and denominator of f have the same degree, 1. The horizontal asymptote is 3 y 3. 1 Copyright 014, 010, 007 Pearson Education, Inc. 35

36 Example: Graphing a Rational Function (continued) Graph: 3x 3 f( x). x Step 6 Plot points between and beyond each x-intercept and vertical asymptotes. The x-intercept is (1,0) The y-intercept is 3 0, x 1 We will evaluate f at x Copyright 014, 010, 007 Pearson Education, Inc x 3 x 5 x 3 vertical asymptote x =

37 Example: Graphing a Rational Function (continued) Graph: 3x 3 f( x). x Step 6 (continued) We evaluate f at the selected points. x f( x) 3x 3 x Copyright 014, 010, 007 Pearson Education, Inc. 37

38 Example: Graphing a Rational Function (continued) 3x 3 Graph: f( x). x Step 7 Graph the function vertical asymptote x = horizontal asymptote y = 3 Copyright 014, 010, 007 Pearson Education, Inc. 38

39 Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is one more than the degree of the denominator. px ( ) In general, if f( x), qx ( ) p and q have no common factors, and the degree of p is one greater than the degree of q, find the slant asymptotes by dividing q(x) into p(x). Copyright 014, 010, 007 Pearson Education, Inc. 39

40 Example: Finding the Slant Asymptotes of a Rational Function Find the slant asymptote of x 5x 7 f( x). x The degree of the numerator,, is exactly one more than the degree of the denominator, 1. To find the equation of the slant asymptote, divide the numerator by the denominator (x 5x 7) ( x ) x 1 x 1 5 Copyright 014, 010, 007 Pearson Education, Inc. 40

41 Example: Finding the Slant Asymptotes of a Rational Function Find the slant asymptote of (x 5x 7) ( x ) x 1 x The equation of the vertical slant asymptote is asymptote x = y = x 1. The graph of f(x) slant asymptote is shown. y = x 1 5 x 5x 7 f( x). x Copyright 014, 010, 007 Pearson Education, Inc. 41

42 Example: Application A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Write the cost function, C, of producing x wheelchairs. Cx ( ) 500, x Copyright 014, 010, 007 Pearson Education, Inc. 4

43 Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Write the average cost function, C, of producing x wheelchairs. 500, x Cx ( ) x Copyright 014, 010, 007 Pearson Education, Inc. 43

44 Example: Application A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Find and interpret C(1000). 500, x Cx ( ) x 500, (1000) C(1000) 1000 The average cost per wheelchair of producing 1000 wheelchairs per month is $ Copyright 014, 010, 007 Pearson Education, Inc. 44

45 Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Find and interpret C(10,000). 500, x Cx ( ) x 500, (10,000) C(10,000) ,000 The average cost per wheelchair of producing 10,000 wheelchairs per month is $450. Copyright 014, 010, 007 Pearson Education, Inc. 45

46 Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Find and interpret C(100,000). 500, x Cx ( ) x 500, (100,000) C(100,000) 100,000 The average cost per wheelchair of producing 100,000 wheelchairs per month is $ Copyright 014, 010, 007 Pearson Education, Inc. 46

47 Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. Find the horizontal asymptote for the average cost function, 500, x Cx ( ). The horizontal x asymptote 500, x 400x 500,000 Cx ( ) is y = 400 x x Copyright 014, 010, 007 Pearson Education, Inc. 47

48 Example: Application (continued) A company is planning to manufacture wheelchairs that are light, fast, and beautiful. The fixed monthly cost will be $500,000 and it will cost $400 to produce each radically innovative chair. The average cost function is 500, x Cx ( ) x The horizontal asymptote for this function is y = 400. Describe what this represents for the company. The cost per wheelchair approaches 400 as more wheelchairs are produced. Copyright 014, 010, 007 Pearson Education, Inc. 48

Mission 1 Simplify and Multiply Rational Expressions

Mission 1 Simplify and Multiply Rational Expressions Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following

More information

2.6. Graphs of Rational Functions. Copyright 2011 Pearson, Inc.

2.6. Graphs of Rational Functions. Copyright 2011 Pearson, Inc. 2.6 Graphs of Rational Functions Copyright 2011 Pearson, Inc. Rational Functions What you ll learn about Transformations of the Reciprocal Function Limits and Asymptotes Analyzing Graphs of Rational Functions

