# Symmetry. A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

Save this PDF as:

Size: px
Start display at page:

Download "Symmetry. A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point (-x, y) is also on the graph."

## Transcription

1 Symmetry When we graphed y =, y = 2, y =, y = 3 3, y =, and y =, we mentioned some of the features of these members of the Library of Functions, the building blocks for much of the study of algebraic functions. Now we go deeper into the study of symmetry, eploring three main categories or types of symmetry. TI-83 note: We will be using a variety of window settings, but unless otherwise indicated, the scales for both the horizontal and vertical aes will be. Symmetry with respect to the y-ais: The first function we ll consider is the squaring function y = 2 whose graph is the parabola shown below. We mentioned before that this is a simple quadratic function, and its main feature is its verte, the point given by (0, 0). This graph clearly has symmetry about the y-ais (the line = 0). Notice the table values in pairs, like (-3, 9) and (3, 9), (-2, 4) and (2, 4), etc., and then read carefully the following generalization: A graph is symmetric with respect to the y-ais if, for every point (, y) on the graph, the point (-, y) is also on the graph. Notice in the table that f(-3) = f(3) = 9 and that f(-2) = f(2) = 4. When f(-) = f() for all real values of in the domain of a function, the function is called an even function, and the graph has symmetry with respect to the y-ais. Another function in the library that has symmetry with respect to the y-ais is the absolute value function, y =, shown in the figure to the right. This, too, is an even function, with points such as (-3, 3) and (3, 3), (-2, 2) and (2, 2), (-, ) and (, ). The origin (0, 0) is its minimum point.

2 Symmetry with respect to the -ais: A function cannot have symmetry with respect to the -ais, so now we turn to a few relations to illustrate this type of symmetry. The graph of y 2 = is shown below. Compare the table for y = 2 with the table below. These two relations have a special relationship; they are inverses of one another. (More on this, later!) y Another relation that has symmetry with respect to the -ais is = y, shown below. This relation includes points such as (0, 0), (, ), (, -), (2, 2), and (2, -2). Again, notice the relationship between this graph and its cousin, the absolute value function, y =. In our two eamples, notice that a substitution of y for y yields essentially the same equation: ( y) 2 = y 2 = and = y = y. Whenever this is the case, the relation has symmetry with respect to the -ais. A graph is symmetric with respect to the -ais if, for every point (, y) on the graph, the point (, -y) is also on the graph. Our third and final type of symmetry to eplore is the first that is not primarily reflectional, but rotational. Here we ll begin with the algebraic generalization and then move into eamples. Symmetry with respect to the origin: A graph is symmetric with respect to the origin if, for every point (, y) on the graph, the point (-, -y) is also on the graph. Here are a few eamples. 2

3 Here are a few eamples. Imagine rotating the graph of y = a full 80 about the origin. All the points in the first quadrant (points like (, ), (2, 2) and (5, 5)) would land on points in the third quadrant (on (-, -), (-2, -2), and (-5, -5)), and vice versa. Notice also that both (, ) and (-, -) are points on the graph, as are (2, 2) and (-2, -2), etc. This graph is of the cubic function, f ( ) = 3. Notice that f(-3) and f(3) are opposites, -27 and 27, and that a similar pattern continues through the table: f(-3) = -f(3)= -27, f(-2) = -f(2) = -8, and so on. In general, f(-) = -f(). y This is the rational function given by y =. Again, notice the clear pattern in the table. For every point (, y) in the table, there s another point (-, -y) also in the table. Also notice that cannot be 0. y In each of our three eamples, f(-) = -f(). When f(-) = -f() for all real values of in the domain of a function, the function is called an odd function and the graph has symmetry with respect to the origin. Reflecting the graph 80 about the origin yields an identical graph. Refer to the following chart as a summary of the odd and even function tests : Odd function Even function f(-) = -f() for all in the domain of f f(-) = f() for all in the domain of f It is best to simply calculate f(-), -f() and compare these results with the original function, f(). 3

