Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts.
|
|
- Godwin Bradford
- 7 years ago
- Views:
Transcription
1 Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts. C. Symmetry: even (f( x) = f(x)) or odd (f( x) = f(x)) function or neither, periodic function. ( ) ( ) D. Asymptotes: horizontal lim f(x) = L and vertical lim x a ±f(x) = ± asymptotes. E. Intervals of Increase or Decrease: Use the I/D Test (first derivative). F. Local Maximum and Minimum Values: Use critical numbers. G. Concavity and Points of Inflection: Compute f (x) and use the Concavity Test. H. Sketch the Curve: Using the information in items A-G, draw the graph. EXAMPLE: Use the guidelines to sketch the curve f(x) = x 1 x+2. Solution: A. Domain: (, 2) ( 2, ) B. Intercepts: (i) x-intercept (y = 0): x 1 x+2 (ii) y-intercept (x = 0): C. Symmetry: = 0 = x 1 = 0 = x = 1 = (1,0) y = = 1 2 (i) This function is not even, since (ii) This function is not odd, since (iii) This function is not periodic. = ( 0, 1 ) 2 f( 1) = = f(1) f( 1) = = f(1) 1
2 D. Asymptotes: (i) Horizontal asymptote: x 1 lim f(x) = lim x+2 = lim x 1 x x+2 x 1 1 x = lim 1+ 2 x = 1 = y = 1 is the horizontal asymptote (ii) Vertical asymptote: x 1 lim x 2 f(x) = lim x 2 x+2 = WORK: = = NOT SMALL SMALL = + BIG = Therefore x = 2 is the vertical asymptote. E. Intervals of Increase or Decrease: We have f (x) = ( ) x 1 = (x 1) (x+2) (x 1)(x+2) x+2 (x+2) 2 = 1 (x+2) (x 1) 1 (x+2) 2 = x+2 x+1 (x+2) 2 = 3 (x+2) 2 > It follows that this function increases on (, 2) and ( 2, ). F. Local Maximum and Minimum Values: This function has no critical numbers, therefore it has no local maximum and minimum values. G. Concavity and Points of Inflection: We have ( ) f 3 (x) = = ( 3(x+2) 2) ( ) = 3 (x+2) 2 = 6(x+2) 3 6 = (x+2) 2 (x+2) It follows that f (x) > 0 if x < 2 and f (x) < 0 if x > 2, therefore f(x) is concave upward on (, 2) and concave downward on ( 2, ) There are no inflection points, since x = 2 is not in the domain. H. Sketch the Curve: 2
3 H. Sketch the Curve: We have EXAMPLE: Use the guidelines to sketch the curve f(x) = 2x2 x 2 1. Solution: A. Domain: B. Intercepts: (i) x-intercepts: (ii) y-intercepts: C. Symmetry: (i) This function is even (?) (ii) This function is odd (?) (iii) This function is periodic (?) 3
4 D. Asymptotes: (i) Horizontal asymptotes: (ii) Vertical asymptotes: E. Intervals of Increase or Decrease: 4
5 F. Local Maximum and Minimum Values: G. Concavity and Points of Inflection: H. Sketch the Curve: 5
6 EXAMPLE: Use the guidelines to sketch the curve f(x) = 2x2 x 2 1. Solution: A. Domain: (, 1) ( 1,1) (1, ) B. Intercepts: (i) x-intercept (y = 0): (ii) y-intercept (x = 0): C. Symmetry: 2x 2 x 2 1 = 0 = 2x2 = 0 = x = 0 = (0,0) (i) This function is even, since y = = 0 = (0,0) f( x) = 2( x)2 ( x) 2 1 = 2x2 x 2 1 = f(x) (ii) This function is not odd, since it is even. (iii) This function is not periodic. D. Asymptotes: (i) Horizontal asymptote: lim 2x 2 x 2 1 = lim 2x 2 x 2 x 2 1 x 2 = lim 2x 2 x 2 x 2 x 2 1 x 2 2 = lim 1 1 = 2 = x 2 y = 2 is the horizontal asymptote (ii) Vertical asymptotes: WORK: 2x 2 lim x 1 + x 2 1 = 2(1.01) 2 (1.01) 2 1 = = + NOT SMALL + SMALL = + BIG = so x = 1 is the first vertical asymptote. Similarly, WORK: 2x 2 lim x 1 x 2 1 = 2( 1.01) 2 ( 1.01) 2 1 = = + NOT SMALL + SMALL = + BIG = so x = 1 is the second vertical asymptote. 6
7 E. Intervals of Increase or Decrease: We have ( ) 2x f 2 (x) = = (2x2 ) (x 2 1) 2x 2 (x 2 1) = 4x(x2 1) 2x 2 2x x 2 1 (x 2 1) 2 (x 2 1) 2 = 4x3 4x 4x 3 (x 2 1) 2 = 4x (x 2 1) It follows that this function increases on (, 1) and ( 1,0) and decreases on (0,1) and (1, ) F. Local Maximum and Minimum Values: x = 0 is the point of local maximum. G. Concavity and Points of Inflection: We have ( ) 4x f (x) = = ( 4x) (x 2 1) 2 ( 4x)((x 2 1) 2 ) (x 2 1) 2 ((x 2 1) 2 ) 2 = 4(x2 1) 2 ( 4x) 2(x 2 1) 2 1 (x 2 1) (x 2 1) 4 = 4(x2 1) 2 ( 4x) 2(x 2 1) 2x (x 2 1) 4 = 4(x2 1) ( 4x) 2 2x (x 2 1) 3 = 4x x 2 (x 2 1) 3 = 12x2 +4 (x 2 1) It follows that f (x) > 0 if x < 1 or x > 1 and f (x) < 0 if 1 < x < 1. Thus f(x) is concave upward on (, 1) and (1, ) and concave downward on ( 1,1) There are no inflection points, since x = ±1 are not in the domain. H. Sketch the Curve: 7
8 H. Sketch the Curve: We have EXAMPLE: Use the guidelines to sketch the curve f(x) = xe x2 /2. Solution: A. Domain: B. Intercepts: (i) x-intercepts: (ii) y-intercepts: C. Symmetry: (i) This function is even (?) (ii) This function is odd (?) (iii) This function is periodic (?) 8