More information

Introduction. A rational function is a quotient of polynomial functions. It can be written in the form

Introduction. A rational function is a quotient of polynomial functions. It can be written in the form RATIONAL FUNCTIONS Introduction A rational function is a quotient of polynomial functions. It can be written in the form where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 2 In general,

More information

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,

More information

Section Properties of Rational Expressions

Section Properties of Rational Expressions 88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:

More information

Chapter 5B - Rational Functions

Chapter 5B - Rational Functions Fry Texas A&M University Math 150 Chapter 5B Fall 2015 143 Chapter 5B - Rational Functions Definition: A rational function is The domain of a rational function is all real numbers, except those values

More information

6.1 Polynomial Functions

6.1 Polynomial Functions 6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and

More information

To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions. Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

More information

Midterm Review. Name: Class: Date: ID: A. Short Answer. 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k.

Midterm Review. Name: Class: Date: ID: A. Short Answer. 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k. Name: Class: Date: ID: A Midterm Review Short Answer 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k. a) b) c) 2. Determine the domain and range of each function.

More information

Rational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions

Rational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W

More information

Vocabulary: I. Inverse Variation: Two variables x and y show inverse variation if they are related as. follows: where a 0

Vocabulary: I. Inverse Variation: Two variables x and y show inverse variation if they are related as. follows: where a 0 8.1: Model Inverse and Joint Variation I. Inverse Variation: Two variables x and y show inverse variation if they are related as follows: where a 0 * In this equation y is said to vary inversely with x.

More information

of multiplicity two. The sign of the polynomial is shown in the table below

of multiplicity two. The sign of the polynomial is shown in the table below 161 Precalculus 1 Review 5 Problem 1 Graph the polynomial function P( ) ( ) ( 1). Solution The polynomial is of degree 4 and therefore it is positive to the left of its smallest real root and to the right

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

3.7 Part 1 Rational Functions

3.7 Part 1 Rational Functions 7 Part 1 Rational Functions Rational functions are used in science and engineering to model complex equations in areas such as 1) fields and forces in physics, 2) electronic circuitry, 3) aerodynamics,

More information

6.1. Rational Expressions and Functions; Multiplying and Dividing. Copyright 2016, 2012, 2008 Pearson Education, Inc. 1

6.1. Rational Expressions and Functions; Multiplying and Dividing. Copyright 2016, 2012, 2008 Pearson Education, Inc. 1 6.1 Rational Expressions and Functions; Multiplying and Dividing 1. Define rational expressions.. Define rational functions and give their domains. 3. Write rational expressions in lowest terms. 4. Multiply

More information

Rational and Radical Functions. College Algebra

Rational and Radical Functions. College Algebra Rational and Radical Functions College Algebra Rational Function A rational function is a function that can be written as the quotient of two polynomial functions P(x) and Q(x) f x = P(x) Q(x) = a )x )

More information

3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,

More information

Chapter 2: Polynomial and Rational Functions

Chapter 2: Polynomial and Rational Functions Chapter 2: Polynomial and Rational Functions Section 2.1 Quadratic Functions Date: Example 1: Sketching the Graph of a Quadratic Function a) Graph f(x) = 3 1 x 2 and g(x) = x 2 on the same coordinate plane.

More information

Making Connections with Rational Functions and Equations

Making Connections with Rational Functions and Equations Section 3.5 Making Connections with Rational Functions and Equations When solving a problem, it's important to read carefully to determine whether a function is being analyzed (Finding key features) or

More information

Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).

Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10). MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and

More information

Exam 1. (2x + 1) 2 9. lim. (rearranging) (x 1 implies x 1, thus x 1 0

Exam 1. (2x + 1) 2 9. lim. (rearranging) (x 1 implies x 1, thus x 1 0 Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus I January 28, 2016 Name: Exam 1 1. Evaluate the it x 1 (2x + 1) 2 9. x 1 (2x + 1) 2 9 4x 2 + 4x + 1 9 = 4x 2 + 4x 8 = 4(x 1)(x

More information

Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions

Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions Date: Objectives: SWBAT (Simplify Rational Expressions) Main Ideas: Assignment: Rational Expression is an expression that can be written

