End Behavior and Symmetry
|
|
- Coral Fisher
- 5 years ago
- Views:
Transcription
1 Algebra 2 Interval Notation Name: Date: Block:
2 X Characteristics of Polynomial Functions Lesson Opener: Graph the function using transformations then identify key characteristics listed below. 1. y x A. Parent function: a, h, k 10 Y Table 1 Table Real zero(s): Domain: y-intercept: Range: Vertex: Interval of Increase: Axis of Symmetry: Interval of Decrease: Extrema:
3 End Behavior and Symmetry End behavior describes what the graph of the function does (y-values) as x goes to or. If a polynomial has Even degree, positive lead coefficient: both ends are up o as x, f ( x) Example: o as x, f ( x) Even degree, negative lead coefficient: both ends are down o as x, f ( x) Example: o as x, f ( x) Odd degree, positive lead coefficient: opposite end behavior with left end down and right end up o as x, f ( x) Example: o as x, f ( x) Odd degree, negative lead coefficient: opposite end behavior with left end up and right end down o as x, f ( x) Example: o as x, f ( x)
4 * Even/Odd/Neither and Symmetry Even functions have symmetry with respect to the y-axis Graphing Example: It s easy to determine if a polynomial is an even function: all the exponents are even If you have a constant term, the exponent is zero and in this case it s considered an even term (NOT ODD) Function Example: Odd functions have symmetry with respect to the origin Graphing Example: It s easy to determine if a polynomial is an odd function: all the exponents are odd Remember, if there is a constant term, the exponent is zero and in this case it s considered an even term (NOT ODD) Function Example: Neither means that the graph might have symmetry but neither with respect to the y-axis nor the origin. Graphing Example: It s easy to determine if a polynomial is a neither function: there is a mixture of even and odd exponents Remember, if there is a constant term, the exponent is zero and in this case it s considered an even term (NOT ODD) Function Example:
5 State the characteristics listed based on the graph of each function. 2 1.) f x x 2x g x x 5x 4 2.) 4 2 End behavior: as x, f ( x) End behavior: as x, f ( x) as x, f ( x) Is the function odd, even, or neither? Explain. as x, f ( x) Is the function odd, even, or neither? Explain.
6 h x x x 3.) 3 j x x 2x 3x 4.) 3 2 End behavior: as x, f ( x) End behavior: as x, f ( x) as x, f ( x) Is the function odd, even, or neither? Explain. as x, f ( x) Is the function odd, even, or neither? Explain.
7 Characteristics Notes Example 1: Odd Degree Identify the characteristics listed below for the given polynomial graph. A. Intercepts (all) x-intercepts: Points on the graph that are on the x-axis. They are also the real zeros of the function. A function may have no x-intercepts, one x-intercept, or several (up to the degree of the function). y-intercept: The single point on the graph that s on the y-axis. Every polynomial graph will have exactly one y-intercept. B. Relative and absolute extrema (specify) Extrema are all about the y-values. The absolute maximum is the highest y-value on a graph. The absolute minimum is the lowest y-value on the graph. An even degree function will have ONE absolute maximum or minimum, it cannot have both an absolute maximum AND minimum. A relative maximum is the highest y-value in that part of the graph, but not the highest of all the graph. A relative minimum is the lowest y-value in that part of the graph, but not the lowest of all the graph. A graph may have several relative maximums or minimums, or it may not have any at all. C. Domain: The domain of a function refers to all possible values of x (real numbers) that have a corresponding value of y. Like asking: What x-values am I allowed to choose? The domain of any polynomial function is all real numbers. This is written in interval notation as,. D. Range: The range of a function refers to any y-values found on the graph. The range of any odd degree function will always be all real numbers, or,. The range of an even degree function with a positive leading coefficient (opens up) will be abs min,. The range of an even degree function with a negative leading coefficient (opens down) will be,abs max.
8 E. Intervals of increase or decrease describe the behavior of the function (y-values) for each section of the domain which is divided by the maximum or minimum points (turning points). Intervals of increase or decrease are always stated using x-values. F. End Behavior As x, f x As x, f x G. Is the function odd, even, or neither? EXPLAIN. If a function has an odd degree AND is symmetric to the origin, then it s an odd function. If a function has an even degree AND is symmetric to the y-axis, then it s an even function. If a function is not symmetric to the origin or the y-axis it is neither (an odd or even function).
