UNIT 2 LINEAR AND EXPONENTIAL RELATIONSHIPS Lesson 1: Graphs As Solution Sets and Function Notation
|
|
|
- Shon Freeman
- 5 years ago
- Views:
Transcription
1 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. {(, ), (, ), (, ), (, ), (, ), (, )}. Find the domain of the first relation. The domain is the set of x-values of the relation. List the domain in numerical order. If any of the values repeat, list them only once. Domain: {,,,,, }. Find the range of the relation. The range is the set of y-values of the relation. List the range in numerical order. If any of the values repeat, list them only once. Range: {,,, } The y-values and each appear in more than one of the ordered pairs, but they are only listed once.. Map the elements in the domain to the corresponding elements in the range. Domain Range SWB pp. ; SRB pp. U- U- U-..
2 Lesson : Graphs As Solution Sets and Function Notation. Analyze the mapping. If there is only one line coming from each element of the domain, the relation is a function. In this case, each element of the domain is paired with exactly one element in the range, so the relation is a function. Notice that some elements in the range have two arrows coming to them. The relation is still a function because each element in the domain has only one line from it. Example Is the relation below a function? Use a mapping diagram to determine your answer. {(, ), (, ), (, ), (, ), (, ), (, )}. Find the domain of the relation. The domain is the set of x-values of the relation. List the domain in numerical order. List repeating values only once. Domain: {,,, } and repeat, but they are only listed once.. Find the range of the relation. The range is the set of y-values of the relation. List the range in numerical order. List repeating values only once. Range: {,,,,, } U-..
3 Lesson : Graphs As Solution Sets and Function Notation. Map the elements in the domain to the corresponding elements in the range. Domain Range. Analyze the mapping. If there is only one line coming from each element of the domain, the relation is a function. In this case, two elements of the domain have multiple lines going to elements in the range. The relation is not a function. Example Use the vertical line test to determine if each relation is a function. Graph A Graph B U-..
4 Lesson : Graphs As Solution Sets and Function Notation. Pass a vertical line over the figure in Graph A. Notice that the line crosses two points on the figure at the same time. The relation is not a function.. Pass a vertical line over the curve in Graph B. Notice that at any time the vertical line only crosses point on the curve at a time. The relation is a function U-..
5 Lesson : Graphs As Solution Sets and Function Notation Example Omar has decided to take yoga classes for one year. The yoga studio costs $ to join and then each yoga class is $. Omar s fees can be represented by the function f(x) = x +. What are the domain and range of the function?. Identify the domain. The domain is the set of x-values that are valid for the function. Omar can take,,,,, etc., classes up to infinity. However, it is unlikely that Omar would take any more than yoga class per day. If Omar never takes a class, the lowest value in the domain would be. If Omar takes a class every day for a year, the highest value in the domain would be. Domain: {,,,,,,, }. Identify the range. The range is the values of y that are valid for the function. The y-values will be the solutions to the function when the domain is substituted. The function is f(x) = x +. Substitute domain values for x in the function to find values for the range. Range: {,,,,,,,, } U-..
6 Lesson : Graphs As Solution Sets and Function Notation Example Identify the domain and range of the function f(x) = x. Use the graph below Identify the domain. The domain is the set of x-values that are valid for the function. This graph goes on infinitely; so, the domain can be any real x-value. Domain: {all real numbers}. Identify the range. The range is the set of y-values that are valid for the function. The range will never go below. In fact, the range never actually reaches. The upper end of the range, however, is limitless. Range: {y > } U-..
