FUNCTIONS. L f(2)= 2. g(3)= _ 3. f(t+l)= _. g(x) ) in for x in the outside function (in this case, f(x)).


 Clifton Cannon
 11 months ago
 Views:
Transcription
1 FUNCTIONS To evaluate a function for a given value, simply plug the value into the function for x. Recall: (f 0 g ) (x) = f(g(x)) OR f[g(x)] read 'Jofg of x" Means to plug the inside function (in this case g(x) ) in for x in the outside function (in this case, f(x)). Example: Given f(x) ==x+ 1 and g(x) = x 4 find f(g(x)). f(g(x)) = f(x4) = (x4) +1 =(x 8x+ 16)+1 =x 16x+3+1 f(g(x)) = X16x + 33 Let f(x) = x+l and g(x) = X1. Find each. L f()=. g(3)= _ 3. f(t+l)= _ 4. f[g()]= _ 5. g[f(m+)]=_~_ 6. [f(x) tg(x) = Let f(x) = sin(x) Find each exactly. 7. f(;)= 8. f(;)= Letj(x)=x, g(x)=x+5, and h(x)=x1. Find each. 9. h[f()]= to. f[g(xl)]= 4
2 INTERCEPTS OF A GRAPH To find the xintercepts, To find the yintercepts, let y = 0 in your equation and solve. let x = 0 in your equation and solve. y Example: Given the function y = x  x  3, fmd all intercepts. xint. (Let y = 0) o =x x3 0=(x3)(x+l) x=l or x=3 xintercepts (1,0) and (3,0) yint. (Letx = 0) y = 0 (0)3 y=3 yintercept (0,3) Find the x and y intercepts for each. 1. y=x y=x+x
3 POINTS OF INTERSECTION Use substitution or elimination method to solve the system of equations. Remember: You are finding a POINT OF INTERSECTION so your answer is an ordered pair. CALCULATOR TIP Remember you can use your calculator to verify your answers below. Graph the two lines then go to CALC (lld Trace) and hit INTERSECT. Example: Find all points of intersection of x  Y = 3 xy=l ELIMINATION METHOD Subtract to eliminate y x x= x x =0 (x)(x+1)=0 x= or x=l Plug in x = and x = lto find y Points of Intersection: (,1) and (1,) SUBSTITUTION METHOD Solve one equation for one variable. y=x 3 y=xl Therefore by substitution x  3 = x 1 xx=0 From here it is the same as the other example Find the point(s) of intersection of the graphs for the given equations. 16. x+y=8 4xy=7 17. x +y=6 x+y=4 18. X=3y y=x1 6
4 DOMAIN AND RANGE Domain  All x values for which a function is defmed (input values) Range  Possible y or Output values EXAMPLE I EXAMPLE Find the domain and range of f(x) Write answers in interval notation. =.J4x1 Q;) ";""4 ~\l d' 1:tA...,. ~ 9"'). ""T'\c..~i \,,~ft.i 6<ln~""'W. fo~.~~ "r....u\ves\"vn.1~lwa..hori~ Cl..)(.h. 'The+vrl'W!~"r \ef+1npj,,,,,q\yt <U.\oocnea.wn+. o.l>honfla. 9. l"'" i 3. ~ ~ur_s,"'1\9ht ~\'(>.\~S M«.'i~ \~ """. Ch"'L~n ;,. "6. $<> ~ \.,., \:0. C~'b::\~~+ Is 0.\\ ~o.\s~ ?~"$. ne. Y'Q~se r~..:lb ~ sat of O>l\~'l't>.\"'S ~ ~ ~,",~+\.w..~'a\.je> 't'''*'' Q.\.~.~. ve.v+\c.t>.,\ o.."i.ts. ~ \. ~"I'~,Jr 'A\._ "*~~*~\ S . "1h~ nisn.$t it>:l..5b~ ~ hl> ['., I],Q.nr't. l~ ~~ ,. +tl ~. DOMAIN For f(x) to be defined 4 x ~ O. This is true when ::;; x ::;; Domain: [,] RANGE The solution to a square root must always be positive thus f (x) must be greater than or equal to O. Range: [0,00) Find the domain and range of each function. Write your answer in INTERVAL notation. 0. f(x)=jx+3 1. f(x)=3sinx. f(x) = xi 7
5 INVERSES To find the inverse of a function, simply switch the x and the y and solve for the new "y" value. Recall fi ( x) is defined as the inverse of f ( x ) Example 1: f{x)=:vx+l Rewrite f{x) as y y = :Vx+ I x = ~y+l Switch x and y Solve for your new y ( X)3 = (~y + 1r Cube both sides x3 =y+l yx 31 fl{x)=x31 Simplify Solve for y Rewrite in inverse notation Find the inverse for each function. 3. f(x) = x+ 1 x 4. f(x)= g(x) = x 6. Y =.J4  x Tfthe graph of f(x) has the point (, 7) then what is one point that will be on the graph of f1 (x)? 8. Explain how the graphs of f(x) and fi (x) compare. 8
