Unit 2 Functions Continuity and End Behavior (Unit 2.3)
|
|
|
- Avis Hopkins
- 5 years ago
- Views:
Transcription
1 Unit 2 Functions Continuity and End Behavior (Unit 2.3) William (Bill) Finch Mathematics Department Denton High School
2 Lesson Goals When you have completed this lesson you will: Identify whether a function is continuous. Identify 3 types of discontinuities. Describe the end-behavior of a function. Continuity / End Behavior 2 / 14
3 Lesson Goals When you have completed this lesson you will: Identify whether a function is continuous. Identify 3 types of discontinuities. Describe the end-behavior of a function. Continuity / End Behavior 2 / 14
4 Lesson Goals When you have completed this lesson you will: Identify whether a function is continuous. Identify 3 types of discontinuities. Describe the end-behavior of a function. Continuity / End Behavior 2 / 14
5 Lesson Goals When you have completed this lesson you will: Identify whether a function is continuous. Identify 3 types of discontinuities. Describe the end-behavior of a function. Continuity / End Behavior 2 / 14
6 Continuity The graph of a function is continuous if it has no breaks or gaps. y x Continuity / End Behavior 3 / 14
7 Discontinuities Three types of discontinuities y y y x x x Infinite discontinuity at x = c Jump discontinuity at x = c Removable discontinuity at x = c Continuity / End Behavior 4 / 14
8 Limits As f (x) approaches some output L as the input x approaches some input c, then the limit of f (x) as x approaches c is L. lim f (x) = L x c Continuity / End Behavior 5 / 14
9 Continuity Test A function f (x) is continuous at x = c if: f (c) exists f (x) approaches the same output value from either side of c (this means lim x c f (x) exists) The output value the function approaches from either side of c is f (c) (this means lim x c f (x) = f (c)) Continuity / End Behavior 6 / 14
10 Continuity Test A function f (x) is continuous at x = c if: f (c) exists f (x) approaches the same output value from either side of c (this means lim x c f (x) exists) The output value the function approaches from either side of c is f (c) (this means lim x c f (x) = f (c)) Continuity / End Behavior 6 / 14
11 Continuity Test A function f (x) is continuous at x = c if: f (c) exists f (x) approaches the same output value from either side of c (this means lim x c f (x) exists) The output value the function approaches from either side of c is f (c) (this means lim x c f (x) = f (c)) Continuity / End Behavior 6 / 14
12 Continuity Test A function f (x) is continuous at x = c if: f (c) exists f (x) approaches the same output value from either side of c (this means lim x c f (x) exists) The output value the function approaches from either side of c is f (c) (this means lim x c f (x) = f (c)) Continuity / End Behavior 6 / 14
13 Example 1 Determine whether f (x) = 1 is continuous at x = 0.5 2x + 1 using the continuity test. If discontinuous, identify the type of discontinuity. Continuity / End Behavior 7 / 14
14 Example 2 Determine whether f (x) = 1 is continuous at x = 1 using x 1 the continuity test. If discontinuous, identify the type of discontinuity. Continuity / End Behavior 8 / 14
15 Example 3 Determine whether f (x) = x 2 is continuous at x = 2 x 2 4 using the continuity test. If discontinuous, identify the type of discontinuity. Continuity / End Behavior 9 / 14
16 End Behavior of a Function The end behavior of a function is a description of the output of the function as x goes left and right towards negative and positive infinity. Left-End Behavior lim f (x) = x Right-End Behavior lim f (x) = x + y x Continuity / End Behavior 10 / 14
17 Example 4 Use the graph of f (x) = x 3 x 2 4x + 4 to describe its end behavior. Continuity / End Behavior 11 / 14
18 Example 5 Use the graph of g(x) = x + 2 to describe its end x 2 x 2 behavior. Continuity / End Behavior 12 / 14
19 Example 6 In physics, the symmetric energy function is E = x 2 + y 2 2 If the y-value is held constant, what happens to the value of symmetric energy when the x-value approaches? Continuity / End Behavior 13 / 14
20 What You Learned You can now: Identify whether a function is continuous. Identify 3 types of discontinuities. Describe the end-behavior of a function. Do problems Chap 1.3 #1-11 odd, odd Continuity / End Behavior 14 / 14
21 What You Learned You can now: Identify whether a function is continuous. Identify 3 types of discontinuities. Describe the end-behavior of a function. Do problems Chap 1.3 #1-11 odd, odd Continuity / End Behavior 14 / 14
22 What You Learned You can now: Identify whether a function is continuous. Identify 3 types of discontinuities. Describe the end-behavior of a function. Do problems Chap 1.3 #1-11 odd, odd Continuity / End Behavior 14 / 14