FOA/Algebra 1. Unit 2B Review - Linear Functions
FOA/Algebra Unit B Review Name: Date: Block: Unit B Review - Linear Functions What you need to know & be able to do. Determine if a relation is a Things to remember Every input only has one output (each
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
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
Graphing 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
ISBN Copyright 2012 J. Weston Walch, Publisher Portland, ME Printed in the United States
Unit 3 4 5 6 7 8 9 0 ISBN 978-0-85-77-8 Copyright 0 J. Weston Walch, Publisher Portland, ME 0403 www.walch.com Printed in the United States of America WALCH EDUCATION Table of Contents Introduction to
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
Unit 2 Functions Continuity and End Behavior (Unit 2.3)
Unit 2 Functions Continuity and End Behavior (Unit 2.3) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Identify whether a function
1.1 - Functions, Domain, and Range
1.1 - Functions, Domain, and Range Lesson Outline Section 1: Difference between relations and functions Section 2: Use the vertical line test to check if it is a relation or a function Section 3: Domain
UNIT 5 QUADRATIC FUNCTIONS Lesson 6: Analyzing Quadratic Functions Instruction
Prerequisite Skills This lesson requires the use of the following skills: factoring quadratic expressions finding the vertex of a quadratic function Introduction We have studied the key features of the
Lesson 4 Representing and Understanding Functions
Lesson 4 Representing and Understanding Functions Key Learning Goals I can represent relations using words, tables of values, mapping diagrams, graphs and equations I can determine if a relation is a function
College 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 (
Section 9.3 Graphing Quadratic Functions
Section 9.3 Graphing Quadratic Functions A Quadratic Function is an equation that can be written in the following Standard Form., where a 0. Every quadratic function has a U-shaped graph called a. If the
Constrained Optimization with Calculus. Background Three Big Problems Setup and Vocabulary
Constrained Optimization with Calculus Background Three Big Problems Setup and Vocabulary Background Information In unit 3, you learned about linear programming, in which all constraints and the objective
1.1 - Functions, Domain, and Range
1.1 - Functions, Domain, and Range Lesson Outline Section 1: Difference between relations and functions Section 2: Use the vertical line test to check if it is a relation or a function Section 3: Domain
Unit: Quadratic Functions
Unit: Quadratic Functions Learning increases when you have a goal to work towards. Use this checklist as guide to track how well you are grasping the material. In the center column, rate your understand
a) 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
Classwork. Exercises Use long division to determine the decimal expansion of. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 8 7
Classwork Exercises 1 5 1. Use long division to determine the decimal expansion of. 2. Use long division to determine the decimal expansion of. 3. Use long division to determine the decimal expansion of.
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
Dependent Independent Input Discrete. X Y Output Continuous
Domain and Range Review Vocabulary Words Dependent Independent Input Discrete X Y Output Continuous Domain is all the values. It is the variable. It is also the of the function. Range is all the values.
Lesson 12: The Graph of the Equation y = f(x)
Classwork In Module 1, you graphed equations such as 4x + y = 10 by plotting the points on the Cartesian coordinate plane that corresponded to all of the ordered pairs of numbers (x, y) that were in the
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
Lesson 10 Rational Functions and Equations
Lesson 10 Rational Functions and Equations Lesson 10 Rational Functions and Equations In this lesson, you will embark on a study of rational functions. Rational functions look different because they are
Algebra I Notes Graphs of Functions OBJECTIVES: F.IF.A.1 Understand the concept of a function and use function notation. F.IF.A.2.
OBJECTIVES: F.IF.A.1 Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element
2.6: 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
Did You Find a Parking Space?