9 Example 2: Even Degree Identify the characteristics listed below for the given polynomial graph. A. Intercepts (all) x-intercepts: Points on the graph that are on the x-axis. They are also the real zeros of the function. A function may have no x-intercepts, one x-intercept, or several (up to the degree of the function). y-intercept: The single point on the graph that s on the y-axis. Every polynomial graph will have exactly one y-intercept. B. Relative and absolute extrema (specify): Extrema are all about the y-values. The absolute maximum is the highest y-value on a graph. The absolute minimum is the lowest y-value on the graph. An even degree function will have ONE absolute maximum or minimum, it cannot have both an absolute maximum AND minimum. A relative maximum is the highest y-value in that part of the graph, but not the highest of all the graph. A relative minimum is the lowest y-value in that part of the graph, but not the lowest of all the graph. A graph may have several relative maximums or minimums, or it may not have any at all. Domain: The domain of a function refers to all possible values of x (real numbers) that have a corresponding value of y. Like asking: What x-values am I allowed to choose? The domain of any polynomial function is all real numbers. This is written in interval notation as,. Range: The range of a function refers to any y-values found on the graph. The range of any odd degree function will always be all real numbers, or,. The range of an even degree function with a positive leading coefficient (opens up) will be abs min,. The range of an even degree function with a negative leading coefficient (opens down) will be,abs max.
10 E. Intervals of increase or decrease describe the behavior of the function (y-values) for each section of the domain which is divided by the maximum or minimum points (turning points). Intervals of increase or decrease are always stated using x-values. F. End Behavior As x, f x As x, f x G. Is the function odd, even, or neither? EXPLAIN. If a function has an odd degree AND is symmetric to the origin, then it s an odd function. If a function has an even degree AND is symmetric to the y-axis, then it s an even function. If a function is not symmetric to the origin or the y-axis it is neither (an odd or even function).
Lesson #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 informationMath-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
More informationGSE Algebra 1 Name Date Block. Unit 3b Remediation Ticket
Unit 3b Remediation Ticket Question: Which function increases faster, f(x) or g(x)? f(x) = 5x + 8; two points from g(x): (-2, 4) and (3, 10) Answer: In order to compare the rate of change (roc), you must
More informationUNIT 8: SOLVING AND GRAPHING QUADRATICS. 8-1 Factoring to Solve Quadratic Equations. Solve each equation:
UNIT 8: SOLVING AND GRAPHING QUADRATICS 8-1 Factoring to Solve Quadratic Equations Zero Product Property For all numbers a & b Solve each equation: If: ab 0, 1. (x + 3)(x 5) = 0 Then one of these is true:
More informationUnit 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
More informationUnit #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
More informationUnit 2 Day 6. Characteristics Of Quadratic, Even, And Odd Functions
Unit 2 Day 6 Characteristics Of Quadratic, Even, And Odd Functions 1 Warm Up 1.) Jenna is trying to invest money into the stock exchange. After some research, she has narrowed it down to two companies.
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 informationUNIT 8 STUDY SHEET POLYNOMIAL FUNCTIONS
UNIT 8 STUDY SHEET POLYNOMIAL FUNCTIONS KEY FEATURES OF POLYNOMIALS Intercepts of a function o x-intercepts - a point on the graph where y is zero {Also called the zeros of the function.} o y-intercepts
More information[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#
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 informationUnit: 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
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 informationUse Derivatives to Sketch the Graph of a Polynomial Function.
Applications of Derivatives Curve Sketching (using derivatives): A) Polynomial Functions B) Rational Functions Lesson 5.2 Use Derivatives to Sketch the Graph of a Polynomial Function. Idea: 1) Identify
More informationAlbertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.
Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2017 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the
More 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 informationSection Functions. Function Notation. Is this a function?