23 What You Learned You can now: Identify whether a function is continuous. Identify 3 types of discontinuities. Describe the end-behavior of a function. Do problems Chap 1.3 #1-11 odd, odd Continuity / End Behavior 14 / 14
24 What You Learned You can now: Identify whether a function is continuous. Identify 3 types of discontinuities. Describe the end-behavior of a function. Do problems Chap 1.3 #1-11 odd, odd Continuity / End Behavior 14 / 14
1-3 Continuity, End Behavior, and Limits
Determine whether each function is continuous at the given x-value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. 1. f (x)
Unit 2 Functions Analyzing Graphs of Functions (Unit 2.2)
Unit 2 Functions Analzing Graphs of Functions (Unit 2.2) William (Bill) Finch Mathematics Department Denton High School Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even
Unit 6 Introduction to Trigonometry The Unit Circle (Unit 6.3)
Unit Introduction to Trigonometr The Unit Circle Unit.) William Bill) Finch Mathematics Department Denton High School Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic
Lesson Goals. Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5) Overview. Overview
Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Graph
Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5)
Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Graph
2-1 Power and Radical Functions
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 15. h(x) = x 3 Evaluate the function for several x-values
Mid-Chapter Quiz: Lessons 2-1 through 2-3
Graph and analyze each function. Describe its domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 2x 3 2 16 1.5 6.75 1 2 0 0 1 2 1.5 6.75
2-1 Power and Radical Functions
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 35. Evaluate the function for several x-values in
5-3 Polynomial Functions
For each graph, a. describe the end behavior, b. determine whether it represents an odd-degree or an even-degree function, and c. state the number of real zeros. 35. a. As the x-values approach negative
10.2 Basic Concepts of Limits
10.2 Basic Concepts of Limits Question 1: How do you evaluate a limit from a table? Question 2: How do you evaluate a limit from a graph? In this chapter, we ll examine the concept of a limit. in its simplest
Unit 6 Introduction to Trigonometry Right Triangle Trigonomotry (Unit 6.1)
Unit 6 Introduction to Trigonometry Right Triangle Trigonomotry (Unit 6.1) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Find
A Crash Course on Limits (in class worksheet)
A Crash Course on Limits (in class worksheet) Many its may be found easily by simple substitution. For example: x x x ( ) = f() = and x = f () = 8 7 and x x = f () = So the first rule is to always TRY
Mid Term Pre Calc Review
Mid Term 2015-13 Pre Calc Review I. Quadratic Functions a. Solve by quadratic formula, completing the square, or factoring b. Find the vertex c. Find the axis of symmetry d. Graph the quadratic function
Study Guide and Review - Chapter 1
State whether each sentence is true or false If false, replace the underlined term to make a true sentence 1 A function assigns every element of its domain to exactly one element of its range A function
End Behavior and Symmetry
Algebra 2 Interval Notation Name: Date: Block: X Characteristics of Polynomial Functions Lesson Opener: Graph the function using transformations then identify key characteristics listed below. 1. y x 2
Practice Test - Chapter 1
Determine whether the given relation represents y as a function of x. 1. y 3 x = 5 When x = 1, y = ±. Therefore, the relation is not one-to-one and not a function. not a function 4. PARKING The cost of
Calculus Limits Images in this handout were obtained from the My Math Lab Briggs online e-book.
Calculus Limits Images in this handout were obtained from the My Math Lab Briggs online e-book. A it is the value a function approaches as the input value gets closer to a specified quantity. Limits are
Pick any positive integer. If the integer is even, divide it by 2. If it is odd,
Equal Groups Multiplying and Dividing Integers Learning Goals In this lesson, you will: Multiply integers. Divide integers. Pick any positive integer. If the integer is even, divide it by 2. If it is odd,
UNIT 2 LINEAR AND EXPONENTIAL RELATIONSHIPS Lesson 1: Graphs As Solution Sets and Function Notation
Lesson : Graphs As Solution Sets and Function Notation Guided Practice.. Example Is the relation below a function? Use a mapping diagram to determine your answer. {(, ), (, ), (, ), (, ), (, ), (, )}.
Pre-Calc Unit 1 Lesson 1
Pre-Calc Unit 1 Lesson 1 The Number System and Set Theory Learning Goal: IWBAT write subsets of the rational, real, and complex number system using set notation and apply set operations on sets of numbers.