4 Eamples: () Test each function for symmetry with respect to the y-ais, the -ais, and the origin. Graph each function to verify your conclusions. (a) y = 2 ( ) 2 = 2, the graph is symmetric with respect to the y-ais. Substituting y for y yields y = 2 or y =, so this function isn t symmetric with 2 respect to the -ais. Similarly, replacing both y for y and for yields y = ( ) 2 or y =, so this function also isn t symmetric with 2 respect to the origin. Solution: Since replacing by - yields an equivalent function y = The graph below verifies symmetry with respect to the y-ais. (b) f() = ( ) 3 Solution: Replacing by yields y = ( ) 3, which is not equivalent to the original function. The graph, therefore, won t have symmetry with respect to the y-ais. Replacing y by y yields -y = won t have symmetry with respect to the -ais. Replacing by and y by y yields -y = - y = ( ) 3 or y = + with respect to the origin. By the graph, it is true that the graph has point symmetry about (,0) which is its inflection point, but it has none of the three symmetries we re considering. ( ) 3 or y = - ( ) 3, so the graph ( ) 3, which simplifies to ( ) 3, so the graph won t have symmetry 4

5 (c) 2 + y 2 = 4 Solution: This circle relation has symmetry with respect to the y-ais, -ais, and the origin. It also has reflectional symmetry over any line passing through the origin and rotational symmetry through any angle with the origin as a fied point. ( ) 2 + y 2 = 4, 2 + (-y) 2 = 4, and (-) 2 + (-y) 2 = 4 are The 3 equations - each equivalent to 2 + y 2 = 4, so we have algebraic verification for symmetry with respect to the y-ais, -ais, and the origin, respectively. (2) Determine algebraically whether the given function is odd, even, or neither. Graph each function to verify your conclusions. (a) f() = 3 Solution: f( ) = 3( ) = 3 f() = 3 Since f() = f(-), the function is even, and its graph has symmetry with respect to the y- ais. Since f(-) -f(), the function is not odd. (b) g() = 2 Solution: g(-) = ( 2 -) = ( 2 ) -g() = - ( 2 ) Since g() g(-), the function is not even. Since g(-) = -g(), the function is odd, and its graph has symmetry with respect to the origin. 5

6 (c) h() = Solution: h(-) = - -h() = Since h() h(-), the function is not even. Since h(-) -h(), the function is not odd. The function is neither even, nor odd. These conclusions are all supported by the graph above. Eercises:. Determine visually whether the corresponding graph is symmetric with respect to the -ais, y-ais, or origin. (a) (b) (c) (d) 6

7 2. Graph each function to determine whether each function may have symmetry with respect to the y-ais, the -ais, and the origin. Then verify your conclusion algebraically. (a) y = (b) y = 3 27 (c) y + 2 = 0 (d) y = 2 4 (e) 2 = y + 5 (f) = y Determine algebraically whether the given function is odd, even, or neither. (a) f( ) = (b) g( ) = + (c) y = (d) h( ) = (e) F() = 5 (f) G( ) = Compare the graphs of f and g. f( ) = 2 3 g( ) = 2 3 (a) Describe the effect of the absolute value sign on the graph. (b) What kind of symmetry do the two functions have in common? 5. Complete the graph below so that it has the indicated type of symmetry. (a) -ais (b) y-ais (c) origin 7