6 EQUATION OF A LINE Slope intercept form: y == mx +b Pointslope form: y  Yl = m( x  Xl) * LEARN! We will use this formula frequently! Vertical line: x = c (slope is undefined) Horizontalline: y = c (slope is 0) Example: Write a linear equation that has a slope of 1; and passes through the point (, 6) Slope intercept form y =.!.x+b Plug in ~ for 17'16 =..!.()+b Plug in the given ordered b=7 I y=x7 Solve for b I Pointslope y+6=(x) 1 y=x7 form Plug in all variables Solve for y 9. Determine the equation of a line passing through the point (5, 3) with an undefined slope. 30. Determine the equation of a line passing through the point (4, ) with a slope of O. 31. Use pointslope form to find the equation of the line passing through the point (0,5) with a slope of /3. 3. Use pointslope form to find a line passing through the point (, 8) and parallel to the line y ::;::: ~ xi Use pointslope form to fmd a line petpendicularto y =x+9 passing through the point (4, 7). 34. Find the equation of a line passing through the points (3, 6) and (1, ). 35. Find the equation of a line with an xintercept (, 0) and a yintercept (0, 3) 9
7 UNIT CIRCLE Yon can determine the sine or the cosine of any standard angle on the unit circle. The xcoordinate of the circle is the cosine and the ycoordinate is the sine of the angle. Recall tangent is defmed as sin/cos or the slope of the line.. 1f I sm= Examples: 1f 1f cos=o tan=und *you must have these memorized OR know how to calculate their values without the use of a calculator. 36. a.) sin1f b.) cos 31f c.) sin ( ;) I{ e.) cos 4 f.) cos(i{) g)cos':: 3 h) sin SI{ 6 1f i) cos 3 J') tan 1f 4 k) tan1f I{ 1) tan3" 4ff m)cos 3 n) sin 1Iff 6 0) tan 71f 4 10
8 TRIGONOMETRIC EQUATIONS Solv~each of the equations for 0 ~ x < 1C 37. sinx=" cosx =.J3 39.4sinzx=3 **Recall sinx = (sinx) **Recall if x = 5 then x =±5 40. cos xlcosx=o *Factor TRANSFORMATION OF FUNCTIONS h(x) = f(x) +c h(x) = f(x)c h(x)=f(x) Vertical shift c units up Vertical shift c units down Reflection over the xaxis h(x) = f(xc) h(x) = f(x+c) Horizontal shift c units right Horizontal shift c units left 41. Given f(x) = x and g(x) = (x  3) +1. How the does the graph of g(x) differ from f(x)? 4. Write an equation for the function that has the shape of f(x) = x3 but moved six units to the left and reflected over the xaxis. 43. If the ordered pair (, 4) is on the graph of f(x), fwd one ordered pair that will be on the following functions: a) f(x)3 b) f(x3) c) f(x) d) f(x)+1 e) f(x) 11
9 VERTICAL ASYMPTOTES Determine the vertical asymptotes for the function. Set the denominator equal to zero to find the xvalue for which the function is undefined. That will be the vertical asymptote given the numerator does not equal 0 also (Remember this is called removable discontinuity). Write a vertical asymptotes as a line in the form x = 118 'i 1 Example: Find the vertical asymptote of y = _1_ x Since when x = the function is in the form 1/0 then the vertical line x = is a vertical asymptote of the function y 6 :1 Y=x 6 I 44. f(x)= x x 45. f(x) = x 4 +x 46. f(x) = x(ix) 4x 47. f(x) = x16 xi 48. f(x) = x +x 49. f(x) = 5x+ 0 x16 1
10 HORIZONTAL ASYMPTOTES Determine the horizontal asymptotes using the three cases below. Case I. Degree of the numerator is less than the degree ofthe denominator. The asymptote is y = O. Example: y == _1_ (As x becomes very large or very negative the value of this function will xi approach 0). Thus there is a horizontal asymptote at y == 0. Case II. Degree of the numerator is the same as the degree of the denominator. The asymptote is the ratio of the lead coefficients. Exmaple: y = X : x 1 (As x becomes very large or very negative the value of this function will 3x +4 approach /3). Thus there is a horizontal asymptote at y == ~. 3 Case ill. Degree of the numerator is greater than the degree of the denominator. There is no horizontal asymptote. The function increases without bound. (lfthe degree of the numerator is exactly 1 more than the degree ofthe denominator, then there exists a slant asymptote, which is determined by long division.) Example: y == X +x 1 (As x becomes very large the value of the function will continue to increase 3x3 and as x becomes very negative the value of the function will also become more negative). Determine an Horizontal Asymptotes. 50. f(x) = x 3 X x+ +x f(x) == 5x3 x +8 4x3x f(x) = (x5) xx *This is very important in the use of limits. * 13