More information

LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS

LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or

More information

. As x gets really large, the last terms drops off and f(x) ½x

. As x gets really large, the last terms drops off and f(x) ½x Pre-AP Algebra 2 Unit 8 -Lesson 3 End behavior of rational functions Objectives: Students will be able to: Determine end behavior by dividing and seeing what terms drop out as x Know that there will be

More information

Math 115 Spring 11 Written Homework 10 Solutions

Math 115 Spring 11 Written Homework 10 Solutions Math 5 Spring Written Homework 0 Solutions. For following its, state what indeterminate form the its are in and evaluate the its. (a) 3x 4x 4 x x 8 Solution: This is in indeterminate form 0. Algebraically,

More information

Horizontal and Vertical Asymptotes from section 2.6

Horizontal and Vertical Asymptotes from section 2.6 Horizontal and Vertical Asymptotes from section 2.6 Definition: In either of the cases f(x) = L or f(x) = L we say that the x x horizontal line y = L is a horizontal asymptote of the function f. Note:

More information

Chapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs

Chapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs 2.6 Limits Involving Infinity; Asymptotes of Graphs Chapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs Definition. Formal Definition of Limits at Infinity.. We say that

More information

MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.

MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)

More information

Polynomial Functions and Models

Polynomial Functions and Models 1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models

More information

2 the maximum/minimum value is ( ).

2 the maximum/minimum value is ( ). Math 60 Ch3 practice Test The graph of f(x) = 3(x 5) + 3 is with its vertex at ( maximum/minimum value is ( ). ) and the The graph of a quadratic function f(x) = x + x 1 is with its vertex at ( the maximum/minimum

More information

Reteach Multiplying and Dividing Rational Expressions

Reteach Multiplying and Dividing Rational Expressions 8-2 Multiplying and Dividing Rational Expressions Examples of rational expressions: 3 x, x 1, and x 3 x 2 2 x 2 Undefined at x 0 Undefined at x 0 Undefined at x 2 When simplifying a rational expression:

More information

Lesson 2.1: Quadratic Functions

Lesson 2.1: Quadratic Functions Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard

More information

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ

More information

Polynomial Degree Leading Coefficient. Sign of Leading Coefficient

Polynomial Degree Leading Coefficient. Sign of Leading Coefficient Chapter 1 PRE-TEST REVIEW Polynomial Functions MHF4U Jensen Section 1: 1.1 Power Functions 1) State the degree and the leading coefficient of each polynomial Polynomial Degree Leading Coefficient y = 2x

More information

Chapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc. Chapter Polynomial and Rational Functions.3 Polynomial Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Identify polynomial functions. Recognize characteristics

More information

1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.

1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph. Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope

More information

Rational Functions. Example 1. Find the domain of each rational function. 1. f(x) = 1 x f(x) = 2x + 3 x 2 4

Rational Functions. Example 1. Find the domain of each rational function. 1. f(x) = 1 x f(x) = 2x + 3 x 2 4 Rational Functions Definition 1. If p(x) and q(x) are polynomials with no common factor and f(x) = p(x) for q(x) 0, then f(x) is called a rational function. q(x) Example 1. Find the domain of each rational

More information

Section 4.1 Polynomial Functions and Models. Copyright 2013 Pearson Education, Inc. All rights reserved

Section 4.1 Polynomial Functions and Models. Copyright 2013 Pearson Education, Inc. All rights reserved Section 4.1 Polynomial Functions and Models Copyright 2013 Pearson Education, Inc. All rights reserved 3 8 ( ) = + (a) f x 3x 4x x (b) ( ) g x 2 x + 3 = x 1 (a) f is a polynomial of degree 8. (b) g is

More information

2.2. Limits Involving Infinity. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

2.2. Limits Involving Infinity. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.2 Limits Involving Infinity Copyright 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Finite Limits as x ± What you ll learn about Sandwich Theorem Revisited Infinite Limits as x a End

More information

PreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College

PreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 6 B) 14 C) 10 D) Does not exist

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 6 B) 14 C) 10 D) Does not exist Assn 3.1-3.3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it exists. 1) Find: lim x -1 6x + 5 5x - 6 A) -11 B) - 1 11 C)