Lesson.4 Skills Practice Name Date Did You Find a Parking Space? Parallel and Perpendicular Lines on the Coordinate Plane Vocabulary Complete the sentence. 1. The point-slope form of the equation of the
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,
Quadratic Functions CHAPTER. 1.1 Lots and Projectiles Introduction to Quadratic Functions p. 31
CHAPTER Quadratic Functions Arches are used to support the weight of walls and ceilings in buildings. Arches were first used in architecture by the Mesopotamians over 4000 years ago. Later, the Romans
Lesson 8 Introduction to Quadratic Functions
Lesson 8 Introduction to Quadratic Functions We are leaving exponential and logarithmic functions behind and entering an entirely different world. As you work through this lesson, you will learn to identify
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
Lesson 8.1 Exercises, pages
Lesson 8.1 Eercises, pages 1 9 A. Complete each table of values. a) -3 - -1 1 3 3 11 8 5-1 - -7 3 11 8 5 1 7 To complete the table for 3, take the absolute value of each value of 3. b) - -3 - -1 1 3 3
2.1 Basics of Functions and Their Graphs
.1 Basics of Functions and Their Graphs Section.1 Notes Page 1 Domain: (input) all the x-values that make the equation defined Defined: There is no division by zero or square roots of negative numbers
4.2 Linear Equations in Point-Slope Form
4.2 Linear Equations in Point-Slope Form Learning Objectives Write an equation in point-slope form. Graph an equation in point-slope form. Write a linear function in point-slope form. Solve real-world
UNIT 5 QUADRATIC FUNCTIONS Lesson 7: Building Functions Instruction
Prerequisite Skills This lesson requires the use of the following skills: multiplying linear expressions factoring quadratic equations finding the value of a in the vertex form of a quadratic equation
Functions, Duality, and Mapping Diagrams. State of Jefferson Math Congress. Oct. 5, Martin Flashman
Functions, Duality, and Mapping Diagrams State of Jefferson Math Congress Oct. 5, 2013 Martin Flashman Professor of Mathematics Humboldt State University flashman@humboldt.edu http://users.humboldt.edu/flashma
Recognizing a Function
Recognizing a Function LAUNCH (7 MIN) Before Why would someone hire a dog walking service? During Do you know exactly what it would cost to hire Friendly Dog Walking? After How does each service encourage
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
1-8 Exploring Transformations
1-8 Exploring Transformations Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Plot each point. D 1. A(0,0) 2. B(5,0) 3. C( 5,0) 4. D(0,5) 5. E(0, 5) 6. F( 5, 5) C A F E B Objectives Apply transformations
Paste them together (lining up the x-axis in each piece) to create the graph of the piecewise-defined function.
Mini-Project 2: Piecewise-Defined Functions A piecewise-defined function is two or more domain-restricted functions combined into a single function (with a large brace). Each of the pieces is a function,
Section 18-1: Graphical Representation of Linear Equations and Functions
Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as
Walt 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
Objectives. Vocabulary. 1-1 Exploring Transformations
Warm Up Plot each point. D Warm Up Lesson Presentation Lesson Quiz 1. A(0,0) 2. B(5,0) 3. C( 5,0) 4. D(0,5) C A B 5. E(0, 5) 6. F( 5, 5) F E Algebra 2 Objectives Apply transformations to points and sets
Sec 4.1 Coordinates and Scatter Plots. Coordinate Plane: Formed by two real number lines that intersect at a right angle.
Algebra I Chapter 4 Notes Name Sec 4.1 Coordinates and Scatter Plots Coordinate Plane: Formed by two real number lines that intersect at a right angle. X-axis: The horizontal axis Y-axis: The vertical
Day 4 Notes- Characteristics of Linear Functions
Day 4 Notes- Characteristics of Linear Functions One key component to fully understanding linear functions is to be able to describe characteristics of the graph and its equation. Important: If a graph
Chapter 12: Quadratic and Cubic Graphs
Chapter 12: Quadratic and Cubic Graphs Section 12.1 Quadratic Graphs x 2 + 2 a 2 + 2a - 6 r r 2 x 2 5x + 8 2y 2 + 9y + 2 All the above equations contain a squared number. They are therefore called quadratic
State the domain and range of the relation. EX: {(-1,1), (1,5), (0,3)} 1 P a g e Province Mathematics Southwest TN Community College
A relation is a set of ordered pairs of real numbers. The domain, D, of a relation is the set of all first coordinates of the ordered pairs in the relation (the xs). The range, R, of a relation is the