8 Solutions:. (a) y-ais (b) -ais (c) y-ais (d) none 2. (a) The quartic function, y = 4 9 2, has symmetry with respect to the -ais, but it doesn t have either of the other two symmetries. y = ( ) 4 9( ) 2 = (b) The cubic function, y = 3 27, has none of the three symmetries. It does, however, have point symmetry with respect to its inflection point, (0, -27). Note: The scale on the y-ais in the graph is 0. (c) The linear function, y + 2 = 0, has symmetry with respect to the origin, but it doesn t have either of the other two symmetries. (-y) + 2(-) = 0 is equivalent to y + 2 = 0. (d) The rational function, y = 2 4, has symmetry with respect to the origin, and it has neither of the other two symmetries. ( y) = ( ) 2 4 ( ) is equivalent to y = 2 4. (e) The quadratic function, y = 2 5, has symmetry with respect to the y-ais and none of the other two symmetries. y = ( ) 2 5 = 2 5 (f) The quadratic relation, = y 2 + 5, has symmetry with respect to the -ais and none of the other two symmetries. = ( y) = y

9 ( ) = f( ) = (a) f( ) = Since f(-) = f(), the function is even; since f(-) -f(), the function isn t odd. ( ) + - = - + g( ) = (b) g( ) = - Since g(-) g() and g(-) -g(), the function is neither even nor odd. ( ) = (c) Let y = f(). f( ) = - = f Since f(-) f(), the function isn t even; since f(-) = -f(), the function is odd. ( ) 3 6 = h( ) = (d) h( ) = 4 Since h(-) h() and h(-) -h(), the function is neither even nor odd. (e) F(-) = 5 -F() = -5 Since F(-) = F(), the function is even; since F(-) -F(), the function isn t odd. ( ) 2 ( ) = (f ) G( ) = 2 = G Since G(-) = G(), the function is even; since G(-) -G(), the function isn t odd. 4. (a) The portion of the graph of f which is below the -ais seems to be reflected upward over the -ais, resulting in the graph of g. (b) Both are even functions and have symmetry with respect to the y-ais. Both f(-) = f() and g(-)= g(). 5. (a) (b) (c) 9

### Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

### Chapter 2 Analysis of Graphs of Functions

Chapter Analysis o Graphs o Functions Chapter Analysis o Graphs o Functions Covered in this Chapter:.1 Graphs o Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry.. Translations

### Exponential Functions

Eponential Functions In this chapter we will study the eponential function and its inverse the logarithmic function. These important functions are indispensable in working with problems that involve population

### Text: A Graphical Approach to College Algebra (Hornsby, Lial, Rockswold)

Students will take Self Tests covering the topics found in Chapter R (Reference: Basic Algebraic Concepts) and Chapter 1 (Linear Functions, Equations, and Inequalities). If any deficiencies are revealed,

### Midterm 1. Solutions

Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function

### Translating Points. Subtract 2 from the y-coordinates

CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that

### Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

### Transformations of y = 1/x 2

Transformations of y = / In Eample on page 6 we saw that some simple rational functions can be graphed by shifting, stretching, and/or reflecting the graph of y = /. Similarly, some rational functions

1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal

### Rational functions are defined for all values of x except those for which the denominator hx ( ) is equal to zero. 1 Function 5 Function

Section 4.6 Rational Functions and Their Graphs Definition Rational Function A rational function is a function of the form that h 0. f g h where g and h are polynomial functions such Objective : Finding

Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x

### Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd

5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts

### 3.5 Summary of Curve Sketching

3.5 Summary of Curve Sketching Follow these steps to sketch the curve. 1. Domain of f() 2. and y intercepts (a) -intercepts occur when f() = 0 (b) y-intercept occurs when = 0 3. Symmetry: Is it even or

### Investigation. So far you have worked with quadratic equations in vertex form and general. Getting to the Root of the Matter. LESSON 9.

DA2SE_73_09.qd 0/8/0 :3 Page Factored Form LESSON 9.4 So far you have worked with quadratic equations in verte form and general form. This lesson will introduce you to another form of quadratic equation,

### Functions: Piecewise, Even and Odd.

Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.

