1.) ( ) Step 1: Factor the numerator and the denominator. Find the domain. is in lowest terms.
|
|
- Calvin Gardner
- 6 years ago
- Views:
Transcription
1 GP3-HW11 College Algebra Sketch the graph of each rational function. 1.) Step 1: Factor the numerator and the denominator. Find the domain. { } Step 2: Rewrite in lowest terms. The rational function is in lowest terms. Step 3: Find the y-intercept, if one exists. Find the x-intercept(s), if any. Plot the points on the graph.
2 Step 4: Find the vertical asymptote(s), if any. Graph each vertical asymptote using a dashed line. The vertical asymptote is the line.
3 Step 5: Find the horizontal asymptote or the oblique asymptote, if one exists. Determine the points, if any, at which the graph intersects this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph intersects the asymptote. The degree of the numerator is equal to the degree of the denominator. The horizontal asymptote is the ratio of the leading coefficients. Thus, the horizontal asymptote is the line.
4 Step 6: Use the zeros of the numerator and denominator to divide the x-axis into intervals. Determine where the graph is above or below the x-axis by choosing a number in each interval and evaluating the output of the function there. Plot the points found. 8
5 Step 7: Analyze the behavior of the graph near each asymptote indicate this behavior on the graph.
6 Step 8: Using the results obtained in Steps 1 thru 7 sketch the graph of the rational function.
7 2.) Step 1: Factor the numerator and the denominator. Find the domain. { } Step 2: Rewrite in lowest terms. The rational function is in lowest terms. Step 3: Find the y-intercept, if one exists. Find the x-intercept(s), if any. Plot the points on the graph.
8 Step 4: Find the vertical asymptote(s), if any. Graph each vertical asymptote using a dashed line.
9 Step 5: Find the horizontal asymptote or the oblique asymptote, if one exists. Determine the points, if any, at which the graph intersects this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph intersects the asymptote. The degree of the numerator is less than the degree of the denominator, so the horizontal asymptote is the line. We have shown in step 3 that the x-intercepts are its horizontal asymptote twice., so this rational function intersects
10 Step 6: Use the zeros of the numerator and denominator to divide the x-axis into intervals. Determine where the graph is above or below the x-axis by choosing a number in each interval and evaluating the output of the function there. Plot the points found.
11 Step 7: Analyze the behavior of the graph near each asymptote indicate this behavior on the graph.
12 Step 8: Using the results obtained in Steps 1 thru 7 sketch the graph of the rational function.
13 3.) Step 1: Factor the numerator and the denominator. Find the domain. Step 2: Rewrite in lowest terms. The rational function is in lowest terms. Step 3: Find the y-intercept, if one exists. Find the x-intercept(s), if any. Plot the points on the graph.
14 Step 4: Find the vertical asymptote(s), if any. Graph each vertical asymptote using a dashed line.
15 Step 5: Find the horizontal asymptote or the oblique asymptote, if one exists. Determine the points, if any, at which the graph intersects this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph intersects the asymptote. The degree of the numerator is greater than the degree of the denominator. It is top heavy by exactly one. There is no horizontal asymptote. There is an oblique asymptote.
16 Step 6: Use the zeros of the numerator and denominator to divide the x-axis into intervals. Determine where the graph is above or below the x-axis by choosing a number in each interval and evaluating the output of the function there. Plot the points found.
17 Step 7: Analyze the behavior of the graph near each asymptote indicate this behavior on the graph.
18 Step 8: Using the results obtained in Steps 1 thru 7 sketch the graph of the rational function.
19 4.) Step 1: Factor the numerator and the denominator. Find the domain. { } Step 2: Rewrite in lowest terms. The rational function is in lowest terms. Step 3: Find the y-intercept, if one exists. Find the x-intercept(s), if any. Plot the points on the graph.