2.2 Graphs Of Functions. Copyright Cengage Learning. All rights reserved.
2.2 Graphs Of Functions Copyright Cengage Learning. All rights reserved. Objectives Graphing Functions by Plotting Points Graphing Functions with a Graphing Calculator Graphing Piecewise Defined Functions
Pre-Calculus Notes: Chapter 3 The Nature of Graphs
Section Families of Graphs Name: Pre-Calculus Notes: Chapter 3 The Nature of Graphs Family of graphs Parent graph A group of graphs that share similar properties The most basic graph that s transformed
S.W.B.A.T: Identify the independent and dependent variable in sentence. Write a function rule for a table and a situation.
Lesson 31 Date: Mr. Jones S.W.B.A.T: Identify the independent and dependent variable in sentence. Write a function rule for a table and a situation. DO NOW 1. If ( ), find f(3). 2. If f(x) = 2x -1, what
ECE 2020B Fundamentals of Digital Design Spring problems, 6 pages Exam Two Solutions 26 February 2014
Problem 1 (4 parts, 21 points) Encoders and Pass Gates Part A (8 points) Suppose the circuit below has the following input priority: I 1 > I 3 > I 0 > I 2. Complete the truth table by filling in the input
Lesson 1: Analyzing Quadratic Functions
UNIT QUADRATIC FUNCTIONS AND MODELING Lesson 1: Analyzing Quadratic Functions Common Core State Standards F IF.7 F IF.8 Essential Questions Graph functions expressed symbolically and show key features
Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola. Day #1
Algebra I Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola Name Period Date Day #1 There are some important features about the graphs of quadratic functions we are going to explore over the
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
Properties of Graphs of Quadratic Functions
H e i g h t (f t ) Lesson 2 Goal: Properties of Graphs of Quadratic Functions Identify the characteristics of graphs of quadratic functions: Vertex Intercepts Domain and Range Axis of Symmetry and use
Unit 1 Algebraic Functions and Graphs
Algebra 2 Unit 1 Algebraic Functions and Graphs Name: Unit 1 Day 1: Function Notation Today we are: Using Function Notation We are successful when: We can Use function notation to evaluate a function This
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
Precalculus 2 Section 10.6 Parametric Equations
Precalculus 2 Section 10.6 Parametric Equations Parametric Equations Write parametric equations. Graph parametric equations. Determine an equivalent rectangular equation for parametric equations. Determine
Situation 3. Parentheses vs. Brackets. Colleen Foy
Situation 3 Parentheses vs. Brackets Colleen Foy Prompt: Students were finding the domain and range of various functions. Most of the students were comfortable with the set builder notation, but when asked
Section 6.1: Quadratic Functions and their Characteristics Vertical Intercept Vertex Axis of Symmetry Domain and Range Horizontal Intercepts
Lesson 6 Quadratic Functions and Equations Lesson 6 Quadratic Functions and Equations We are leaving exponential functions behind and entering an entirely different world. As you work through this lesson,
Transformations with Fred Functions Day 1
Transformations with Fred Functions Day 1 KEY/TEACHER NOTES To the right is a graph of a Fred function. We can use Fred functions to explore transformations in the coordinate plane. Fred is any generic
Functions. Copyright Cengage Learning. All rights reserved.
Functions Copyright Cengage Learning. All rights reserved. 2.2 Graphs Of Functions Copyright Cengage Learning. All rights reserved. Objectives Graphing Functions by Plotting Points Graphing Functions with
SLOPE A MEASURE OF STEEPNESS through 2.1.4
SLOPE A MEASURE OF STEEPNESS 2.1.2 through 2.1.4 Students used the equation = m + b to graph lines and describe patterns in previous courses. Lesson 2.1.1 is a review. When the equation of a line is written
Data Analysis & Probability
Unit 5 Probability Distributions Name: Date: Hour: Section 7.2: The Standard Normal Distribution (Area under the curve) Notes By the end of this lesson, you will be able to Find the area under the standard
Sketching graphs of polynomials
Sketching graphs of polynomials We want to draw the graphs of polynomial functions y = f(x). The degree of a polynomial in one variable x is the highest power of x that remains after terms have been collected.