More information

Section 0.2 & 0.3 Worksheet. Types of Functions

Section 0.2 & 0.3 Worksheet. Types of Functions MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2

More information

CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section Multiplying and Dividing Rational Expressions

CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section Multiplying and Dividing Rational Expressions Name Objectives: Period CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section 8.3 - Multiplying and Dividing Rational Expressions Multiply and divide rational expressions. Simplify rational expressions,

More information

Mini Lecture 2.1 Introduction to Functions

Mini Lecture 2.1 Introduction to Functions Mini Lecture.1 Introduction to Functions 1. Find the domain and range of a relation.. Determine whether a relation is a function. 3. Evaluate a function. 1. Find the domain and range of the relation. a.

More information

CHAPTER 4: Polynomial and Rational Functions

CHAPTER 4: Polynomial and Rational Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.

MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)

More information

Limits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes

Limits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition

More information

AP Calculus Summer Homework

AP Calculus Summer Homework Class: Date: AP Calculus Summer Homework Show your work. Place a circle around your final answer. 1. Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.

More information

Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal

Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Asymptote Example 2: Real-World Example: Use Graphs

More information

Chapter 2 Formulas and Definitions:

Chapter 2 Formulas and Definitions: Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)

More information

Advanced Math Quiz Review Name: Dec Use Synthetic Division to divide the first polynomial by the second polynomial.

Advanced Math Quiz Review Name: Dec Use Synthetic Division to divide the first polynomial by the second polynomial. Advanced Math Quiz 3.1-3.2 Review Name: Dec. 2014 Use Synthetic Division to divide the first polynomial by the second polynomial. 1. 5x 3 + 6x 2 8 x + 1, x 5 1. Quotient: 2. x 5 10x 3 + 5 x 1, x + 4 2.

More information

PreCalculus: Semester 1 Final Exam Review

PreCalculus: Semester 1 Final Exam Review Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain

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. Calculus I - Homework Chapter 2 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the graph is the graph of a function. 1) 1)

More information

MTH30 Review Sheet. y = g(x) BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE

MTH30 Review Sheet. y = g(x) BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH0 Review Sheet. Given the functions f and g described by the graphs below: y = f(x) y = g(x) (a)

More information

College Algebra Notes

College Algebra Notes Metropolitan Community College Contents Introduction 2 Unit 1 3 Rational Expressions........................................... 3 Quadratic Equations........................................... 9 Polynomial,

More information

Math 95 Practice Final Exam

Math 95 Practice Final Exam Part 1: No Calculator Evaluating Functions and Determining their Domain and Range The domain of a function is the set of all possible inputs, which are typically x-values. The range of a function is the

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational

More information

Math 1314 Lesson 1: Prerequisites. Example 1: Simplify and write the answer without using negative exponents:

Math 1314 Lesson 1: Prerequisites. Example 1: Simplify and write the answer without using negative exponents: Math 1314 Lesson 1: Prerequisites 1. Exponents 1 m n n n m Recall: x = x = x n x Example 1: Simplify and write the answer without using negative exponents: a. x 5 b. ( x) 5 Example : Write as a radical:

More information

College Algebra. George Voutsadakis 1. LSSU Math 111. Lake Superior State University. 1 Mathematics and Computer Science

College Algebra. George Voutsadakis 1. LSSU Math 111. Lake Superior State University. 1 Mathematics and Computer Science College Algebra George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 111 George Voutsadakis (LSSU) College Algebra December 2014 1 / 71 Outline 1 Higher Degree

More information

GUIDED NOTES 5.6 RATIONAL FUNCTIONS

GUIDED NOTES 5.6 RATIONAL FUNCTIONS GUIDED NOTES 5.6 RATIONAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identify

More information

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) x 8. C) y = x + 3 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) x 8. C) y = x + 3 2 Precalculus Fall Final Exam Review Name Date Period MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplify the expression. Assume that the variables

More information

Student: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed

More information

1.2. Functions and Their Properties. Copyright 2011 Pearson, Inc.

1.2. Functions and Their Properties. Copyright 2011 Pearson, Inc. 1.2 Functions and Their Properties Copyright 2011 Pearson, Inc. What you ll learn about Function Definition and Notation Domain and Range Continuity Increasing and Decreasing Functions Boundedness Local