Section 1-21 Functions and Their Properties Section 1-21 function definition and notation domain and range continuity increasing/decreasing boundedness local and absolute extrema symmetry asymptotes end
More 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 informationUNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions. Instruction. Guided Practice Example 1
Guided Practice Example 1 A local store s monthly revenue from T-shirt sales is modeled by the function f(x) = 5x 2 + 150x 7. Use the equation and graph to answer the following questions: At what prices
More informationMore Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a
More Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a Example: Solve using the square root property. a) x 2 144 = 0 b) x 2 + 144 = 0 c) (x + 1) 2 = 12 Completing
More informationIntegers and Absolute Value. Unit 1 Lesson 5
Unit 1 Lesson 5 Students will be able to: Understand integers and absolute value Key Vocabulary: An integer Positive number Negative number Absolute value Opposite Integers An integer is a positive or
More information2/22/ Transformations but first 1.3 Recap. Section Objectives: Students will know how to analyze graphs of functions.
1 2 3 4 1.4 Transformations but first 1.3 Recap Section Objectives: Students will know how to analyze graphs of functions. 5 Recap of Important information 1.2 Functions and their Graphs Vertical line
More informationPolynomial Functions Graphing Investigation Unit 3 Part B Day 1. Graph 1: y = (x 1) Graph 2: y = (x 1)(x + 2) Graph 3: y =(x 1)(x + 2)(x 3)
Part I: Polynomial Functions when a = 1 Directions: Polynomial Functions Graphing Investigation Unit 3 Part B Day 1 1. For each set of factors, graph the zeros first, then use your calculator to determine
More informationSection 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
More informationAlgebra II Quadratic Functions
1 Algebra II Quadratic Functions 2014-10-14 www.njctl.org 2 Ta b le o f C o n te n t Key Terms click on the topic to go to that section Explain Characteristics of Quadratic Functions Combining Transformations
More informationSECTION 1.2 (e-book 2.3) Functions: Graphs & Properties
SECTION 1.2 (e-book 2.3) Functions: Graphs & Properties Definition (Graph Form): A function f can be defined by a graph in the xy-plane. In this case the output can be obtained by drawing vertical line
More informationSection 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
More informationCHAPTER 6 Quadratic Functions
CHAPTER 6 Quadratic Functions Math 1201: Linear Functions is the linear term 3 is the leading coefficient 4 is the constant term Math 2201: Quadratic Functions Math 3201: Cubic, Quartic, Quintic Functions
More informationChapter 2: Polynomial and Rational Functions Power Standard #7
Chapter 2: Polynomial and Rational s Power Standard #7 2.1 Quadratic s Lets glance at the finals. Learning Objective: In this lesson you learned how to sketch and analyze graphs of quadratic functions.
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 informationQuadratic 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
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 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 informationAlgebraically Speaking Chalkdust Algebra 1 Fall Semester
Algebraically Speaking Chalkdust Algebra 1 Fall Semester Homework Assignments: Chapter 1 The Real Number System: Lesson 1.1 - Real Numbers: Order and Absolute Value Do the following problems: # 1 9 Odd,
More informationFoundations 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,
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 informationAlgebra 2CP S1 Final Exam Information. Your final exam will consist of two parts: Free Response and Multiple Choice
Algebra 2CP Name Algebra 2CP S1 Final Exam Information Your final exam will consist of two parts: Free Response and Multiple Choice Part I: Free Response: Five questions, 10 points each (50 points total),
More information5-3 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
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 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 informationDependent Variable Independent Variable dependent variable : independent variable function: domain ran ge
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
More informationALGEBRA 2 W/ TRIGONOMETRY MIDTERM REVIEW
Name: Block: ALGEBRA W/ TRIGONOMETRY MIDTERM REVIEW Algebra 1 Review Find Slope and Rate of Change Graph Equations of Lines Write Equations of Lines Absolute Value Functions Transformations Piecewise Functions
More information1.1 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
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 informationGraphing Polynomial Functions: The Leading Coefficient Test and End Behavior: For an n th n. degree polynomial function a 0, 0, then
Graphing Polynomial Functions: The Leading Coefficient Test and End Behavior: For an n th n n 1 degree polynomial function a 0, n If n is even and an 0, then f x a x a x a x a with n n 1 1 0 x If n is
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 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 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 informationMath 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
More informationObtaining Information from a Function s Graph.