### STRAND: ALGEBRA Unit 3 Solving Equations

CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

### Lesson 8.3 Exercises, pages

Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5 - b) = 5 c) = - d) =. Sketch broken lines to represent the vertical and horizontal

### REVIEW, pages

REVIEW, pages 69 697 8.. Sketch a graph of each absolute function. Identif the intercepts, domain, and range. a) = ƒ - + ƒ b) = ƒ ( + )( - ) ƒ 8 ( )( ) Draw the graph of. It has -intercept.. Reflect, in

A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which

### Objectives. By the time the student is finished with this section of the workbook, he/she should be able

QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

Curve Sketching 1. Domain. Intercepts. Symmetry. Asymptotes 5. Intervals of Increase or Decrease 6. Local Maimum and Minimum Values 7. Concavity and Points of Inflection 8. Sketch the curve MATH 110 Lecture

### Methods to Solve Quadratic Equations

Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree

### 5-1. Lesson Objective. Lesson Presentation Lesson Review

5-1 Using Transformations to Graph Quadratic Functions Lesson Objective Transform quadratic functions. Describe the effects of changes in the coefficients of y = a(x h) 2 + k. Lesson Presentation Lesson

### 2 Analysis of Graphs of

ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

ALGEBRA 2 SCOPE AND SEQUENCE UNIT 1 2 3 4 5 6 DATES & NO. OF DAYS 8/22-9/16 19 days 9/19-9/30 9 days 10/3-10/14 9 days 10/20-10/25 4 days 10/27-12/2 17 days 12/5-12/16 10 days UNIT NAME Foundations for

### Section 1.3: Transformations of Graphs

CHAPTER 1 A Review of Functions Section 1.3: Transformations of Graphs Vertical and Horizontal Shifts of Graphs Reflecting, Stretching, and Shrinking of Graphs Combining Transformations Vertical and Horizontal

MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens

### Review of Key Concepts: 1.2 Characteristics of Polynomial Functions

Review of Key Concepts: 1.2 Characteristics of Polynomial Functions Polynomial functions of the same degree have similar characteristics The degree and leading coefficient of the equation of the polynomial

### Systems of Equations Involving Circles and Lines

Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system

### Algebra II A Final Exam

Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.

### Functions and their Graphs

Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers

### Review Exercises. Review Exercises 83

Review Eercises 83 Review Eercises 1.1 In Eercises 1 and, sketch the lines with the indicated slopes through the point on the same set of the coordinate aes. Slope 1. 1, 1 (a) (b) 0 (c) 1 (d) Undefined.,

### Rockhurst High School Algebra 1 Topics

Rockhurst High School Algebra 1 Topics Chapter 1- Pre-Algebra Skills Simplify a numerical expression using PEMDAS. Substitute whole numbers into an algebraic expression and evaluate that expression. Substitute

### FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### x 2. 4x x 4x 1 x² = 0 x = 0 There is only one x-intercept. (There can never be more than one y-intercept; do you know why?)

Math Learning Centre Curve Sketching A good graphing calculator can show ou the shape of a graph, but it doesn t alwas give ou all the useful information about a function, such as its critical points and

### ) represents the original function, will a, r 1

Transformations of Quadratic Functions Focus on Content: Download Problem Sets Problem Set #1 1. In the graph below, the quadratic function displayed on the right has been reflected over the y-axis producing

### ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

### APPLICATIONS OF DIFFERENTIATION

4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

### The Distance Formula and the Circle

10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

### Lesson 36 MA 152, Section 3.1

Lesson 36 MA 5, Section 3. I Quadratic Functions A quadratic function of the form y = f ( x) = ax + bx + c, where a, b, and c are real numbers (general form) has the shape of a parabola when graphed. The

### College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

### + k, and follows all the same rules for determining. y x 4. = + c. ( ) 2

The Quadratic Function The quadratic function is another parent function. The equation for the quadratic function is and its graph is a bowl-shaped curve called a parabola. The point ( 0,0) is called the

### G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

### Roots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)

Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign

### 1.5 Shifting, Reflecting, and Stretching Graphs

7_00.qd /7/0 0: AM Page 7. Shifting, Reflecting, and Stretching Graphs Section. Shifting, Reflecting, and Stretching Graphs 7 Summar of Graphs of Parent Functions One of the goals of this tet is to enable