More information

CHAPTER 4: Polynomial and Rational Functions

CHAPTER 4: Polynomial and Rational Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

Chapter 9 BLM Answers

Chapter 9 BLM Answers Chapter 9 BLM Answers BLM 9 Prerequisite Skills. a) x 5x4 x x x x x x 5x x 5x 5x 4x 4x 0 g(x) x 5x 4. a) g(5 g) g(f 4g) c) (x 5)(x ) d) (x )(x 5) e) (a )(a ) f) (x 0.)(x 0.) g) 5(xy )(xy ). a),, and h(x)

More information

Math 1120 Calculus Test 3

Math 1120 Calculus Test 3 March 27, 2002 Your name The first 7 problems count 5 points each Problems 8 through 11 are multiple choice and count 7 points each and the final ones counts as marked In the multiple choice section, circle

More information

Part I: Multiple Choice Questions

Part I: Multiple Choice Questions Name: Part I: Multiple Choice Questions. What is the slope of the line y=3 A) 0 B) -3 ) C) 3 D) Undefined. What is the slope of the line perpendicular to the line x + 4y = 3 A) -/ B) / ) C) - D) 3. Find

More information

Review 1. 1 Relations and Functions. Review Problems

Review 1. 1 Relations and Functions. Review Problems Review 1 1 Relations and Functions Objectives Relations; represent a relation by coordinate pairs, mappings and equations; functions; evaluate a function; domain and range; operations of functions. Skills

More information

Sect Complex Numbers

Sect Complex Numbers 161 Sect 10.8 - Complex Numbers Concept #1 Imaginary Numbers In the beginning of this chapter, we saw that the was undefined in the real numbers since there is no real number whose square is equal to a

More information

Homework 1. 3x 12, 61.P (x) = 3t 21 Section 1.2

Homework 1. 3x 12, 61.P (x) = 3t 21 Section 1.2 Section 1.1 Homework 1 (34, 36) Determine whether the equation defines y as a function of x. 34. x + h 2 = 1, 36. y = 3x 1 x + 2. (40, 44) Find the following for each function: (a) f(0) (b) f(1) (c) f(

More information

MHF4U. Advanced Functions Grade 12 University Mitchell District High School. Unit 3 Rational Functions & Equations 6 Video Lessons

MHF4U. Advanced Functions Grade 12 University Mitchell District High School. Unit 3 Rational Functions & Equations 6 Video Lessons MHF4U Advanced Functions Grade 12 University Mitchell District High School Unit 3 Rational Functions & Equations 6 Video Lessons Allow no more than 15 class days for this unit! This includes time for review

More information

Answers. 2. List all theoretically possible rational roots of the polynomial: P(x) = 2x + 3x + 10x + 14x ) = A( x 4 + 3x 2 4)

Answers. 2. List all theoretically possible rational roots of the polynomial: P(x) = 2x + 3x + 10x + 14x ) = A( x 4 + 3x 2 4) CHAPTER 5 QUIZ Tuesday, April 1, 008 Answers 5 4 1. P(x) = x + x + 10x + 14x 5 a. The degree of polynomial P is 5 and P must have 5 zeros (roots). b. The y-intercept of the graph of P is (0, 5). The number

More information

b n x n + b n 1 x n b 1 x + b 0

b n x n + b n 1 x n b 1 x + b 0 Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)

More information

Graphing Rational Functions

Graphing Rational Functions Unit 1 R a t i o n a l F u n c t i o n s Graphing Rational Functions Objectives: 1. Graph a rational function given an equation 2. State the domain, asymptotes, and any intercepts Why? The function describes

More information

f (x) = x 2 Chapter 2 Polynomial Functions Section 4 Polynomial and Rational Functions Shapes of Polynomials Graphs of Polynomials the form n

f (x) = x 2 Chapter 2 Polynomial Functions Section 4 Polynomial and Rational Functions Shapes of Polynomials Graphs of Polynomials the form n Chapter 2 Functions and Graphs Section 4 Polynomial and Rational Functions Polynomial Functions A polynomial function is a function that can be written in the form a n n 1 n x + an 1x + + a1x + a0 for