20 Step 4: Find the vertical asymptote(s), if any. Graph each vertical asymptote using a dashed line.
21 Step 5: Find the horizontal asymptote or the oblique asymptote, if one exists. Determine the points, if any, at which the graph intersects this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph intersects the asymptote. The degree of the numerator is greater than the degree of the denominator, so there is no horizontal asymptote. Since its top heavy by exactly one, there is an oblique asymptote. The oblique asymptote is the line
22 Step 6: Use the zeros of the numerator and denominator to divide the x-axis into intervals. Determine where the graph is above or below the x-axis by choosing a number in each interval and evaluating the output of the function there. Plot the points found..25
23 Step 7: Analyze the behavior of the graph near each asymptote indicate this behavior on the graph. Step 8: Using the results obtained in Steps 1 thru 7 sketch the graph of the rational function.
Section Rational Functions and Inequalities. A rational function is a quotient of two polynomials. That is, is a rational function if
Section 6.1 --- Rational Functions and Inequalities A rational function is a quotient of two polynomials. That is, is a rational function if =, where and are polynomials and is not the zero polynomial.
More informationMath 121. Graphing Rational Functions Fall 2016
Math 121. Graphing Rational Functions Fall 2016 1. Let x2 85 x 2 70. (a) State the domain of f, and simplify f if possible. (b) Find equations for the vertical asymptotes for the graph of f. (c) For each
More informationCHAPTER 4: Polynomial and Rational Functions
171S 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 information2-3 Graphing Rational Functions
2-3 Graphing Rational Functions Factor What are the end behaviors of the Graph? Sketch a graph How to identify the intercepts, asymptotes and end behavior of a rational function. How to sketch the graph
More information16 Rational Functions Worksheet
16 Rational Functions Worksheet Concepts: The Definition of a Rational Function Identifying Rational Functions Finding the Domain of a Rational Function The Big-Little Principle The Graphs of Rational
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.7 Graphs of Rational Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze and
More information2-4 Graphing Rational Functions
2-4 Graphing Rational Functions Factor What are the zeros? What are the end behaviors? How to identify the intercepts, asymptotes, and end behavior of a rational function. How to sketch the graph of a
More information3.5D Graphing Rational Functions
3.5D Graphing Rational Functions A. Strategy 1. Find all asymptotes (vertical, horizontal, oblique, curvilinear) and holes for the function. 2. Find the and intercepts. 3. Plot the and intercepts, draw
More information5.2 Properties of Rational functions
5. Properties o Rational unctions A rational unction is a unction o the orm n n1 polynomial p an an 1 a1 a0 k k1 polynomial q bk bk 1 b1 b0 Eample 3 5 1 The domain o a rational unction is the set o all
More informationMath Sections 4.4 and 4.5 Rational Functions. 1) A rational function is a quotient of polynomial functions:
1) A rational function is a quotient of polynomial functions: 2) Explain how you find the domain of a rational function: a) Write a rational function with domain x 3 b) Write a rational function with domain
More informationCHAPTER 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 informationRational Functions HONORS PRECALCULUS :: MR. VELAZQUEZ
Rational Functions HONORS PRECALCULUS :: MR. VELAZQUEZ Definition of Rational Functions Rational Functions are defined as the quotient of two polynomial functions. This means any rational function can
More informationSection 5.1 Polynomial Functions & Models Polynomial Function
Week 8 Handout MAC 1105 Professor Niraj Wagh J Section 5.1 Polynomial Functions & Models Polynomial Function A polynomial function is of the form: f (x) = a n x n + a n 1 x n 1 +... + a 1 x 1 + a 0 where
More informationGraphing Techniques. Domain (, ) Range (, ) Squaring Function f(x) = x 2 Domain (, ) Range [, ) f( x) = x 2
Graphing Techniques In this chapter, we will take our knowledge of graphs of basic functions and expand our ability to graph polynomial and rational functions using common sense, zeros, y-intercepts, stretching
More informationObjectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function
SECTIONS 3.5: Rational Functions Objectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function I. Rational Functions A rational function is a
More informationx 16 d( x) 16 n( x) 36 d( x) zeros: x 2 36 = 0 x 2 = 36 x = ±6 Section Yes. Since 1 is a polynomial (of degree 0), P(x) =
9 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS Section -. Yes. Since is a polynomial (of degree 0), P() P( ) is a rational function if P() is a polynomial.. A vertical asymptote is a vertical line a that
More informationMath-3 Lesson 3-6 Analyze Rational functions The Oblique Asymptote