Sec.4.1 Increasing and Decreasing Functions
U4L1: Sec.4.1 Increasing and Decreasing Functions A function is increasing on a particular interval if for any, then. Ie: As x increases,. A function is decreasing on a particular interval if for any,
Quadratic Functions. *These are all examples of polynomial functions.
Look at: f(x) = 4x-7 f(x) = 3 f(x) = x 2 + 4 Quadratic Functions *These are all examples of polynomial functions. Definition: Let n be a nonnegative integer and let a n, a n 1,..., a 2, a 1, a 0 be real
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
4.2 and 4.6 filled in notes.notebook. December 08, Integration. Copyright Cengage Learning. All rights reserved.
4 Integration Copyright Cengage Learning. All rights reserved. 1 4.2 Area Copyright Cengage Learning. All rights reserved. 2 Objectives Use sigma notation to write and evaluate a sum. Understand the concept
Section 7.2 Characteristics of Quadratic Functions
Section 7. Characteristics of Quadratic Functions A QUADRATIC FUNCTION is a function of the form " # $ N# 1 & ;# & 0 Characteristics Include:! Three distinct terms each with its own coefficient:! An x
Graph 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
[The following questions were adapted from Polygraph: Parabolas, Part 2]
![[The following questions were adapted from Polygraph: Parabolas, Part 2]](/thumbs/83/88001318.jpg "[The following questions were adapted from Polygraph: Parabolas, Part 2]")
Opening Exploration 1. Go to https://student.desmos.com and use the class code: to play Polygraph: Parabolas. A description of the game is given below. [The following questions were adapted from https://teacher.desmos.com/activitybuilder/custom/574f12421390db611564fa32#
ECE 2020B Fundamentals of Digital Design Spring problems, 6 pages Exam Two 26 February 2014
Instructions: This is a closed book, closed note exam. Calculators are not permitted. If you have a question, raise your hand and I will come to you. Please work the exam in pencil and do not separate
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
MAT 102 Introduction to Statistics Chapter 6. Chapter 6 Continuous Probability Distributions and the Normal Distribution
MAT 102 Introduction to Statistics Chapter 6 Chapter 6 Continuous Probability Distributions and the Normal Distribution 6.2 Continuous Probability Distributions Characteristics of a Continuous Probability
UNIT 4 MODELING AND ANALYZING EXPONENTIAL FUNCTIONS Lesson 1: Creating Exponential Equations
Guided Practice Eample 1 If a pendulum swings to 9% of its previous height on each swing and starts out at a height of 6 cm, what is the equation that models this scenario? What is its graph? 1. Read the
Mini-Project 1: The Library of Functions and Piecewise-Defined Functions
Name Course Days/Start Time Mini-Project 1: The Library of Functions and Piecewise-Defined Functions Part 2: Piecewise-Defined Functions A piecewise-defined function is two or more functions combined into
EXERCISE SET 10.2 MATD 0390 DUE DATE: INSTRUCTOR
EXERCISE SET 10. STUDENT MATD 090 DUE DATE: INSTRUCTOR You have studied the method known as "completing the square" to solve quadratic equations. Another use for this method is in transforming the equation
Topics. Introduction to Repetition Structures Often have to write code that performs the same task multiple times. Controlled Loop
Topics C H A P T E R 4 Repetition Structures Introduction to Repetition Structures The for Loop: a Count- Sentinels Nested Loops Introduction to Repetition Structures Often have to write code that performs
UNIT 3B CREATING AND GRAPHING EQUATIONS Lesson 4: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing multiple equations on a graphing calculator graphing quadratic equations graphing linear equations Introduction A system
Finding a Global Maximum. Method 2: Sawtooth Method (Shubert-Piyavskii Method)
Finding a Global Maximum Method 2: Sawtooth Method (Shubert-Piyavskii Method) Requirements The sawtooth method requires two things that aren t always obvious: 1. Left and right boundaries (from common
Lesson 6-2: Function Operations
So numbers not only have a life but they have relationships well actually relations. There are special relations we call functions. Functions are relations for which each input has one and only one output.