More information

Semester Review Packet

Semester Review Packet MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree

More information

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.6 Rational Functions and Asymptotes Copyright Cengage Learning. All rights reserved. What You Should Learn Find the

More information

( ) = 1 x. g( x) = x3 +2

( ) = 1 x. g( x) = x3 +2 Rational Functions are ratios (quotients) of polynomials, written in the form f x N ( x ) and D x ( ) are polynomials, and D x ( ) does not equal zero. The parent function for rational functions is f x

More information

Polynomial and Rational Functions. Chapter 3

Polynomial and Rational Functions. Chapter 3 Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30

More information

Calculus 221 worksheet

Calculus 221 worksheet Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function

More information

Lecture 7: Sections 2.3 and 2.4 Rational and Exponential Functions. Recall that a power function has the form f(x) = x r where r is a real number.

Lecture 7: Sections 2.3 and 2.4 Rational and Exponential Functions. Recall that a power function has the form f(x) = x r where r is a real number. L7-1 Lecture 7: Sections 2.3 and 2.4 Rational and Exponential Functions Recall that a power function has the form f(x) = x r where r is a real number. f(x) = x 1/2 f(x) = x 1/3 ex. Sketch the graph of

More information

Mathematics for Business and Economics - I. Chapter 5. Functions (Lecture 9)

Mathematics for Business and Economics - I. Chapter 5. Functions (Lecture 9) Mathematics for Business and Economics - I Chapter 5. Functions (Lecture 9) Functions The idea of a function is this: a correspondence between two sets D and R such that to each element of the first set,

More information

4.3 Division of Polynomials

4.3 Division of Polynomials 4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed

More information

Name Advanced Math Functions & Statistics. Non- Graphing Calculator Section A. B. C.

Name Advanced Math Functions & Statistics. Non- Graphing Calculator Section A. B. C. 1. Compare and contrast the following graphs. Non- Graphing Calculator Section A. B. C. 2. For R, S, and T as defined below, which of the following products is undefined? A. RT B. TR C. TS D. ST E. RS

More information

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2) Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements

More information

= lim. (1 + h) 1 = lim. = lim. = lim = 1 2. lim

= lim. (1 + h) 1 = lim. = lim. = lim = 1 2. lim Math 50 Exam # Solutions. Evaluate the following its or explain why they don t exist. (a) + h. h 0 h Answer: Notice that both the numerator and the denominator are going to zero, so we need to think a

More information

Practice Test - Chapter 2

Practice Test - Chapter 2 Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several

More information

Functions and Equations

Functions and Equations Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Euclid eworkshop # Functions and Equations c 006 CANADIAN

More information

Name: Class: Date: A. 70 B. 62 C. 38 D. 46

Name: Class: Date: A. 70 B. 62 C. 38 D. 46 Class: Date: Test 2 REVIEW Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Divide: (4x 2 49y 2 ) (2x 7y) A. 2x 7y B. 2x 7y C. 2x 7y D. 2x 7y 2. What is

More information

1. Definition of a Polynomial

1. Definition of a Polynomial 1. Definition of a Polynomial What is a polynomial? A polynomial P(x) is an algebraic expression of the form Degree P(x) = a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 3 x 3 + a 2 x 2 + a 1 x + a 0 Leading

More information

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14 Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)

More information

Practice Test - Chapter 2

Practice Test - Chapter 2 Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several

More information

10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions

10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The

More information

1.1 : (The Slope of a straight Line)

1.1 : (The Slope of a straight Line) 1.1 : (The Slope of a straight Line) Equations of Nonvertical Lines: A nonvertical line L has an equation of the form y mx b. The number m is called the slope of L and the point (0, b) is called the y-intercept.

More information

INTERNET MAT 117. Solution for the Review Problems. (1) Let us consider the circle with equation. x 2 + 2x + y 2 + 3y = 3 4. (x + 1) 2 + (y + 3 2

INTERNET MAT 117. Solution for the Review Problems. (1) Let us consider the circle with equation. x 2 + 2x + y 2 + 3y = 3 4. (x + 1) 2 + (y + 3 2 INTERNET MAT 117 Solution for the Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (i) Group

More information

UMUC MATH-107 Final Exam Information

UMUC MATH-107 Final Exam Information UMUC MATH-07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from

More information

(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)

(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks) 1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of

More information