Dependent Variable Independent Variable dependent variable : independent variable function: domain ran ge
|
|
- Cleopatra Garrett
- 5 years ago
- Views:
Transcription
1 FUNCTIONS The values of one variable often depend on the values for another: The temperature at which water boils depends on elevation (the boiling point drops as you go up). The amount by which your savings will grow in a year depends on the interest rate offered by the bank. The area of a circle depends on the circle s radius. Dependent Variable Independent Variable dependent variable : the variable quantity that depends on the value of the another variable called the independent variable. function: a rule that assigns to each element in one set a unique element in another set. The sets may be sets of any kind and do not have to be the same. A function is like a machine that assigns a unique output to every allowable input. The inputs make up the domain of the function; the outputs make up the range. 1
2 When we define a function y f ( x) with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of x-values for which the formula gives real y-values the socalled natural domain. If we want to restrict the domain, we must say so. The domain of y x is understood to be the entire set of real numbers. We must write y x, x 0 if we want to restrict the function to positive values of x. The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed, or half-open, finite or infinite. The endpoints of an interval make up the interval s boundary and are called boundary points. The remaining points make up the interval s interior and are called interior points. Closed intervals contain their boundary points. Open intervals contain no boundary points. Every point of an open interval is an interior point of the interval. Name Graph Inequality Notation Interval Notation The set of all real numbers The set of numbers greater than a The set of numbers greater than a or equal to a The set of numbers less than b The set of numbers less than b or equal to b Open interval ab Closed interval ab Closed at a and open at b Open at a and closed at b Domain Range Domain Range
3 Recognize that the graph is reasonable. See all the important characteristics of the graph. Interpret those characteristics. Recognize grapher failure. Domain Range Domain Range EVEN FUNCTIONS AND ODD FUNCTIONS--SYMM The graphs of even and odd functions have important symmetry properties. For every x in the function s domain, a function is an Even function of x if Odd function of x if The graph is symmetric about the y-axis. The graph is symmetric about the origin. The graph of y x (an even function) is symmetric about the y-axis. 3 The graph of y x (an odd function) is symmetric about the origin. 3
4 Determine whether the function is even, odd, or neither. Try to answer without writing anything (except the answer). 1) y x 4 4 ) 3) y x 4) y x1 y x 1 While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domains. Piecewise functions let us make functions that do anything we want. A doctor s fee is based on the length of time. Up to 6 minutes costs $50. Over 6 to 15 minutes costs $80. Over 15 minutes costs $80 plus $5 per minute above 15 minutes Which we can write like this: $ 50 if t 6 f ( t) $ 80 if t 6 and t 15 $ 80 $ 5( t15 ) if t 15 a) If you were there for 1 minutes, what would the fee be? b) If you were there for 0 minutes, what would the fee be? Here is another piecewise function., if x 1 hx ( ) x, if x1 a) What is h(-1)? c) What is h(4)? b) What is h(1)? d) What is h(-5)? 4
5 Graphing Piecewise-Defined Functions Graph x, if x0 y f ( x) x, if 0 x 1 1, if x 1. Writing Formulas for Piecewise Functions Write a formula for the function y f ( x) whose graph consists of the two line segments in Figure f( x) Write a piecewise formula for the function. f( x) 5
6 The absolute value function y x is defined piecewise by the formula x x, if x0 x, if x 0. Domain Range, 0, The function is even, and symmetric about the y-axis. g( x) x h k vertex at ( hk, ) h 0 right; k 0 up h 0 left ; k 0 down Using Transformations (a) Draw the graph of the function. Then find its (b) domain and (c) range. 1) f ( x) x 1 ) f ( x) 3 x Domain Range Domain Range 6