Math- Lesson - Analyze Rational functions The Oblique Asymptote Quiz: a What is the domain? b Where are the holes? c What is the vertical asymptote? y 4 8 8 a -, b = c = - Last time Zeroes of the numerator
More informationCollege Algebra. Fifth Edition. James Stewart Lothar Redlin Saleem Watson
College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson 4 Polynomial and Rational Functions 4.6 Rational Functions Rational Functions A rational function is a function of the form Px (
More informationRational Functions Video Lecture. Sections 4.4 and 4.5
Rational Functions Video Lecture Sections 4.4 and 4.5 Course Learning Objectives: 1)Demonstrate an understanding of functional attributes such as domain and range. Determine these attributes for a function
More informationx 2 + 3, r 4(x) = x2 1
Math 121 (Lesieutre); 4.2: Rational functions; September 1, 2017 1. What is a rational function? It s a function of the form p(x), where p(x) and q(x) are both polynomials. In other words, q(x) something
More information3.5. Rational Functions: Graphs, Applications, and Models. 3.5 Rational Functions: Graphs, Applications, and Models 3.6 Variation
3 Polynomial and Rational Functions 3 Polynomial and Rational Functions 3.5 Rational Functions: s, Applications, and Models 3.6 Variation Sections 3.5 3.6 2008 Pearson Addison-Wesley. All rights reserved
More information4.3 Rational Thinking
RATIONAL EXPRESSIONS & FUNCTIONS -4.3 4.3 Rational Thinking A Solidify Understanding Task The broad category of functions that contains the function!(#) = & ' is called rational functions. A rational number
More informationHonors Advanced Algebra Unit 4: Rational & Radical Relationships January 12, 2017 Task 18: Graphing Rational Functions
Honors Advanced Algebra Name Unit 4: Rational & Radical Relationships January 1, 017 Task 18: Graphing Rational Functions MGSE9 1.F.IF.7 Graph functions expressed symbolically and show key features of
More informationGraphing Rational Functions
Graphing Rational Functions Return to Table of Contents 109 Vocabulary Review x-intercept: The point where a graph intersects with the x-axis and the y-value is zero. y-intercept: The point where a graph
More informationThe Graph of a Rational Function. R x
Precalculus.7 Notes The Graph of a Rational Function Analyzing the Graph of a Rational Function 1. Completely factor the numerator and denominator.. List the key features of the graph. Domain: Set the
More informationDomain: The domain of f is all real numbers except those values for which Q(x) =0.
Math 1330 Section.3.3: Rational Functions Definition: A rational function is a function that can be written in the form P() f(), where f and g are polynomials. Q() The domain of the rational function such
More informationSection 4.4 Rational Functions and Their Graphs. 1, the line x = 0 (y-axis) is its vertical asymptote.
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, 16 is a rational function.
More information2-5 Rational Functions
Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any. 3. f (x) = The function is undefined at the real zeros of the denominator b(x) = (x + 3)(x 4). The real
More informationSection 4.4 Rational Functions and Their Graphs
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.
More information3.5. Rational Functions: Graphs, Applications, and Models
3.5 Rational Functions: s, Applications, and Models The Reciprocal Function The Function Asympototes Steps for ing Rational Functions Rational Function Models Copyright 2008 Pearson Addison-Wesley. All
More information3.7 Rational Functions. Copyright Cengage Learning. All rights reserved.
3.7 Rational Functions Copyright Cengage Learning. All rights reserved. Objectives Rational Functions and Asymptotes Transformations of y = 1/x Asymptotes of Rational Functions Graphing Rational Functions
More informationSection 2.3 (e-book 4.1 & 4.2) Rational Functions
Section 2.3 (e-book 4.1 & 4.2) Rational Functions Definition 1: The ratio of two polynomials is called a rational function, i.e., a rational function has the form, where both and are polynomials. Remark
More informationMath 1330 Section : Rational Functions Definition: A rational function is a function that can be written in the form f ( x ), where
2.3: Rational Functions P( x ) Definition: A rational function is a function that can be written in the form f ( x ), where Q( x ) and Q are polynomials, consists of all real numbers x such that You will
More information2.6: Rational Functions and Their Graphs
2.6: Rational Functions and Their Graphs Rational Functions are quotients of polynomial functions. The of a rational expression is all real numbers except those that cause the to equal. Example 1 (like
More informationThe domain of any rational function is all real numbers except the numbers that make the denominator zero or where q ( x)
We will look at the graphs of these functions, eploring their domain and end behavior. College algebra Class notes Rational Functions with Vertical, Horizontal, and Oblique Asymptotes (section 4.) Definition:
More information. 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 informationLogin your clickers & NO calculators. Get the 4.1 checkpoint from the brown table and answer the questions.