You used set notation to denote elements, subsets, and complements. (Lesson 0-1)
You used set notation to denote elements, subsets, and complements. (Lesson 0-1) Describe subsets of real numbers. Identify and evaluate functions and state their domains. set-builder notation interval
Mathematical Focus 1 Exponential functions adhere to distinct properties, including those that limit the values of what the base can be.
Situation: Restrictions on Exponential Functions Prepared at the University of Georgia in Dr. Wilson s EMAT 500 Class July 5, 013 Sarah Major Prompt: A teacher prompts her students to turn in their homework
1-5 Parent Functions and Transformations
Describe the following characteristics of the graph of each parent function: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. 1.
Trigonometric Fourier series
Trigonometric Fourier series Recall that two vectors are orthogonal if their inner product is zero Suppose we consider all functions with at most a finite number of discontinuities ined on the interval
Unit 1 Introduction to Precalculus Rectangular Coordinates (Unit 1.1)
Unit 1 Introduction to Precalculus Rectangular Coordinates (Unit 1.1) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Understand
Continuity and Tangent Lines for functions of two variables
Continuity and Tangent Lines for functions of two variables James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 4, 2014 Outline 1 Continuity
Math 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
MATH 137 : Calculus 1 for Honours Mathematics. Online Assignment #5. Limits and Continuity of Functions
1 Instructions: MATH 137 : Calculus 1 for Honours Mathematics Online Assignment #5 Limits and Continuity of Functions Due by 9:00 pm on WEDNESDAY, June 13, 2018 Weight: 2% This assignment includes the
Mid-Chapter Quiz: Lessons 1-1 through 1-4
Determine whether each relation represents y as a function of x. 1. 3x + 7y = 21 This equation represents y as a function of x, because for every x-value there is exactly one corresponding y-value. The
Section 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
Warm Up MM2A5. 1. Graph f(x) = x 3 and make a table for the function. 2. Graph g(x) = (x) and make a table for the function
MM2A5 Warm Up 1. Graph f(x) = x 3 and make a table for the function 2. Graph g(x) = (x) and make a table for the function 3. What do you notice between the functions in number 1? 4. What do you notice
Foundations of Math II
Foundations of Math II Unit 6b: Toolkit Functions Academics High School Mathematics 6.6 Warm Up: Review Graphing Linear, Exponential, and Quadratic Functions 2 6.6 Lesson Handout: Linear, Exponential,
1-1 Sets of Numbers. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2
1-1 Sets of Numbers Warm Up Lesson Presentation Lesson Quiz Warm Up Write in decimal form. 1. 4.5 2. 3. Write as a decimal approximation. 1.414 Order from least to greatest. 4. 10, 5, 10, 0, 5 10, 5, 0,
2.4 Polynomial and Rational Functions
Polnomial Functions Given a linear function f() = m + b, we can add a square term, and get a quadratic function g() = a 2 + f() = a 2 + m + b. We can continue adding terms of higher degrees, e.g. we can
A Rational Existence Introduction to Rational Functions
Lesson. Skills Practice Name Date A Rational Eistence Introduction to Rational Functions Vocabular Write the term that best completes each sentence.. A rational function is an function that can be written
Learning Log Title: CHAPTER 3: ARITHMETIC PROPERTIES. Date: Lesson: Chapter 3: Arithmetic Properties
Chapter 3: Arithmetic Properties CHAPTER 3: ARITHMETIC PROPERTIES Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Chapter 3: Arithmetic Properties Date: Lesson: Learning Log Title:
Graphs of Rational Functions
Objectives Lesson 5 Graphs of Rational Functions the table. near 0, we evaluate fix) to the left and right of x = 0 as shown in Because the denominator is zero when x = 0, the domain off is all real numbers
Calculus Limits Images in this handout were obtained from the My Math Lab Briggs online e-book.