7 FUNCTIONS The values of one variable often depend on the values for another: The temperature at which water boils depends on elevation (the boiling point drops as you go up). The amount by which your savings will grow in a year depends on the interest rate offered by the bank. The area of a circle depends on the circle s radius. Dependent Variable Boling temperature of water Amount of interest Area of a circle Independent Variable elevation Interest rate radius dependent variable : the variable quantity that depends on the value of the another variable called the independent variable. function: a rule that assigns to each element in one set a unique element in another set. The sets may be sets of any kind and do not have to be the same. A function is like a machine that assigns a unique output to every allowable input. The inputs make up the domain of the function; the outputs make up the range. a) b) 7
8 When we define a function y f ( x) with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of x-values for which the formula gives real y-values the socalled natural domain. If we want to restrict the domain, we must say so. The domain of y x is understood to be the entire set of real numbers. We must write y x, x 0 if we want to restrict the function to positive values of x. The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed, or half-open, finite or infinite. The endpoints of an interval make up the interval s boundary and are called boundary points. The remaining points make up the interval s interior and are called interior points. Closed intervals contain their boundary points. Open intervals contain no boundary points. Every point of an open interval is an interior point of the interval. Name Graph Inequality Notation Interval Notation The set of all real numbers The set of numbers greater than a The set of numbers greater than a or equal to a The set of numbers less than b The set of numbers less than b or equal to b Open interval ab Closed interval ab Closed at a and open at b Open at a and closed at b Domain Range, 0 0,, 0 0, Domain Range, 0, 0 8
9 Recognize that the graph is reasonable. See all the important characteristics of the graph. Interpret those characteristics. Recognize grapher failure. Domain Range, 0, Domain Range, 0, EVEN FUNCTIONS AND ODD FUNCTIONS--SYMM The graphs of even and odd functions have important symmetry properties. For every x in the function s domain, a function is an Even function of x if Odd function of x if The graph is symmetric about the y-axis. The graph is symmetric about the origin. The graph of y x (an even function) is symmetric about the y-axis. 3 The graph of y x (an odd function) is symmetric about the origin. 9
10 Determine whether the function is even, odd, or neither. Try to answer without writing anything (except the answer). 1) y x 4 4 ) 3) y x 4) y x1 y x x f ( x) x 4 4 x x f( x) 1 1 f( x) x x x 1 x 1 f( x) EVEN EVEN ODD NEITHER While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domains. Piecewise functions let us make functions that do anything we want. A doctor s fee is based on the length of time. Up to 6 minutes costs $50. Over 6 to 15 minutes costs $80. Over 15 minutes costs $80 plus $5 per minute above 15 minutes Which we can write like this: $ 50 if t 6 f ( t) $ 80 if t 6 and t 15 $ 80 $ 5( t15 ) if t 15 a) If you were there for 1 minutes, what would the fee be? b) If you were there for 0 minutes, what would the fee be? $80 $80 $5(0 15) $105 Here is another piecewise function., if x 1 hx ( ) x, if x1 a) What is h(-1)? c) What is h(4)? 4 b) What is h(1)? d) What is h(-5)? 10
11 Graphing Piecewise-Defined Functions Graph x, if x0 y f ( x) x, if 0 x 1 1, if x 1. Writing Formulas for Piecewise Functions Write a formula for the function y f ( x) whose graph consists of the two line segments in Figure x, 0 x1 f( x) x 1, 1 x. Write a piecewise formula for the function. x, 0 x f( x) 5 x, x
12 The absolute value function y x is defined piecewise by the formula x x, if x0 x, if x 0. Domain Range, 0, The function is even, and symmetric about the y-axis. g( x) x h k vertex at ( hk, ) h 0 right; k 0 up h 0 left ; k 0 down Using Transformations (b) Draw the graph of the function. Then find its (b) domain and (c) range. 1) f ( x) x 1 ) f ( x) 3 x Domain Range, 1, Domain Range,, 1
1.2 Functions and Graphs
Section.2 Functions and Graphs 3.2 Functions and Graphs You will be able to use the language, notation, and graphical representation of functions to epress relationships between variable quantities. Function,
More informationExam 2 Review. 2. What the difference is between an equation and an expression?
Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? 2. What the difference is between an equation and an expression? 3. How to tell if an equation is linear? 4. How
More informationProperties 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
More information1-3 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)
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Pre-Calculus Mid Term Review. January 2014 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the graph of the function f, plotted with a solid
More informationEnd 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
More informationMAT121: SECTION 2.7 ANALYZING GRAPHS AND PIECEWISE FUNCTIONS
MAT121: SECTION 2.7 ANALYZING GRAPHS AND PIECEWISE FUNCTIONS SYMMETRY, EVEN, ODD A graph can be symmetric about the x-axis, y-axis, or the origin (y = x). If a mirror is placed on those lines, the graph
More informationWarm-Up. Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) 2) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,2)
Warm-Up Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) ) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,) 8.4 Graph and Write Equations of Ellipses What are the major parts of
More informationMath Analysis Chapter 1 Notes: Functions and Graphs
Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 1-1 Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian coordinate system) Practice: Label each on the
More information1.1 calculator viewing window find roots in your calculator 1.2 functions find domain and range (from a graph) may need to review interval notation
1.1 calculator viewing window find roots in your calculator 1.2 functions find domain and range (from a graph) may need to review interval notation functions vertical line test function notation evaluate
More informationState 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
More informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More informationMath Analysis Chapter 1 Notes: Functions and Graphs
Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 1-1 Graphs; Section 1- Basics of Functions and Their Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian
More informationPiecewise Defined Functions
Piecewise Defined Functions Most of the functions that we ve looked at this semester can be expressed as a single equation. For example, f(x) =3x 2 5x +2,org(x) = x 1, or h(x) =e 3x 1. Sometimes an equation
More informationLesson Plan #31. Class: Geometry Date: Tuesday November 27 th, 2018
Lesson Plan #31 1 Class: Geometry Date: Tuesday November 27 th, 2018 Topic: Properties of parallel lines? Aim: What are some properties of parallel lines? Objectives: HW #31: 1) Students will be able to
More informationTest 3 review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Test 3 review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Approximate the coordinates of each turning point by graphing f(x) in the standard viewing
More informationVoluntary State Curriculum Algebra II
Algebra II Goal 1: Integration into Broader Knowledge The student will develop, analyze, communicate, and apply models to real-world situations using the language of mathematics and appropriate technology.
More informationCHAPTER 2: More on Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3
More informationModule 1. Name: Date: Period: Find the following function values. 4. Find the following: Domain. Range. The graph is increasing over the interval
Name: Date: Period: Algebra Fall Final Exam Review My Exam Date Is : Module 1 Find the following function values. f(x) = 3x + g(x) = x h(x) = x 3 1. g(f(x)). h(3) g(3) 3. g(f()) 4. Find the following:
More informationSection 1.1 The Distance and Midpoint Formulas
Section 1.1 The Distance and Midpoint Formulas 1 y axis origin x axis 2 Plot the points: ( 3, 5), (0,7), ( 6,0), (6,4) 3 Distance Formula y x 4 Finding the Distance Between Two Points Find the distance
More informationCCNY Math Review Chapter 2: Functions
CCN Math Review Chapter : Functions Section.1: Functions.1.1: How functions are used.1.: Methods for defining functions.1.3: The graph of a function.1.: Domain and range.1.5: Relations, functions, and
More informationUnit 12 Special Functions
Algebra Notes Special Functions Unit 1 Unit 1 Special Functions PREREQUISITE SKILLS: students should be able to describe a relation and a function students should be able to identify the domain and range
More information3. Solve the following. Round to the nearest thousandth.
This review does NOT cover everything! Be sure to go over all notes, homework, and tests that were given throughout the semester. 1. Given g ( x) i, h( x) x 4x x, f ( x) x, evaluate the following: a) f
More informationP.5-P.6 Functions & Analyzing Graphs of Functions p.58-84
P.5-P.6 Functions & Analyzing Graphs of Functions p.58-84 Objectives: Determine whether relations between two variables are functions. Use function notation and evaluate functions. Find the domains of
More informationSolve the following system of equations. " 2x + 4y = 8 # $ x 3y = 1. 1 cont d. You try:
1 Solve the following system of equations. " 2x + 4y = 8 # $ x 3y = 1 Method 1: Substitution 1. Solve for x in the second equation. 1 cont d Method 3: Eliminate y 1. Multiply first equation by 3 and second
More informationUnit 6 Quadratic Functions
Unit 6 Quadratic Functions 12.1 & 12.2 Introduction to Quadratic Functions What is A Quadratic Function? How do I tell if a Function is Quadratic? From a Graph The shape of a quadratic function is called
More informationUnit 1 Quadratic Functions
Unit 1 Quadratic Functions This unit extends the study of quadratic functions to include in-depth analysis of general quadratic functions in both the standard form f ( x) = ax + bx + c and in the vertex
More informationMath 370 Exam 1 Review Name. Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x.
Math 370 Exam 1 Review Name Determine whether the relation is a function. 1) {(-6, 6), (-6, -6), (1, 3), (3, -8), (8, -6)} Not a function The x-value -6 corresponds to two different y-values, so this relation
More information2.1 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
More information3, 10,( 2, 4) Name. CP Algebra II Midterm Review Packet Unit 1: Linear Equations and Inequalities. Solve each equation. 3.