Math 121. Graphing Rational Functions Fall 2016
Math 121. Graphing Rational Functions Fall 2016 1. Let x2 85 x 2 70. (a) State the domain of f, and simplify f if possible. (b) Find equations for the vertical asymptotes for the graph of f. (c) For each
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.3 New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions. NEW
More informationSection 7.6 Graphs of the Sine and Cosine Functions
Section 7.6 Graphs of the Sine and Cosine Functions We are going to learn how to graph the sine and cosine functions on the xyplane. Just like with any other function, it is easy to do by plotting points.
More informationTrigonometric Graphs Dr. Laura J. Pyzdrowski
1 Names: About this Laboratory In this laboratory, we will examine trigonometric functions and their graphs. Upon completion of the lab, you should be able to quickly sketch such functions and determine
More informationGraphing Rational Functions
Graphing Rational Functions Return to Table of Contents 109 Vocabulary Review xintercept: The point where a graph intersects with the xaxis and the yvalue is zero. yintercept: The point where a graph
More informationCLEP PreCalculus. 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 PreCalculus Section : Time 0 Minutes 50 Questions For each question below, choose the best answer from the choices given. An online graphing calculator (noncas) is allowed to be used for this section..
More information4.2 Graphing Inverse Trigonometric Functions
4.2 Graphing Inverse Trigonometric Functions Learning Objectives Understand the meaning of restricted domain as it applies to the inverses of the six trigonometric functions. Apply the domain, range and
More informationSection 6.2 Graphs of the Other Trig Functions
Section 62 Graphs of the Other Trig Functions 369 Section 62 Graphs of the Other Trig Functions In this section, we will explore the graphs of the other four trigonometric functions We ll begin with the
More informationMath Analysis Chapter 1 Notes: Functions and Graphs
Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 11 Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian coordinate system) Practice: Label each on the
More informationSummer Review for Students Entering PreCalculus with Trigonometry. TI84 Plus Graphing Calculator is required for this course.
1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios and Pythagorean Theorem 4. Multiplying and Dividing Rational Expressions
More informationGraphing Trigonometric Functions: Day 1
Graphing Trigonometric Functions: Day 1 PreCalculus 1. Graph the six parent trigonometric functions.. Apply scale changes to the six parent trigonometric functions. Complete the worksheet Exploration:
More information23 Graphing Rational Functions
23 Graphing Rational Functions Factor What are the end behaviors of the Graph? Sketch a graph How to identify the intercepts, asymptotes and end behavior of a rational function. How to sketch the graph
More informationMath Analysis Chapter 1 Notes: Functions and Graphs
Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 11 Graphs; Section 1 Basics of Functions and Their Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian
More informationCore Mathematics 3 Functions
http://kumarmaths.weebly.com/ Core Mathematics 3 Functions Core Maths 3 Functions Page 1 Functions C3 The specifications suggest that you should be able to do the following: Understand the definition of
More informationFunction f. Function f 1
Page 1 REVIEW (1.7) What is an inverse function? Do all functions have inverses? An inverse function, f 1, is a kind of undoing function. If the initial function, f, takes the element a to the element
More informationFunctions. Edexcel GCE. Core Mathematics C3
Edexcel GCE Core Mathematics C Functions Materials required for examination Mathematical Formulae (Green) Items included with question papers Nil Advice to Candidates You must ensure that your answers
More informationThis is called the horizontal displacement of also known as the phase shift.