Calculus Limits Images in this handout were obtained from the My Math Lab Briggs online e-book. A it is the value a function approaches as the input value gets closer to a specified quantity. Limits are
1.1 Pearson Modeling and Equation Solving
Date:. Pearson Modeling and Equation Solving Syllabus Objective:. The student will solve problems using the algebra of functions. Modeling a Function: Numerical (data table) Algebraic (equation) Graphical
GRAPHING CALCULATOR - WINDOW SIZING
Section 1.1 GRAPHING CALCULATOR - WINDOW SIZING WINDOW BUTTON. Xmin= Xmax= Xscl= Ymin= Ymax= Yscl= Xres=resolution, smaller number= clearer graph Larger number=quicker graphing Xscl=5, Yscal=1 Xscl=10,
Finding 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
Math-3 Lesson 1-7 Analyzing the Graphs of Functions
Math- Lesson -7 Analyzing the Graphs o Functions Which unctions are symmetric about the y-axis? cosx sin x x We call unctions that are symmetric about the y -axis, even unctions. Which transormation is
Example 1: Use the graph of the function f given below to find the following. a. Find the domain of f and list your answer in interval notation
When working with the graph of a function, the inputs (the elements of the domain) are always the values on the horizontal ais (-ais) and the outputs (the elements of the range) are always the values on
Function Transformations and Symmetry
CHAPTER Function Transformations and Symmetry The first well-documented postal system was in ancient Rome, where mail was carried by horsedrawn carriages and ox-drawn wagons. The US Postal Service delivers
Section 2.5: Continuity
Section 2.5: Continuity 1. The Definition of Continuity We start with a naive definition of continuity. Definition 1.1. We say a function f() is continuous if we can draw its graph without lifting out
Graphs of the Circular Functions. Copyright 2017, 2013, 2009 Pearson Education, Inc.
4 Graphs of the Circular Functions Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 4.3 Graphs of the Tangent and Cotangent Functions Graph of the Tangent Function Graph of the Cotangent Function Techniques
Unit 1 and Unit 2 Concept Overview
Unit 1 and Unit 2 Concept Overview Unit 1 Do you recognize your basic parent functions? Transformations a. Inside Parameters i. Horizontal ii. Shift (do the opposite of what feels right) 1. f(x+h)=left
Rational 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
Charting new territory: Formulating the Dalivian coordinate system
Parabola Volume 53, Issue 2 (2017) Charting new territory: Formulating the Dalivian coordinate system Olivia Burton and Emma Davis 1 Numerous coordinate systems have been invented. The very first and most
Limits, Continuity, and Asymptotes
LimitsContinuity.nb 1 Limits, Continuity, and Asymptotes Limits Limit evaluation is a basic calculus tool that can be used in many different situations. We will develop a combined numerical, graphical,
Interior Penalty Functions. Barrier Functions A Problem and a Solution
Interior Penalty Functions Barrier Functions A Problem and a Solution Initial Steps A second type of penalty function begins with an initial point inside the feasible region, which is why these procedures
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the intervals on which the function is continuous. 2 1) y = ( + 5)2 + 10 A) (-, ) B)
2-5 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
This 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:
Lesson 2b Functions and Function Operations
As we continue to work with more complex functions it is important that we are comfortable with Function Notation, opertions on Functions and opertions involving more than one function. In this lesson,
Radical Functions. Attendance Problems. Identify the domain and range of each function.
Page 1 of 12 Radical Functions Attendance Problems. Identify the domain and range of each function. 1. f ( x) = x 2 + 2 2. f ( x) = 3x 3 Use the description to write the quadratic function g based on the
Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103. Chapter 2. Sets
Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103 Chapter 2 Sets Slides are adopted from Discrete Mathematics and It's Applications Kenneth H.
IMC Unit 4 LINES Study Guide
IMC Unit 4 LINES Study Guide In this unit, you will continue to explore linear equations in two variables. You will learn how to find the equation of a line from its graph. You will then learn how to write
Michael Moody School of Pharmacy University of London 29/39 Brunswick Square London WC1N 1AX, U.K.
This material is provided for educational use only. The information in these slides including all data, images and related materials are the property of : Michael Moody School of Pharmacy University of
Determine if the lines defined by the given equations are parallel, perpendicular, or neither. 1) -4y = 2x + 5
Review test 3 -College Algebra Math1314 - Spring 2017 - Houston Community College Name Date MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine
GUIDED NOTES 3.5 TRANSFORMATIONS OF FUNCTIONS
GUIDED NOTES 3.5 TRANSFORMATIONS OF FUNCTIONS LEARNING OBJECTIVES In this section, you will: Graph functions using vertical and horizontal shifts. Graph functions using reflections about the x-axis and
Functions and Graphs. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.
Functions and Graphs The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type
Graphing 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
Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Section 12 Variables and Expressions
Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Please watch Section 12 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item67.cfm
,?...?, the? or? s are for any holes or vertical asymptotes.