Obtaining Information from a Function s Graph Summary about using closed dots, open dots, and arrows on the graphs 1 A closed dot indicate that the graph does not extend beyond this point and the point
More informationThe x-intercept can be found by setting y = 0 and solving for x: 16 3, 0
y=-3/4x+4 and y=2 x I need to graph the functions so I can clearly describe the graphs Specifically mention any key points on the graphs, including intercepts, vertex, or start/end points. What is the
More informationSec.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,
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 informationMath 2 Final Exam Study Guide. Translate down 2 units (x, y-2)
Math 2 Final Exam Study Guide Name: Unit 2 Transformations Translation translate Slide Moving your original point to the left (-) or right (+) changes the. Moving your original point up (+) or down (-)
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 informationExample 1: Given below is the graph of the quadratic function f. Use the function and its graph to find the following: Outputs
Quadratic Functions: - functions defined by quadratic epressions (a 2 + b + c) o the degree of a quadratic function is ALWAYS 2 - the most common way to write a quadratic function (and the way we have
More informationGraphing Rational Functions
Graphing Rational Functions Return to Table of Contents 109 Vocabulary Review x-intercept: The point where a graph intersects with the x-axis and the y-value is zero. y-intercept: The point where a graph
More information3.1 Investigating Quadratic Functions in Vertex Form
Math 2200 Date: 3.1 Investigating Quadratic Functions in Vertex Form Degree of a Function - refers to the highest exponent on the variable in an expression or equation. In Math 1201, you learned about
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 informationProperties 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
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 informationDetermine 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
More informationSection Rational Functions and Inequalities. A rational function is a quotient of two polynomials. That is, is a rational function if
Section 6.1 --- Rational Functions and Inequalities A rational function is a quotient of two polynomials. That is, is a rational function if =, where and are polynomials and is not the zero polynomial.
More informationAlgebra 2 Semester 1 (#2221)
Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the 2016-2017 Course Guides for the following course: Algebra 2 Semester
More informationAccelerated Precalculus 1.2 (Intercepts and Symmetry) Day 1 Notes. In 1.1, we discussed using t-charts to help graph functions. e.g.
Accelerated Precalculus 1.2 (Intercepts and Symmetry) Day 1 Notes In 1.1, we discussed using t-charts to help graph functions. e.g., Graph: y = x 3 What are some other strategies that can make graphing
More informationThe Graph of a Rational Function. R x
Precalculus.7 Notes The Graph of a Rational Function Analyzing the Graph of a Rational Function 1. Completely factor the numerator and denominator.. List the key features of the graph. Domain: Set the
More informationGraphing Techniques. Domain (, ) Range (, ) Squaring Function f(x) = x 2 Domain (, ) Range [, ) f( x) = x 2
Graphing Techniques In this chapter, we will take our knowledge of graphs of basic functions and expand our ability to graph polynomial and rational functions using common sense, zeros, y-intercepts, stretching
More informationIntroduction : 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
More informationNOTES: ALGEBRA FUNCTION NOTATION
STARTER: 1. Graph f by completing the table. f, y -1 0 1 4 5 NOTES: ALGEBRA 4.1 FUNCTION NOTATION y. Graph f 4 4 f 4 4, y --5-4 - - -1 0 1 y A Brief Review of Function Notation We will be using function
More informationVertex maximum or minimum Axis of Symmetry OPENS: UP MINIMUM
5.1 GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM & MUTIPLYING BINOMIALS Standard Form of a Quadratic: y ax bx c or f x ax bx c ex. y x 5x 13 a= b= c=. Every function/graph in the Quadratic family originates
More informationName: Date: Absolute Value Transformations
Name: Date: Absolute Value Transformations Vocab: Absolute value is the measure of the distance awa from zero on a number line. Since absolute value is the measure of distance it can never be negative!