Login your clickers & NO calculators. Get the 4.1 checkpoint from the brown table and answer the questions. Nov 3 4:18 PM 1 Do you plan on doing chapter 3 test corrections? Yes No Nov 3 4:19 PM 1 Algebra
More informationRational functions, like rational numbers, will involve a fraction. We will discuss rational functions in the form:
Name: Date: Period: Chapter 2: Polynomial and Rational Functions Topic 6: Rational Functions & Their Graphs Rational functions, like rational numbers, will involve a fraction. We will discuss rational
More informationAlgebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational Functions; Task 5.3.2
1 TASK 5.3.2: FUNCTIONS AND THEIR QUOTIENTS Solutions 1. Graph the following functions and their quotient. (Hint: Put Function 1 in Y1=, Function 2 in Y2=, then make Y3= Y1/Y2. Change the graph style for
More informationLesson 2.4 Exercises, pages
Lesson. Eercises, pages 13 10 A 3. Sketch the graph of each function. ( - )( + 1) a) = b) = + 1 ( )( 1) 1 (- + )( - ) - ( )( ) 0 0 The function is undefined when: 1 There is a hole at 1. The function can
More information1) A rational function is a quotient of polynomial functions:
Math 165 - Sections 4.4 and 4.5 Rational Functions 1) A rational function is a quotient of polynomial functions: 2) Explain how you find the domain of a rational function: a) Write a rational function
More informationFinding Asymptotes KEY
Unit: 0 Lesson: 0 Discontinuities Rational functions of the form f ( are undefined at values of that make 0. Wherever a rational function is undefined, a break occurs in its graph. Each such break is called
More informationRATIONAL EQUATIONS AND FUNCTIONS
ALGEBRA II CHAPTER 9 NOTES RATIONAL EQUATIONS AND FUNCTIONS Name Algebra II 9. Graphing Simple Rational Functions Day One Today I am graphing simple rational functions. I am successful today when I can
More informationChapter 9 Review. By Charlie and Amy
Chapter 9 Review By Charlie and Amy 9.1- Inverse and Joint Variation- Explanation There are 3 basic types of variation: direct, indirect, and joint. Direct: y = kx Inverse: y = (k/x) Joint: y=kxz k is
More information= ( )= To find the domain, we look at the vertical asymptote(s) (where denominator equals zero) , =0
Precalculus College Algebra Review for Final Name It is also a good idea to go back through your old tests and quizzes to review. 1. Find (+1) given ()=3 +1 2. Determine () given ()=+2 and ()= (+1)=3(+1)
More informationExploring 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: Graph and analyze this function:
More informationf( x ), or a solution to the equation f( x) 0. You are already familiar with ways of solving
The Bisection Method and Newton s Method. If f( x ) a function, then a number r for which f( r) 0 is called a zero or a root of the function f( x ), or a solution to the equation f( x) 0. You are already
More information9.1A Exploring Rational Functions Using Transformations
PC 30 9.A Eploring Rational Functions Using Transformations To Identify and Sketch the graph of a Rational Function with Denominator Degree One Using Transformations. RATIONAL FUNCTION: A RATIONAL FUNCTION
More information2.3 Graph Sketching: Asymptotes and Rational Functions Math 125
.3 Graph Sketching: Asymptotes and Rational Functions Math 15.3 GRAPH SKETCHING: ASYMPTOTES AND RATIONAL FUNCTIONS All the functions from the previous section were continuous. In this section we will concern
More informationSection Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
More informationDate Lesson Text TOPIC Homework. Simplifying Rational Expressions Pg. 246 # 2-5, 7
UNIT RATIONAL FUNCTIONS EQUATIONS and INEQUALITIES Date Lesson Tet TOPIC Homework Oct. 7.0 (9).0 Simplifing Rational Epressions Pg. 6 # -, 7 Oct. 9. (0). Graphs of Reciprocal Functions Pg. #,,, doso, 6,
More informationSection 3.6 Rational Functions
Section 3.6 Rational Functions DEFINITION: A rational function is a function of the form r() = P() Q() where P() and Q() are polynomials with Q() 0. EXAMPLE: f() = 1, g() = 7 4+3, h() = 2 +5+11 2 4 + 3
More informationGRAPHING RATIONAL FUNCTIONS DAY 2 & 3. Unit 12