Name: Period: Pre-Cal AB: Unit 14: Rational Functions Monday Tuesday Block Friday 16 17 18/19 0 end of 9 weeks Graphing Rational Graphing Rational Partial Fractions QUIZ 3 Conic Sections (ON Friday s Quiz)
6.3 Creating and Comparing Quadratics
6.3 Creating and Comparing Quadratics Just like with exponentials and linear functions, to be able to compare quadratics, we ll need to be able to create equation forms of the quadratic functions. Let
Graphing Trig Functions - Sine & Cosine
Graphing Trig Functions - Sine & Cosine Up to this point, we have learned how the trigonometric ratios have been defined in right triangles using SOHCAHTOA as a memory aid. We then used that information
Properties of a Function s Graph
Section 3.2 Properties of a Function s Graph Objective 1: Determining the Intercepts of a Function An intercept of a function is a point on the graph of a function where the graph either crosses or touches
Chapter 04: Instruction Sets and the Processor organizations. Lesson 20: RISC and converged Architecture
Chapter 04: Instruction Sets and the Processor organizations Lesson 20: RISC and converged Architecture 1 Objective Learn the RISC architecture Learn the Converged Architecture 2 Reduced Instruction Set
(2 1) What does evaluate to? What does evaluate to? What does evaluate to?
Order of Operations (Time 20 minutes) Math is a language, just like English, Spanish, or any other language. We use nouns, like "bread", "tomato", "mustard" and "cheese" to describe physical objects. Math
South County Secondary School AP Calculus BC
South County Secondary School AP Calculus BC Summer Assignment For students entering Calculus BC in the Fall of 8 This packet will be collected for a grade at your first class. The material covered in
Computational Foundations of Cognitive Science
Computational Foundations of Cognitive Science Lecture 16: Models of Object Recognition Frank Keller School of Informatics University of Edinburgh keller@inf.ed.ac.uk February 23, 2010 Frank Keller Computational
Math 3 Coordinate Geometry Part 2 Graphing Solutions
Math 3 Coordinate Geometry Part 2 Graphing Solutions 1 SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY The solution of two linear equations is the point where the two lines intersect. For example, in the graph
Family of Functions Lesson
Family of Functions Lesson Introduction: Show pictures of family members to illustrate that even though family members are different (in most cases) they have very similar characteristics (DNA). Today
= ( )= 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)
Welcome to class! Have a great day! I ll miss you while I m gone :)
Welcome to class! 1. You have a substitute (Ms. Williams) today while I am at training for Nonviolent Crisis Intervention. I have very high expectations for you in my absence 2. I know I didn t put the
Simplifying Square Root Expressions[In Class Version][Algebra 1 Honors].notebook August 26, Homework Assignment. Example 5 Example 6.
![Simplifying Square Root Expressions[In Class Version][Algebra 1 Honors].notebook August 26, Homework Assignment. Example 5 Example 6.](/thumbs/96/127077659.jpg "Simplifying Square Root Expressions[In Class Version][Algebra 1 Honors].notebook August 26, Homework Assignment. Example 5 Example 6.")
Homework Assignment The following examples have to be copied for next class Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 7 Example 8 Example 9 Example 10 Example 11 Example 12 The
Distributions of Continuous Data
C H A P T ER Distributions of Continuous Data New cars and trucks sold in the United States average about 28 highway miles per gallon (mpg) in 2010, up from about 24 mpg in 2004. Some of the improvement
Smooth rounded corner. Smooth rounded corner. Smooth rounded corner
3.2 Graphs of Higher Degree Polynomial Functions Definition of a 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 defined by
Introduction : Identifying Key Features of Linear and Exponential Graphs
Introduction Real-world contexts that have two variables can be represented in a table or graphed on a coordinate plane. There are many characteristics of functions and their graphs that can provide a
Helping Students Understand Pre-Algebra
Helping Students Understand Pre-Algebra By Barbara Sandall, Ed.D., & Mary Swarthout, Ph.D. COPYRIGHT 2005 Mark Twain Media, Inc. ISBN 10-digit: 1-58037-294-5 13-digit: 978-1-58037-294-7 Printing No. CD-404021
Learning Objectives. Continuous Random Variables & The Normal Probability Distribution. Continuous Random Variable
Learning Objectives Continuous Random Variables & The Normal Probability Distribution 1. Understand characteristics about continuous random variables and probability distributions 2. Understand the uniform
MAFS Algebra 1. Quadratic Functions. Day 17 - Student Packet
MAFS Algebra 1 Quadratic Functions Day 17 - Student Packet Day 17: Quadratic Functions MAFS.912.F-IF.3.7a, MAFS.912.F-IF.3.8a I CAN graph a quadratic function using key features identify and interpret
A-SSE.1.1, A-SSE.1.2-
Putnam County Schools Curriculum Map Algebra 1 2016-2017 Module: 4 Quadratic and Exponential Functions Instructional Window: January 9-February 17 Assessment Window: February 20 March 3 MAFS Standards
UNIT 3 EXPRESSIONS AND EQUATIONS Lesson 3: Creating Quadratic Equations in Two or More Variables
Guided Practice Example 1 Find the y-intercept and vertex of the function f(x) = 2x 2 + x + 3. Determine whether the vertex is a minimum or maximum point on the graph. 1. Determine the y-intercept. The
Outputs. Inputs. the point where the graph starts, as we ll see on Example 1.