Rational 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
Y. Butterworth Lehmann & 9.2 Page 1 of 11
Pre Chapter 9 Coverage Quadratic (2 nd Degree) Form a type of graph called a parabola Form of equation we'll be dealing with in this chapter: y = ax 2 + c Sign of a determines opens up or down "+" opens
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
9.1: GRAPHING QUADRATICS ALGEBRA 1
9.1: GRAPHING QUADRATICS ALGEBRA 1 OBJECTIVES I will be able to graph quadratics: Given in Standard Form Given in Vertex Form Given in Intercept Form What does the graph of a quadratic look like? https://www.desmos.com/calculator
Developmental Math An Open Program Unit 7 Geometry First Edition
Developmental Math An Open Program Unit 7 Geometry First Edition Lesson 1 Basic Geometric Concepts and Figures TOPICS 7.1.1 Figures in 1 and 2 Dimensions 1 Identify and define points, lines, line segments,
Name: Chapter 7 Review: Graphing Quadratic Functions
Name: Chapter Review: Graphing Quadratic Functions A. Intro to Graphs of Quadratic Equations: = ax + bx+ c A is a function that can be written in the form = ax + bx+ c where a, b, and c are real numbers
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
Planting the Seeds Exploring Cubic Functions
295 Planting the Seeds Exploring Cubic Functions 4.1 LEARNING GOALS In this lesson, you will: Represent cubic functions using words, tables, equations, and graphs. Interpret the key characteristics of
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.
Aspects of Geometry. Finite models of the projective plane and coordinates
Review Sheet There will be an exam on Thursday, February 14. The exam will cover topics up through material from projective geometry through Day 3 of the DIY Hyperbolic geometry packet. Below are some
Lesson 24 - Exploring Graphical Transformations and Composite Functions
(A) Lesson Objectives a. Review composite functions and how it can be represented numerically, algebraically and graphically. b. Introduce graphical transformations c. Understand that graphical transformations
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
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
Multi-step transformations
October 6, 2016 Transformations (section 1.6) Day 4 page 1 Multi-step transformations Objective: Apply transformations involving multiple steps or multiple substitutions. Upcoming: We will have a test
Direction Fields; Euler s Method
Direction Fields; Euler s Method It frequently happens that we cannot solve first order systems dy (, ) dx = f xy or corresponding initial value problems in terms of formulas. Remarkably, however, this
x + 1 = 2 Write x as a fraction in the form a, where a and b are positive integers with no common factor other than 1. Pass on the value of b a.
A1 Solve for x the equation x + 1 = 2 x. Write x as a fraction in the form a, where a and b are positive b integers with no common factor other than 1. Pass on the value of b a. A3 ABCD is a square that
MATH STUDENT BOOK. 12th Grade Unit 4
MATH STUDENT BOOK th Grade Unit Unit GRAPHING AND INVERSE FUNCTIONS MATH 0 GRAPHING AND INVERSE FUNCTIONS INTRODUCTION. GRAPHING 5 GRAPHING AND AMPLITUDE 5 PERIOD AND FREQUENCY VERTICAL AND HORIZONTAL
Creating a Crosstab Query in Design View
Procedures LESSON 31: CREATING CROSSTAB QUERIES Using the Crosstab Query Wizard box, click Crosstab Query Wizard. 5. In the next Crosstab Query the table or query on which you want to base the query. 7.
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
. 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