1 GRAPHING RATIONAL FUNCTIONS DAY 2 & 3 Unit 12 2 Warm up! Analyze the graph Domain: Range: Even/Odd Symmetry: End behavior: Increasing: Decreasing: Intercepts: Vertical Asymptotes: Horizontal Asymptotes:
More informationBegin Notes Immediately. Look at Example Below!!! Glue in Notebook
Begin Notes Immediately Look at Eample Below!!! Glue in Notebook Graphing Rational Functions The Parent Function can be transformed by using f( ) 1 f ( ) a k h What do a, h and k represent? a the vertical
More informationf (x ) ax b cx d Solving Rational Equations Pg. 285 # 1, 3, 4, (5 7)sodo, 11, 12, 13
UNIT RATIONAL FUNCTIONS EQUATIONS and INEQUALITIES Date Lesson Tet TOPIC Homework Oct. 7.0 (9).0 Simplifing Rational Epressions Pg. 6 # -, 7 Oct. 8. (0). Graphs of Reciprocal Functions Pg. #,,, doso, 6,
More information3.6-Rational Functions & Their Graphs
.6-Rational Functions & Their Graphs What is a Rational Function? A rational function is a function that is the ratio of two polynomial functions. This definition is similar to a rational number which
More informationa) y = x 3 + 3x 2 2 b) = UNIT 4 CURVE SKETCHING 4.1 INCREASING AND DECREASING FUNCTIONS
UNIT 4 CURVE SKETCHING 4.1 INCREASING AND DECREASING FUNCTIONS We read graphs as we read sentences: left to right. Plainly speaking, as we scan the function from left to right, the function is said to
More informationGoal: Graph rational expressions by hand and identify all important features
Goal: Graph rational expressions by hand and identify all important features Why are we doing this? Rational expressions can be used to model many things in our physical world. Understanding the features
More informationAlbertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.
Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2017 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the
More informationAlgebra Domains of Rational Functions
Domains of Rational Functions Rational Expressions are fractions with polynomials in both the numerator and denominator. If the rational expression is a function, it is a Rational Function. Finding the
More informationRadical and Rational Function Exam Questions
Radical and Rational Function Exam Questions Name: ANSWERS 2 Multiple Choice 1. Identify the graph of the function x y. x 2. Given the graph of y f x, what is the domain of x f? a. x R b. 2 x 2 c. x 2
More informationAH Properties of Functions.notebook April 19, 2018
Functions Rational functions are of the form where p(x) and q(x) are polynomials. If you can sketch a function without lifting the pencil off the paper, it is continuous. E.g. y = x 2 If there is a break
More informationSection 2-7. Graphs of Rational Functions
Section 2-7 Graphs of Rational Functions Section 2-7 rational functions and domain transforming the reciprocal function finding horizontal and vertical asymptotes graphing a rational function analyzing
More informationNumerator Degree < Denominator Degree
Polynomial, Radical, and Rational Functions Eample 1 Numerator Degree < Denominator Degree Predict if any asymptotes or holes are present in the graph of each rational function. Use a graphing calculator
More informationSection 5.4 Properties of Rational Functions
Rational Function A rational function is a function of the form R(xx) = P(xx), where P(xx)and Q(xx) are polynomial Q(xx) functions and Q(xx) 0. Domain is the set of all real numbers xx except the value(s)
More informationVertical Line Test a relationship is a function, if NO vertical line intersects the graph more than once
Algebra 2 Chapter 2 Domain input values, X (x, y) Range output values, Y (x, y) Function For each input, there is exactly one output Example: Vertical Line Test a relationship is a function, if NO vertical
More informationPRECALCULUS I/MATH 126 (2188) SHANNON MYERS
PRECALCULUS I/MATH 126 (2188) SHANNON MYERS π 100 POINTS POSSIBLE π YOUR WORK MUST SUPPORT YOUR ANSWER FOR FULL CREDIT TO BE AWARDED π YOU MAY USE A SCIENTIFIC AND/OR A TI-83/84/85/86 CALCULATOR π PROVIDE
More informationGraph Sketching. Review: 1) Interval Notation. Set Notation Interval Notation Set Notation Interval Notation. 2) Solving Inequalities