### Calculus and Vectors. Day Lesson Title Math Learning Goals Expectations 1, 2, 3,

Unit Applying Properties of Derivatives Calculus and Vectors Lesson Outline Day Lesson Title Math Learning Goals Epectations 1,,, The Second Derivative (Sample Lessons Included) 4 Curve Sketching from

### The Conic Sections. The Parabola. Section 3.1. Copyright 2013, 2009, 2005, 2002 Pearson Education, Inc.

The Conic Sections Section 3.1 The Parabola Definition: Parabola A parabola is the set of all points in the plane that are equidistant from a fixed line (the directrix) and a fixed point not on the line

### Portable Assisted Study Sequence ALGEBRA IIA

SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

### Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

### Inverse Relations and Functions

11.1 Inverse Relations and Functions 11.1 OBJECTIVES 1. Find the inverse of a relation 2. Graph a relation and its inverse 3. Find the inverse of a function 4. Graph a function and its inverse 5. Identif

### Conic Sections. 1. The type of graph can be determined by looking at the terms.

Conic Sections General Quadratic Equation in Two Variables The general quadratic equation in two variables can be written as A + By + Cy + D + Ey + F = 0 where at least one of the variables A, B, C is

### Notes on Curve Sketching. B. Intercepts: Find the y-intercept (f(0)) and any x-intercepts. Skip finding x-intercepts if f(x) is very complicated.

Notes on Curve Sketching The following checklist is a guide to sketching the curve y = f(). A. Domain: Find the domain of f. B. Intercepts: Find the y-intercept (f(0)) and any -intercepts. Skip finding

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### Algebra II Semester Exam Review Sheet

Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph

### Name Class Date. Deriving the Equation of a Parabola

Name Class Date Parabolas Going Deeper Essential question: What are the defining features of a parabola? Like the circle, the ellipse, and the hperbola, the parabola can be defined in terms of distance.

### Precalculus with Limits Larson Hostetler. `knill/mathmovies/ Assessment Unit 1 Test

Unit 1 Real Numbers and Their Properties 14 days: 45 minutes per day (1 st Nine Weeks) functions using graphs, tables, and symbols Representing & Classifying Real Numbers Ordering Real Numbers Absolute

Section 3.1 Quadratic Functions 1 3.1 Quadratic Functions Functions Let s quickl review again the definition of a function. Definition 1 A relation is a function if and onl if each object in its domain

### How are the properties of real numbers and the order of operations used to simplify an expression and/or solve an algebraic equation or inequality?

Topic: Algebra I Review Key Learning: Review Algebraic concepts of single-variable expressions and equations using the order of operations, sets of real numbers and the properties of real numbers. How

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

### Packet 1 for Unit 2 Intercept Form of a Quadratic Function. M2 Alg 2

Packet 1 for Unit Intercept Form of a Quadratic Function M Alg 1 Assignment A: Graphs of Quadratic Functions in Intercept Form (Section 4.) In this lesson, you will: Determine whether a function is linear

### The Graph of a Linear Equation

4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### 5.3 Graphing Cubic Functions

Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1

### Math Rational Functions

Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Chapter 8. Examining Rational Functions. You see rational functions written, in general, in the form of a fraction: , where f and g are polynomials

Chapter 8 Being Respectful of Rational Functions In This Chapter Investigating domains and related vertical asymptotes Looking at limits and horizontal asymptotes Removing discontinuities of rational functions

0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

### Section 1-5 Functions: Graphs and Transformations

1-5 Functions: Graphs and Transformations 61 (D) Write a brief description of the relationship between tire pressure and mileage, using the terms increasing, decreasing, local maximum, and local minimum

### Polynomial and Rational Functions

Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

Table of Contents Sequence List 368-102215 Level 1 Level 5 1 A1 Numbers 0-10 63 H1 Algebraic Expressions 2 A2 Comparing Numbers 0-10 64 H2 Operations and Properties 3 A3 Addition 0-10 65 H3 Evaluating

. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the

### CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

### Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties.