We ve seen how to work with functions algebraically, by finding domains as well as function values. In this set of notes we ll be working with functions graphically, and we ll see how to find the domain
C E N T E R A T H O U S T O N S C H O O L of H E A L T H I N F O R M A T I O N S C I E N C E S. Image Operations II
T H E U N I V E R S I T Y of T E X A S H E A L T H S C I E N C E C E N T E R A T H O U S T O N S C H O O L of H E A L T H I N F O R M A T I O N S C I E N C E S Image Operations II For students of HI 5323
Calculus I Review Handout 1.3 Introduction to Calculus - Limits. by Kevin M. Chevalier
Calculus I Review Handout 1.3 Introduction to Calculus - Limits by Kevin M. Chevalier We are now going to dive into Calculus I as we take a look at the it process. While precalculus covered more static
Control Statements Loops
CS 117 Spring 2004 Nested if statements if-else statements can reside within other if-else statements nested if statements Control Statements Loops April 26, 2004 Example (pseudocode) Get interest rate
5008: Computer Architecture HW#2
5008: Computer Architecture HW#2 1. We will now support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be
Grade 6 Integers. Answer the questions. Choose correct answer(s) from the given choices. For more such worksheets visit
ID : cn6integers [1] Grade 6 Integers For more such worksheets visit www.edugain.com Answer the questions (1) If a and b are two integers such that a is the predecessor of b, then what is the value of
Chapter P: Preparation for Calculus
1. Which of the following is the correct graph of y = x x 3? E) Copyright Houghton Mifflin Company. All rights reserved. 1 . Which of the following is the correct graph of y = 3x x? E) Copyright Houghton
AP Calculus AB Unit 2 Assessment
Class: Date: 203-204 AP Calculus AB Unit 2 Assessment Multiple Choice Identify the choice that best completes the statement or answers the question. A calculator may NOT be used on this part of the exam.
Modeling web-crawlers on the Internet with random walksdecember on graphs11, / 15
Modeling web-crawlers on the Internet with random walks on graphs December 11, 2014 Modeling web-crawlers on the Internet with random walksdecember on graphs11, 2014 1 / 15 Motivation The state of the
TIPS4Math Grade 5 Collect, Organize and Interpret Numerical Data
Collect, Organize and Interpret Numerical Data Overall Expectations Students will: Collect and organize discrete or continuous primary data and secondary data and display the data using charts and graphs,
CISC 1400 Discrete Structures
CISC 1400 Discrete Structures Chapter 4 Relations CISC1400 Yanjun Li 1 1 Relation A relation is a connection between objects in one set and objects in another set (possibly the same set). age is a relation
Programming for Experimental Research. Flow Control
Programming for Experimental Research Flow Control FLOW CONTROL In a simple program, the commands are executed one after the other in the order they are typed. Many situations require more sophisticated
Unit 8: Recursion. Dave Abel. April 6th, 2016
Unit 8: Recursion Dave Abel April 6th, 2016 1 Takeaway Repeated self reference, or recursion, is everywhere, in the world and in computation! It s simple, beautiful, and incredibly powerful. 2 Outline
Cardinality of Sets. Washington University Math Circle 10/30/2016
Cardinality of Sets Washington University Math Circle 0/0/06 The cardinality of a finite set A is just the number of elements of A, denoted by A. For example, A = {a, b, c, d}, B = {n Z : n } = {,,, 0,,,
1 EquationsofLinesandPlanesin 3-D
1 EquationsofLinesandPlanesin 3-D Recall that given a point P (a, b, c), one can draw a vector from the origin to P. Such a vector is called the position vector of the point P and its coordinates are a,