Name CP Algebra II Midterm Review Packet 018-019 Unit 1: Linear Equations and Inequalities Solve each equation. 1. x. x 4( x 5) 6x. 8x 5(x 1) 5 4. ( k ) k 4 5. x 4 x 6 6. V lhw for h 7. x y b for x z Find
More informationAlgebra 2 Chapter Relations and Functions
Algebra 2 Chapter 2 2.1 Relations and Functions 2.1 Relations and Functions / 2.2 Direct Variation A: Relations What is a relation? A of items from two sets: A set of values and a set of values. What does
More informationUnit 1: Sections Skill Set
MthSc 106 Fall 2011 Calculus of One Variable I : Calculus by Briggs and Cochran Section 1.1: Review of Functions Unit 1: Sections 1.1 3.3 Skill Set Find the domain and range of a function. 14, 17 13, 15,
More informationAlgebra II. Slide 1 / 181. Slide 2 / 181. Slide 3 / 181. Conic Sections Table of Contents
Slide 1 / 181 Algebra II Slide 2 / 181 Conic Sections 2015-04-21 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 181 Review of Midpoint and Distance Formulas Introduction
More information1.1 THIS IS LINES 1.2 FUNCTIONS
GOOGLE SHEETS 1.1 THIS IS LINES 1.2 FUNCTIONS I CAN LEARN HOW TO EVALUATE FUNCTIONS AND FIND THEIR DOMAINS. I HAVE A VIDEO POSTED ONLINE THAT HELPS YOU THROUGH THE MIRE OF GOOGLE SHEETS. ON THE VIDEO I
More informationQUESTIONS 1 10 MAY BE DONE WITH A CALCULATOR QUESTIONS ARE TO BE DONE WITHOUT A CALCULATOR. Name
QUESTIONS 1 10 MAY BE DONE WITH A CALCULATOR QUESTIONS 11 5 ARE TO BE DONE WITHOUT A CALCULATOR Name 2 CALCULATOR MAY BE USED FOR 1-10 ONLY Use the table to find the following. x -2 2 5-0 7 2 y 12 15 18
More informationSection 1.5 Transformation of Functions
Section 1.5 Transformation of Functions 61 Section 1.5 Transformation of Functions Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs and equations
More informationUnit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form
Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form Imagine the path of a basketball as it leaves a player s hand and swooshes through the net. Or, imagine the path of an Olympic diver
More informationMAC Learning Objectives. Transformation of Graphs. Module 5 Transformation of Graphs. - A Library of Functions - Transformation of Graphs
MAC 1105 Module 5 Transformation of Graphs Learning Objectives Upon completing this module, you should be able to: 1. Recognize the characteristics common to families of functions. 2. Evaluate and graph
More informationMAC Module 5 Transformation of Graphs. Rev.S08
MAC 1105 Module 5 Transformation of Graphs Learning Objectives Upon completing this module, you should be able to: 1. Recognize the characteristics common to families of functions. 2. Evaluate and graph
More informationPre-Calculus 11: Final Review
Pre-Calculus 11 Name: Block: FORMULAS Sequences and Series Pre-Calculus 11: Final Review Arithmetic: = + 1 = + or = 2 + 1 Geometric: = = or = Infinite geometric: = Trigonometry sin= cos= tan= Sine Law:
More informationREVIEW FOR THE FIRST SEMESTER EXAM
Algebra II Honors @ Name Period Date REVIEW FOR THE FIRST SEMESTER EXAM You must NEATLY show ALL of your work ON SEPARATE PAPER in order to receive full credit! All graphs must be done on GRAPH PAPER!
More informationThe notion of functions
Chapter 1 The notion of functions Textbook Chapter 1 1.1 The concept of functions Although the concept of functions was invented a very long time ago, it is very easy today to gain an intuitive notion
More informationSection 1.5 Transformation of Functions
6 Chapter 1 Section 1.5 Transformation of Functions Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs and equations in order to explain or
More informationCollege Pre Calculus A Period. Weekly Review Sheet # 1 Assigned: Monday, 9/9/2013 Due: Friday, 9/13/2013
College Pre Calculus A Name Period Weekly Review Sheet # 1 Assigned: Monday, 9/9/013 Due: Friday, 9/13/013 YOU MUST SHOW ALL WORK FOR EVERY QUESTION IN THE BOX BELOW AND THEN RECORD YOUR ANSWERS ON THE
More informationChapter 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
More information+ bx + c = 0, you can solve for x by using The Quadratic Formula. x
Math 33B Intermediate Algebra Fall 01 Name Study Guide for Exam 4 The exam will be on Friday, November 9 th. You are allowed to use one 3" by 5" index card on the exam as well as a scientific calculator.