Polynomial functions mc-ty-polynomial-2009-1 Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties. In order to master the techniques

### Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

### lim lim x l 1 sx 1 and so the line x 1 is a vertical asymptote. f x 2xsx 1 x 2 1 (2sx 1) x 1

SECTION 3.4 CURE SKETCHING 3.4 CURE SKETCHING EXAMPLE A Sketch the graph of f x. sx A. Domain x x 0 x x, B. The x- and y-intercepts are both 0. C. Symmetry: None D. Since x l sx there is no horizontal

### Algebra 2: Q1 & Q2 Review

Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

### Essential Question How can you describe the graph of the equation Ax + By = C? Number of adult tickets. adult

3. Graphing Linear Equations in Standard Form Essential Question How can ou describe the graph of the equation A + B = C? Using a Table to Plot Points Work with a partner. You sold a total of \$16 worth

### CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided

### Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

### 4.4 CURVE SKETCHING. there is no horizontal asymptote. Since sx 1 l 0 as x l 1 and f x is always positive, we have

SECTION 4.4 CURE SKETCHING 4.4 CURE SKETCHING EXAMPLE A Sketch the graph of f x. sx A. Domain x x 0 x x, B. The x- and y-intercepts are both 0. C. Symmetry: None D. Since x l sx there is no horizontal

### Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

### N.CN.7, A.CED.1, 2, 3, N.Q.2, A.SSE.1,

Learning Targets: I can solve interpret key features of quadratic functions from different form. I can choose a method to solve, and then, solve a quadratic equation and explain my reasoning. #1 4 For

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots. Using your examples above, answer the following:

Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots Name Period Objective 1: Understanding Square roots Defining a SQUARE ROOT: Square roots are like a division problem but both factors must

### Example #1: f(x) = x 2. Sketch the graph of f(x) and determine if it passes VLT and HLT. Is the inverse of f(x) a function?

Unit 3 Eploring Inverse trig. functions Standards: F.BF. Find inverse functions. F.BF.d (+) Produce an invertible function from a non invertible function by restricting the domain. F.TF.6 (+) Understand

### Identify examples of field properties: commutative, associative, identity, inverse, and distributive.

Topic: Expressions and Operations ALGEBRA II - STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers

### 8 Graphs of Quadratic Expressions: The Parabola

8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be

### A Library of Parent Functions. Linear and Squaring Functions. Writing a Linear Function. Write the linear function f for which f 1 3 and f 4 0.

0_006.qd 66 /7/0 Chapter.6 8:0 AM Page 66 Functions and Their Graphs A Librar of Parent Functions What ou should learn Identif and graph linear and squaring functions. Identif and graph cubic, square root,

### Section 1.1 Real numbers. Set Builder notation. Interval notation

Section 1.1 Real numbers Set Builder notation Interval notation Functions a function is the set of all possible points y that are mapped to a single point x. If when x=5 y=4,5 then it is not a function

### Math Analysis. A2. Explore the "menu" button on the graphing calculator in order to locate and use functions.

St. Mary's College High School Math Analysis Introducing the Graphing Calculator A. Using the Calculator for Order of Operations B. Using the Calculator for simplifying fractions, decimal notation and

### INVERSE TRIGONOMETRIC FUNCTIONS

Chapter INVERSE TRIGONOMETRIC FUNCTIONS Overview Inverse function Inverse of a function f eists, if the function is one-one and onto, ie, bijective Since trigonometric functions are many-one over their

### MTH 3005 - Calculus I Week 8: Limits at Infinity and Curve Sketching

MTH 35 - Calculus I Week 8: Limits at Infinity and Curve Sketching Adam Gilbert Northeastern University January 2, 24 Objectives. After reviewing these notes the successful student will be prepared to