9 D. Asymptotes: (i) Horizontal asymptotes: (ii) Vertical asymptotes: E. Intervals of Increase or Decrease: F. Local Maximum and Minimum Values: 9
10 G. Concavity and Points of Inflection: H. Sketch the Curve: 10
11 EXAMPLE: Use the guidelines to sketch the curve f(x) = xe x2 /2. Solution: A. Domain: (, ) B. Intercepts: (i) x-intercept (y = 0): (ii) y-intercept (x = 0): C. Symmetry: (i) This function is not even, since (ii) This function is odd, since (iii) This function is not periodic. D. Asymptotes: (i) Horizontal asymptote: lim /2 xe x2 = lim xe x2 /2 = 0 = x = 0 = (0,0) y = 0 e 02 /2 = 0 = (0,0) f( 1) = ( 1)e ( 1)2 /2 1 e 12 /2 = f(1) f( x) = ( x)e ( x)2 /2 = xe x2 /2 = f(x) x e x2 /2 = lim Therefore y = 0 is the horizontal asymptote. (ii) No vertical asymptotes. x (e x2 /2 ) = lim E. Intervals of Increase or Decrease: We have f (x) = 1 e x2 /2 (x 2 /2) = lim ( xe x2 /2) = x e x2 /2 +x(e x2 /2 ) = e x2 /2 +xe x2 /2 ) ( x2 2 1 e x2 /2 x = 0 = e x2 /2 +xe x2 /2 ( x) = (1 x 2 )e x2 / One can see that this function increases on (1, 1) and decreases on (, 1) and (1, ) F. Local Maximum and Minimum Values: x = 1 is local minimum and x = 1 is local maximum 11
12 G. Concavity and Points of Inflection: We have ( f (x) = (1 x 2 )e /2) ( ) x2 = (1 x 2 ) e x2 /2 +(1 x 2 ) e x2 /2 = 2xe x2 /2 +(1 x 2 )e x2 /2 = 2xe x2 /2 x(1 x 2 )e x2 /2 ) ( x2 2 = ( 2 1+x 2 )xe x2 /2 = (x 2 3)xe x2 / It follows that f (x) > 0 if 3 < x < 0 or x > 3 and f (x) < 0 if x < 3 or 0 < x < 3. Thus f(x) is and concave upward on ( 3,0) and ( 3, ) concave downward on (, 3) and (0, 3) There are three inflection points x = 0,± 3. 12
5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
More informationEquations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0
College Algebra Review Problems for Final Exam Equations #1-10 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve
More informationMA107 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 informationUnit 3 - Lesson 3. MM3A2 - Logarithmic Functions and Inverses of exponential functions
Math Instructional Framework Time Frame Unit Name Learning Task/Topics/ Themes Standards and Elements Lesson Essential Questions Activator Unit 3 - Lesson 3 MM3A2 - Logarithmic Functions and Inverses of
More informationAlgebra 2 Notes AII.7 Functions: Review, Domain/Range. Function: Domain: Range:
Name: Date: Block: Functions: Review What is a.? Relation: Function: Domain: Range: Draw a graph of a : a) relation that is a function b) relation that is NOT a function Function Notation f(x): Names the
More informationWeek 1: Functions and Equations
Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter
More informationSlope-Intercept Equation. Example
1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine
More informationPARABOLAS AND THEIR FEATURES
STANDARD FORM PARABOLAS AND THEIR FEATURES If a! 0, the equation y = ax 2 + bx + c is the standard form of a quadratic function and its graph is a parabola. If a > 0, the parabola opens upward and the
More information1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some
Section 3.1: First Derivative Test Definition. Let f be a function with domain D. 1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some open interval containing c. The number
More information2.5 Transformations of Functions
2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [
More informationSection 2.7 One-to-One Functions and Their Inverses
Section. One-to-One Functions and Their Inverses One-to-One Functions HORIZONTAL LINE TEST: A function is one-to-one if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More informationChapter 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 informationCalculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim
More information7.1 Graphs of Quadratic Functions in Vertex Form
7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called
More informationa. 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 informationMSLC 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 informationSection 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 informationMAT12X Intermediate Algebra
MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions
More informationPlot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.
Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line