More informationUNIT 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
More informationQUADRATIC FUNCTIONS: MINIMUM/MAXIMUM POINTS, USE OF SYMMETRY. 7.1 Minimum/Maximum, Recall: Completing the square
CHAPTER 7 QUADRATIC FUNCTIONS: MINIMUM/MAXIMUM POINTS, USE OF SYMMETRY 7.1 Minimum/Maximum, Recall: Completing the square The completing the square method uses the formula x + y) = x + xy + y and forces
More informationMAC 1105 Fall Term 2018
MAC 1105 Fall Term 2018 Each objective covered in MAC 1105 is listed below. Along with each objective is the homework list used with MyMathLab (MML) and a list to use with the text (if you want to use
More informationAlgebra II Notes Unit Two: Linear Equations and Functions
Syllabus Objectives:.1 The student will differentiate between a relation and a function.. The student will identify the domain and range of a relation or function.. The student will derive a function rule
More informationLesson 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
More informationUnit Essential Questions: Does it matter which form of a linear equation that you use?
Unit Essential Questions: Does it matter which form of a linear equation that you use? How do you use transformations to help graph absolute value functions? How can you model data with linear equations?
More information1-2 Analyzing Graphs of Functions and Relations
Use the graph of each function to estimate the indicated function values. Then confirm the estimate algebraically. Round to the nearest hundredth, if necessary. The function value at x = 1 appears to be
More informationLesson #6: Basic Transformations with the Absolute Value Function
Lesson #6: Basic Transformations with the Absolute Value Function Recall: Piecewise Functions Graph:,, What parent function did this piecewise function create? The Absolute Value Function Algebra II with
More information1.8 Coordinate Geometry. Copyright Cengage Learning. All rights reserved.
1.8 Coordinate Geometry Copyright Cengage Learning. All rights reserved. Objectives The Coordinate Plane The Distance and Midpoint Formulas Graphs of Equations in Two Variables Intercepts Circles Symmetry
More informationDOWNLOAD PDF BIG IDEAS MATH VERTICAL SHRINK OF A PARABOLA
Chapter 1 : BioMath: Transformation of Graphs Use the results in part (a) to identify the vertex of the parabola. c. Find a vertical line on your graph paper so that when you fold the paper, the left portion
More informationPrecalculus Notes Unit 1 Day 1
Precalculus Notes Unit Day Rules For Domain: When the domain is not specified, it consists of (all real numbers) for which the corresponding values in the range are also real numbers.. If is in the numerator
More informationMath 8 Honors Coordinate Geometry part 3 Unit Updated July 29, 2016
Review how to find the distance between two points To find the distance between two points, use the Pythagorean theorem. The difference between is one leg and the difference between and is the other leg.
More informationUNIT 1: NUMBER LINES, INTERVALS, AND SETS
ALGEBRA II CURRICULUM OUTLINE 2011-2012 OVERVIEW: 1. Numbers, Lines, Intervals and Sets 2. Algebraic Manipulation: Rational Expressions and Exponents 3. Radicals and Radical Equations 4. Function Basics
More informationChapter 2: Introduction to Functions
Chapter 2: Introduction to Functions Lesson 1: Introduction to Functions Lesson 2: Function Notation Lesson 3: Composition of Functions Lesson 4: Domain and Range Lesson 5: Restricted Domain Lesson 6:
More informationSlide 1 / 96. Linear Relations and Functions
Slide 1 / 96 Linear Relations and Functions Slide 2 / 96 Scatter Plots Table of Contents Step, Absolute Value, Piecewise, Identity, and Constant Functions Graphing Inequalities Slide 3 / 96 Scatter Plots
More informationALGEBRA II A CURRICULUM OUTLINE
ALGEBRA II A CURRICULUM OUTLINE 2013-2014 OVERVIEW: 1. Linear Equations and Inequalities 2. Polynomial Expressions and Equations 3. Rational Expressions and Equations 4. Radical Expressions and Equations
More information1 Review of Functions Symmetry of Functions; Even and Odd Combinations of Functions... 42
Contents 0.1 Basic Facts...................................... 8 0.2 Factoring Formulas.................................. 9 1 Review of Functions 15 1.1 Functions.......................................