Lesson. Graph Sketching Review: ) Interval Notation Set Notation Interval Notation Set Notation Interval Notation a) { R / < < 5} b) I (, 3) ( 3, ) c){ R} d) I (, ] (0, ) e){ R / > 5} f) I [ 3,5) ) Solving
More informationName: Rational Functions 2.1H. Set Topic: Simplifying rational expressions & operations on rational expressions
Name: Rational Functions 2.1H Ready, Set, Go! Ready Topic: Polynomial division Use division to determine if the given linear term is a factor of the polynomial. If it is a linear factor, then find the
More informationReview for Mastery Using Graphs and Tables to Solve Linear Systems
3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations
More informationThe x-intercept can be found by setting y = 0 and solving for x: 16 3, 0
y=-3/4x+4 and y=2 x I need to graph the functions so I can clearly describe the graphs Specifically mention any key points on the graphs, including intercepts, vertex, or start/end points. What is the
More informationslope rise run Definition of Slope
The Slope of a Line Mathematicians have developed a useful measure of the steepness of a line, called the slope of the line. Slope compares the vertical change (the rise) to the horizontal change (the
More informationWhat is the reasonable domain of this volume function? (c) Can there exist a volume of 0? (d) Estimate a maximum volume for the open box.
MA 15800 Lesson 11 Summer 016 E 1: From a rectangular piece of cardboard having dimensions 0 inches by 0 inches, an open bo is to be made by cutting out identical squares of area from each corner and,
More informationRadical Functions Review
Radical Functions Review Specific Outcome 3 Graph and analyze radical functions (limited to functions involving one radical) Acceptable Standard sketch and analyze (domain, range, invariant points, - and
More informationGraphs and transformations 4G
Graphs and transformations 4G a f(x + ) is a translation by one unit to the left. d A (0, ), B ( ),0, C (, 4), D (, 0) A (, ), B (0, 0), C (, 4), D (5, 0) e f(x) is a stretch with scale factor b f(x) 4
More informationAP Calculus Summer Review Packet
AP Calculus Summer Review Packet Name: Date began: Completed: **A Formula Sheet has been stapled to the back for your convenience!** Email anytime with questions: danna.seigle@henry.k1.ga.us Complex Fractions
More informationIntroduction to Rational Functions Group Activity 5 Business Project Week #8
MLC at Boise State 013 Defining a Rational Function Introduction to Rational Functions Group Activity 5 Business Project Week #8 f x A rational function is a function of the form, where f x and g x are
More informationPart I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each.
Math 106/108 Final Exam Page 1 Part I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each. 1. Factor completely. Do not solve. a) 2x
More informationThis handout will discuss three kinds of asymptotes: vertical, horizontal, and slant.
CURVE SKETCHING This is a handout that will help you systematically sketch functions on a coordinate plane. This handout also contains definitions of relevant terms needed for curve sketching. ASYMPTOTES:
More informationSection Functions. Function Notation. Is this a function?
Section 1-21 Functions and Their Properties Section 1-21 function definition and notation domain and range continuity increasing/decreasing boundedness local and absolute extrema symmetry asymptotes end
More informationCh. 8.7 Graphs of Rational Functions Learning Intentions: Identify characteristics of the graph of a rational function from its equation.
Ch. 8.7 Graphs of Rational Functions Learning Intentions: Identify characteristics of the graph of a rational function from its equation. Learn to write the equation of a rational function from its graph.
More informationEvaluating Rational Functions
. Name Date A C T I V I T Y 7 Evaluating Rational Functions A rational function is a function that can be expressed as f(x) = h(x) g(x), where h(x) and g(x) are both polynomials and g(x) is not the zero
More informationWorking with Rational Expressions
Working with Rational Expressions Return to Table of Contents 4 Goals and Objectives Students will simplify rational expressions, as well as be able to add, subtract, multiply, and divide rational expressions.