More informationMA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
More informationLecture 3: Derivatives and extremes of functions
Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16
More informationRational Functions, Limits, and Asymptotic Behavior
Unit 2 Rational Functions, Limits, and Asymptotic Behavior Introduction An intuitive approach to the concept of a limit is often considered appropriate for students at the precalculus level. In this unit,
More informationWARM UP EXERCSE. 2-1 Polynomials and Rational Functions
WARM UP EXERCSE Roots, zeros, and x-intercepts. x 2! 25 x 2 + 25 x 3! 25x polynomial, f (a) = 0! (x - a)g(x) 1 2-1 Polynomials and Rational Functions Students will learn about: Polynomial functions Behavior
More information100. In general, we can define this as if b x = a then x = log b
Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,
More informationCalculus with Parametric Curves
Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS. Copyright Cengage Learning. All rights reserved.
3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions.
More information1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.
1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal
More informationQUADRATIC EQUATIONS AND FUNCTIONS
Douglas College Learning Centre QUADRATIC EQUATIONS AND FUNCTIONS Quadratic equations and functions are very important in Business Math. Questions related to quadratic equations and functions cover a wide
More informationLecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)
Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function
More informationList the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated
MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More informationSection 3.1 Quadratic Functions and Models
Section 3.1 Quadratic Functions and Models DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the
More informationSimplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationEXPONENTIAL FUNCTIONS 8.1.1 8.1.6
EXPONENTIAL FUNCTIONS 8.1.1 8.1.6 In these sections, students generalize what they have learned about geometric sequences to investigate exponential functions. Students study exponential functions of the
More informationPower functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd
5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts
More informationFor additional information, see the Math Notes boxes in Lesson B.1.3 and B.2.3.
EXPONENTIAL FUNCTIONS B.1.1 B.1.6 In these sections, students generalize what they have learned about geometric sequences to investigate exponential functions. Students study exponential functions of the
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More information1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most
More informationMore Quadratic Equations
More Quadratic Equations Math 99 N1 Chapter 8 1 Quadratic Equations We won t discuss quadratic inequalities. Quadratic equations are equations where the unknown appears raised to second power, and, possibly
More informationCalculus 1: Sample Questions, Final Exam, Solutions
Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.
More informationExamples 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 informationMathematics. 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 informationSome Lecture Notes and In-Class Examples for Pre-Calculus:
Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
More informationAP CALCULUS AB 2007 SCORING GUIDELINES (Form B)
AP CALCULUS AB 2007 SCORING GUIDELINES (Form B) Question 4 Let f be a function defined on the closed interval 5 x 5 with f ( 1) = 3. The graph of f, the derivative of f, consists of two semicircles and
More information1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
More information1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
More informationGraphs of Polar Equations
Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate
More informationBEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.
BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. I. GENERALITIES There are 3 common methods to solve quadratic inequalities. Therefore, students sometimes are confused to select the fastest and the best
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More informationis the degree of the polynomial and is the leading coefficient.
Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...
More informationMATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010
MATH 11 FINAL EXAM FALL 010-011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic
More informationCHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises
CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =
More informationSection 4.5 Exponential and Logarithmic Equations
Section 4.5 Exponential and Logarithmic Equations Exponential Equations An exponential equation is one in which the variable occurs in the exponent. EXAMPLE: Solve the equation x = 7. Solution 1: We have
More informationFunctions Modeling Change: A Precalculus Course. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Functions Modeling Change: A Precalculus Course Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 PREFACE This supplement consists of my lectures of a freshmen-level mathematics class offered
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationNotes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions.
Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions. Basic Functions In several sections you will be applying shifts
More informationProcedure 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 informationPre-Calculus Math 12 First Assignment
Name: Pre-Calculus Math 12 First Assignment This assignment consists of two parts, a review of function notation and an introduction to translating graphs of functions. It is the first work for the Pre-Calculus
More informationMATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationThe degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL 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
More informationPre Calculus Math 40S: Explained!
www.math0s.com 97 Conics Lesson Part I The Double Napped Cone Conic Sections: There are main conic sections: circle, ellipse, parabola, and hyperbola. It is possible to create each of these shapes by passing
More informationhttps://williamshartunionca.springboardonline.org/ebook/book/27e8f1b87a1c4555a1212b...
of 19 9/2/2014 12:09 PM Answers Teacher Copy Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Students
More information2-5 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 informationLIMITS 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 informationWriting the Equation of a Line in Slope-Intercept Form
Writing the Equation of a Line in Slope-Intercept Form Slope-Intercept Form y = mx + b Example 1: Give the equation of the line in slope-intercept form a. With y-intercept (0, 2) and slope -9 b. Passing
More informationExponential Functions. Exponential Functions and Their Graphs. Example 2. Example 1. Example 3. Graphs of Exponential Functions 9/17/2014
Eponential Functions Eponential Functions and Their Graphs Precalculus.1 Eample 1 Use a calculator to evaluate each function at the indicated value of. a) f ( ) 8 = Eample In the same coordinate place,
More informationIntroduction to Quadratic Functions
Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2
More informationSection 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 informationMA.8.A.1.2 Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem. Constant Rate of Change/Slope
MA.8.A.1.2 Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem Constant Rate of Change/Slope In a Table Relationships that have straight-lined graphs
More informationGraphing 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 informationInverse Functions and Logarithms
Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a one-to-one function if it never takes on the same value twice; that
More informationM 1310 4.1 Polynomial Functions 1
M 1310 4.1 Polynomial Functions 1 Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a, a,..., a, a, a n n1 2 1 0, be real numbers, with a
More informationGraphing Linear Equations in Two Variables
Math 123 Section 3.2 - Graphing Linear Equations Using Intercepts - Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the
More information6.2 Solving Nonlinear Equations
6.2. SOLVING NONLINEAR EQUATIONS 399 6.2 Solving Nonlinear Equations We begin by introducing a property that will be used extensively in this and future sections. The zero product property. If the product
More informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationFunctions: Piecewise, Even and Odd.
Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.
More information4.4 Transforming Circles
Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More information2.2 Derivative as a Function
2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x
More informationAverage rate of change of y = f(x) with respect to x as x changes from a to a + h:
L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationcorrect-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationGraphing calculators Transparencies (optional)
What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity Teacher Version Lesson Objective: Length of Activity: Students will: Recognize piecewise functions and the notation used
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationSingle Variable Calculus. Early Transcendentals
Single Variable Calculus Early Transcendentals This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License To view a copy of this license, visit http://creativecommonsorg/licenses/by-nc-sa/30/
More informationGetting to know your TI-83
Calculator Activity Intro Getting to know your TI-83 Press ON to begin using calculator.to stop, press 2 nd ON. To darken the screen, press 2 nd alternately. To lighten the screen, press nd 2 alternately.
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1
More informationLecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization
Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued
More informationPRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationPolynomial 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 informationf(x) = g(x), if x A h(x), if x B.
1. Piecewise Functions By Bryan Carrillo, University of California, Riverside We can create more complicated functions by considering Piece-wise functions. Definition: Piecewise-function. A piecewise-function
More informationKevin James. MTHSC 102 Section 1.5 Exponential Functions and Models
MTHSC 102 Section 1.5 Exponential Functions and Models Exponential Functions and Models Definition Algebraically An exponential function has an equation of the form f (x) = ab x. The constant a is called
More information0 0 such that f x L whenever x a
Epsilon-Delta Definition of the Limit Few statements in elementary mathematics appear as cryptic as the one defining the limit of a function f() at the point = a, 0 0 such that f L whenever a Translation:
More information