More informationMath 1113 Notes - Functions Revisited
Math 1113 Notes - Functions Revisited Philippe B. Laval Kennesaw State University February 14, 2005 Abstract This handout contains more material on functions. It continues the material which was presented
More informationUnit 2: Functions and Graphs
AMHS Precalculus - Unit 16 Unit : Functions and Graphs Functions A function is a rule that assigns each element in the domain to exactly one element in the range. The domain is the set of all possible
More informationSection 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
More informationYEAR 7 SCHEME OF WORK - EXTENSION
YEAR 7 SCHEME OF WORK - EXTENSION Autumn Term 1 Number Skills Spring Term 1 Angles and Shape Summer Term 1 Multiplicative Reasoning Analysing and displaying data Decimals Perimeter, Area and Volume Half
More informationCourse Number: Course Title: Geometry
Course Number: 1206310 Course Title: Geometry RELATED GLOSSARY TERM DEFINITIONS (89) Altitude The perpendicular distance from the top of a geometric figure to its opposite side. Angle Two rays or two line
More informationLesson 20: Four Interesting Transformations of Functions
Student Outcomes Students apply their understanding of transformations of functions and their graphs to piecewise functions. Lesson Notes In Lessons 17 19 students study translations and scalings of functions
More informationYou 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
More information2.4. A LIBRARY OF PARENT FUNCTIONS
2.4. A LIBRARY OF PARENT FUNCTIONS 1 What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal function. Identify and graph step and
More informationFinal Exam Review Algebra Semester 1
Final Exam Review Algebra 015-016 Semester 1 Name: Module 1 Find the inverse of each function. 1. f x 10 4x. g x 15x 10 Use compositions to check if the two functions are inverses. 3. s x 7 x and t(x)
More informationThe notion of functions
Chapter 1 The notion of functions Textbook Chapter 1 1.1 The concept of functions Although the concept of functions was invented a very long time ago, it is very easy today to gain an intuitive notion
More informationSECTION 1.3: BASIC GRAPHS and SYMMETRY
(Section.3: Basic Graphs and Symmetry).3. SECTION.3: BASIC GRAPHS and SYMMETRY LEARNING OBJECTIVES Know how to graph basic functions. Organize categories of basic graphs and recognize common properties,
More informationSample tasks from: Algebra Assessments Through the Common Core (Grades 6-12)
Sample tasks from: Algebra Assessments Through the Common Core (Grades 6-12) A resource from The Charles A Dana Center at The University of Texas at Austin 2011 About the Dana Center Assessments More than
More informationMini-Lecture 3.1 Graphing Equations
Copyright 0 Pearson Education, Inc. Mini-Lecture. Graphing Equations. Plot ordered pairs.. Determine whether an ordered pair of numbers is a solution to an equation in two variables.. Graph linear equations.
More informationPlanar graphs. Chapter 8
Chapter 8 Planar graphs Definition 8.1. A graph is called planar if it can be drawn in the plane so that edges intersect only at vertices to which they are incident. Example 8.2. Different representations
More informationYork Public Schools Subject Area: Mathematics Course: 6 th Grade Math NUMBER OF DAYS TAUGHT DATE
6.1.1.d 6.EE.A.1 Represent large numbers using exponential notation (i.e.. 10x10x10x10x10) (Review PV chart first) Write evaluate numerical expressions involving whole number exponents 5 days August, Lesson
More informationOther Functions and their Inverses
CHAPTER Other Functions and their Inverses Water tanks have been used throughout human history to store water for consumption. Many municipal water tanks are placed on top of towers so that water drawn
More informationCCSSM Curriculum Analysis Project Tool 1 Interpreting Functions in Grades 9-12
Tool 1: Standards for Mathematical ent: Interpreting Functions CCSSM Curriculum Analysis Project Tool 1 Interpreting Functions in Grades 9-12 Name of Reviewer School/District Date Name of Curriculum Materials:
More informationDowker Notation and Arc Presentation
Dowker Notation and Arc Presentation Jennifer Waters Department of Mathematics California State University- Channel Islands Camarillo, Ca 93012 August 18, 2009 Abstract In this paper we will discuss Cubic
More informationMATHia Unit MATHia Workspace Overview CCSS
1 Function Overview Searching for Patterns Exploring and Analyzing Patterns Comparing Familiar Function Representations Students watch a video about a well-known mathematician creating an expression for
More informationChapter 10. Exploring Conic Sections
Chapter 10 Exploring Conic Sections Conics A conic section is a curve formed by the intersection of a plane and a hollow cone. Each of these shapes are made by slicing the cone and observing the shape
More informationPlatte County School District #2 Mathematics Standards 4th Grade School Year
Platte County School District #2 Mathematics Standards 4th Grade 2017-2018 School Year Standard 4.OA.A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that
More informationNBT 4 Fluently add and subtract multi-digit whole number using the standard algorithm.