More information1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check
Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's
More information2. From General Form: y = ax 2 + bx + c # of x-intercepts determined by the, D =
Alg2H 5-3 Using the Discriminant, x-intercepts, and the Quadratic Formula WK#6 Lesson / Homework --Complete without calculator Read p.181-p.186. Textbook required for reference as well as to check some
More informationInequalities and you 3
Inequalities and you 3 NAME: This worksheet will provide practice for solving absolute value, polynomial, and rational inequalities. We will also work on understanding why the procedures work. We will
More informationQuadratic Equations. Learning Objectives. Quadratic Function 2. where a, b, and c are real numbers and a 0
Quadratic Equations Learning Objectives 1. Graph a quadratic function using transformations. Identify the vertex and axis of symmetry of a quadratic function 3. Graph a quadratic function using its vertex,
More informationAlgebra II Radical Equations
1 Algebra II Radical Equations 2016-04-21 www.njctl.org 2 Table of Contents: Graphing Square Root Functions Working with Square Roots Irrational Roots Adding and Subtracting Radicals Multiplying Radicals
More informationFebruary 8 th February 12 th. Unit 6: Polynomials & Introduction to Quadratics
Algebra I February 8 th February 12 th Unit 6: Polynomials & Introduction to Quadratics Jump Start 1) Use the elimination method to solve the system of equations below. x + y = 2 3x + y = 8 2) Solve: 13
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 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 informationQuadratic Functions Dr. Laura J. Pyzdrowski
1 Names: (8 communication points) About this Laboratory A quadratic function in the variable x is a polynomial where the highest power of x is 2. We will explore the domains, ranges, and graphs of quadratic
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 informationGRAPHING POLYNOMIALS DAY 2 U N I T 1 1
GRAPHING POLYNOMIALS DAY 2 U N I T 1 1 ODD/EVEN DEGREE POLYNOMIAL Odd degree polynomial: A polynomial whose largest power is an odd integer Even degree polynomial : A polynomial whose largest power is
More informationMath 1314 Lesson 12 Curve Analysis (Polynomials) This lesson will cover analyzing polynomial functions using GeoGebra.
Math 1314 Lesson 12 Curve Analysis (Polynomials) This lesson will cover analyzing polynomial functions using GeoGebra. Suppose your company embarked on a new marketing campaign and was able to track sales
More informationPOLYNOMIALS Graphing Polynomial Functions Common Core Standard
K Polynomials, Lesson 6, Graphing Polynomial Functions (r. 2018) POLYNOMIALS Graphing Polynomial Functions Common Core Standard Next Generation Standard F-BF.3 Identify the effect on the graph of replacing
More information5.1 Introduction to the Graphs of Polynomials
Math 3201 5.1 Introduction to the Graphs of Polynomials In Math 1201/2201, we examined three types of polynomial functions: Constant Function - horizontal line such as y = 2 Linear Function - sloped line,
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 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 informationSummer Math Assignments for Students Entering Algebra II
Summer Math Assignments for Students Entering Algebra II Purpose: The purpose of this packet is to review pre-requisite skills necessary for the student to be successful in Algebra II. You are expected
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 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 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 information1.) ( ) Step 1: Factor the numerator and the denominator. Find the domain. is in lowest terms.
GP3-HW11 College Algebra Sketch the graph of each rational function. 1.) Step 1: Factor the numerator and the denominator. Find the domain. { } Step 2: Rewrite in lowest terms. The rational function is
More informationRational functions, like rational numbers, will involve a fraction. We will discuss rational functions in the form:
Name: Date: Period: Chapter 2: Polynomial and Rational Functions Topic 6: Rational Functions & Their Graphs Rational functions, like rational numbers, will involve a fraction. We will discuss rational
More informationQuadratics Functions: Review
Quadratics Functions: Review Name Per Review outline Quadratic function general form: Quadratic function tables and graphs (parabolas) Important places on the parabola graph [see chart below] vertex (minimum
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 informationSummer Math Assignments for Students Entering Integrated Math
Summer Math Assignments for Students Entering Integrated Math Purpose: The purpose of this packet is to review pre-requisite skills necessary for the student to be successful in Integrated Math. You are
More informationSlide 1 / 180. Radicals and Rational Exponents
Slide 1 / 180 Radicals and Rational Exponents Slide 2 / 180 Roots and Radicals Table of Contents: Square Roots Intro to Cube Roots n th Roots Irrational Roots Rational Exponents Operations with Radicals
More informationALGEBRA 1 NOTES. Quarter 3. Name: Block
2016-2017 ALGEBRA 1 NOTES Quarter 3 Name: Block Table of Contents Unit 8 Exponent Rules Exponent Rules for Multiplication page 4 Negative and Zero Exponents page 8 Exponent Rules Involving Quotients page
More informationStandard Form of Quadratic Functions
Math Objectives Students will be able to predict how a specific change in the value of a will affect the shape of the graph of the quadratic ax bx c. Students will be able to predict how a specific change
More information