More informationSession 3. Rational and Radical Equations. Math 30-1 R 3. (Revisit, Review and Revive)
Session 3 Rational and Radical Equations Math 30-1 R 3 (Revisit, Review and Revive) Rational Functions Review Specific Outcome 14 Graph and analyze rational functions (limited to numerators and denominators
More informationMath 111 Lecture Notes
A rational function is of the form R() = p() q() where p and q are polnomial functions. A rational function is undefined where the denominator equals zero, as this would cause division b zero. The zeros
More informationUnit 1: Sections Skill Set
MthSc 106 Fall 2011 Calculus of One Variable I : Calculus by Briggs and Cochran Section 1.1: Review of Functions Unit 1: Sections 1.1 3.3 Skill Set Find the domain and range of a function. 14, 17 13, 15,
More informationAlgebra 2 Notes Name: Section 8.4 Rational Functions. A function is a function whose rule can be written as a of. 1 x. =. Its graph is a, f x
Algebra Notes Name: Section 8. Rational Functions DAY ONE: A function is a function whose rule can be written as a of two polynomials. The parent rational function is f. Its graph is a, which has two separate
More informationModule 12 Rational Functions and Rational Equations
MAC 1105 Module 12 Rational Functions and Rational Equations Learning Objective Upon completing this module, you should be able to: 1. Identify a rational function and state its domain. 2. Find and interpret
More informationMath Stuart Jones. 4.3 Curve Sketching
4.3 Curve Sketching In this section, we combine much of what we have talked about with derivatives thus far to draw the graphs of functions. This is useful in many situations to visualize properties of
More informationMAC What is a Rational Function? Module 12. Rational Functions and Rational Equations. Learning Objective
MAC 1105 Module 12 Rational Functions and Rational Equations Learning Objective Upon completing this module, you should be able to: 1. Identify a rational function and state its domain. 2. Find and interpret
More informationMath 370 Exam 1 Review Name. Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.
Math 370 Exam 1 Review Name Determine whether the relation is a function. 1) {(-6, 6), (-6, -6), (1, 3), (3, -8), (8, -6)} Not a function The x-value -6 corresponds to two different y-values, so this relation
More informationSection 4.3. Graphing Exponential Functions
Graphing Exponential Functions Graphing Exponential Functions with b > 1 Graph f x = ( ) 2 x Graphing Exponential Functions by hand. List input output pairs (see table) Input increases by 1 and output
More informationRational Functions. Definition A rational function can be written in the form. where N(x) and D(x) are
Rational Functions Deinition A rational unction can be written in the orm () N() where N() and D() are D() polynomials and D() is not the zero polynomial. *To ind the domain o a rational unction we must
More informationYou should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1
Name GRAPHICAL REPRESENTATION OF DATA: You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1 ) and (x, y ) is x1 x y1 y,.
More information7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) Goals: Graph rational functions Find the zeros and y-intercepts Identify and understand the asymptotes Match graphs to their equations
More informationAP Calculus AB Summer Review Packet
AP Calculus AB Summer Review Packet Mr. Burrows Mrs. Deatherage 1. This packet is to be handed in to your Calculus teacher on the first day of the school year. 2. All work must be shown on separate paper
More informationWalt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC
Walt Whitman High School SUMMER REVIEW PACKET For students entering AP CALCULUS BC Name: 1. This packet is to be handed in to your Calculus teacher on the first day of the school year.. All work must be
More informationGraphs of Exponential
Graphs of Exponential Functions By: OpenStaxCollege As we discussed in the previous section, exponential functions are used for many realworld applications such as finance, forensics, computer science,
More informationCURVE SKETCHING EXAM QUESTIONS
CURVE SKETCHING EXAM QUESTIONS Question 1 (**) a) Express f ( x ) in the form ( ) 2 f x = x + 6x + 10, x R. f ( x) = ( x + a) 2 + b, where a and b are integers. b) Describe geometrically the transformations
More informationWarm-Up. Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) 2) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,2)
Warm-Up Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) ) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,) 8.4 Graph and Write Equations of Ellipses What are the major parts of
More information