Lincoln lementary School Curriculum Prioritization and Mapping Timeline Topic Priority Standard Learning Targets August (15 instructional Days) August August August Numbers and Base Ten Concepts Numbers
More informationPrecalculus Chapter 2A Practice Guide Name
Precalculus Chapter A Practice Guide Name Day 1 Day.1 (page 96). (page 108 ).3 (page 1) 15,1,,3,7,33 37,4,49,50,5,55 17,30,38,47,53,61 67,85 Day 3 43,48,51,68 1,4,6,7,13,16,18,19.4 Worksheets.5 (page 145)
More informationPart I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each.
Math 106/108 Final Exam Page 1 Part I. Problems in this section are mostly short answer and multiple choice. Little partial credit will be given. 5 points each. 1. Factor completely. Do not solve. a) 2x
More informationCLEP Pre-Calculus. Section 1: Time 30 Minutes 50 Questions. 1. According to the tables for f(x) and g(x) below, what is the value of [f + g]( 1)?
CLEP Pre-Calculus Section : Time 0 Minutes 50 Questions For each question below, choose the best answer from the choices given. An online graphing calculator (non-cas) is allowed to be used for this section..
More informationAH Properties of Functions.notebook April 19, 2018
Functions Rational functions are of the form where p(x) and q(x) are polynomials. If you can sketch a function without lifting the pencil off the paper, it is continuous. E.g. y = x 2 If there is a break
More informationFunction 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
More informationExit Ticket. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson Lesson 14: Ordered Pairs
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 6 3 Lesson 14: Ordered Pairs 1. On the map below, the fire department and the hospital have one matching coordinate. Determine the proper order of the ordered
More informationMid 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
More information1 of 21 8/6/2018, 8:17 AM
1 of 1 8/6/018, 8:17 AM Student: Date: Instructor: Alfredo Alvarez Course: Math 1314 Summer 018 Assignment: math 131437 Free Response with Help 51 1. Solve the equation by factoring. 9x + 1x 8 = 0 The
More informationSet 5, Total points: 100 Issued: week of
Prof. P. Koumoutsakos Prof. Dr. Jens Walther ETH Zentrum, CLT F 1, E 11 CH-809 Zürich Models, Algorithms and Data (MAD): Introduction to Computing Spring semester 018 Set 5, Total points: 100 Issued: week
More information8. The triangle is rotated around point D to create a new triangle. This looks like a rigid transformation.
2.1 Transformations in the Plane 1. True 2. True 3. False 4. False 5. True 6. False 7. True 8. The triangle is rotated around point D to create a new triangle. This looks like a rigid transformation. 9.
More information4.3 Quadratic functions and their properties
4.3 Quadratic functions and their properties A quadratic function is a function defined as f(x) = ax + x + c, a 0 Domain: the set of all real numers x-intercepts: Solutions of ax + x + c = 0 y-intercept:
More informationMath 2 Coordinate Geometry Part 3 Inequalities & Quadratics
Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics 1 DISTANCE BETWEEN TWO POINTS - REVIEW To find the distance between two points, use the Pythagorean theorem. The difference between x 1 and x
More informationAlgebra I Notes Absolute Value Functions Unit 04c
OBJECTIVES: F.IF.B.4 Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables
More informationHonors Precalculus: Solving equations and inequalities graphically and algebraically. Page 1
Solving equations and inequalities graphically and algebraically 1. Plot points on the Cartesian coordinate plane. P.1 2. Represent data graphically using scatter plots, bar graphs, & line graphs. P.1
More informationMath 112 Spring 2016 Midterm 2 Review Problems Page 1
Math Spring Midterm Review Problems Page. Solve the inequality. The solution is: x x,,,,,, (E) None of these. Which one of these equations represents y as a function of x? x y xy x y x y (E) y x 7 Math
More information