sin (x) GRAPHS OF TRIGONOMETRIC FUNCTIONS Definitions A function f is said to be periodic if there is a positive number p such that f(x + p) = f(x) for all values of x. The smallest positive number p for
More informationUsing Fundamental Identities. Fundamental Trigonometric Identities. Reciprocal Identities. sin u 1 csc u. sec u. sin u Quotient Identities
3330_050.qxd /5/05 9:5 AM Page 374 374 Chapter 5 Analytic Trigonometry 5. Using Fundamental Identities What you should learn Recognize and write the fundamental trigonometric identities. Use the fundamental
More informationTEKS Clarification Document. Mathematics Precalculus
TEKS Clarification Document Mathematics Precalculus 2012 2013 111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 912. Source: The provisions of this 111.31 adopted
More informationGraphs of Rational Functions
Objectives Lesson 5 Graphs of Rational Functions the table. near 0, we evaluate fix) to the left and right of x = 0 as shown in Because the denominator is zero when x = 0, the domain off is all real numbers
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 informationYou are not expected to transform y = tan(x) or solve problems that involve the tangent function.
In this unit, we will develop the graphs for y = sin(x), y = cos(x), and later y = tan(x), and identify the characteristic features of each. Transformations of y = sin(x) and y = cos(x) are performed and
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 informationFinal Exam Review Algebra Semester 1
Final Exam Review Algebra 015016 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 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 informationTI Nspire Testing Instructions
TI Nspire Testing Instructions Table of Contents How to Nsolve How to Check Compositions of Functions How to Verify Compositions of Functions How to Check Factoring How to Use Graphs to Backward Factor
More informationMath 113 Exam 1 Practice
Math Exam Practice January 6, 00 Exam will cover sections 6.6.5 and 7.7.5 This sheet has three sections. The first section will remind you about techniques and formulas that you should know. The second
More information2.2 Limit of a Function and Limit Laws
Limit of a Function and Limit Laws Section Notes Page Let s look at the graph y What is y()? That s right, its undefined, but what if we wanted to find the y value the graph is approaching as we get close
More information91 GCSE Maths. GCSE Mathematics has a Foundation tier (Grades 1 5) and a Higher tier (Grades 4 9).
91 GCSE Maths GCSE Mathematics has a Foundation tier (Grades 1 5) and a Higher tier (Grades 4 9). In each tier, there are three exams taken at the end of Year 11. Any topic may be assessed on each of
More informationRational Functions HONORS PRECALCULUS :: MR. VELAZQUEZ
Rational Functions HONORS PRECALCULUS :: MR. VELAZQUEZ Definition of Rational Functions Rational Functions are defined as the quotient of two polynomial functions. This means any rational function can
More information8.6 Other Trigonometric Functions
8.6 Other Trigonometric Functions I have already discussed all the trigonometric functions and their relationship to the sine and cosine functions and the x and y coordinates on the unit circle, but let
More information102 Circles. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra2 2
102 Circles Warm Up Lesson Presentation Lesson Quiz Holt Algebra2 2 Warm Up Find the slope of the line that connects each pair of points. 1. (5, 7) and ( 1, 6) 1 6 2. (3, 4) and ( 4, 3) 1 Warm Up Find
More informationVoluntary State Curriculum Algebra II
Algebra II Goal 1: Integration into Broader Knowledge The student will develop, analyze, communicate, and apply models to realworld situations using the language of mathematics and appropriate technology.
More informationIntro. To Graphing Linear Equations
Intro. To Graphing Linear Equations The Coordinate Plane A. The coordinate plane has 4 quadrants. B. Each point in the coordinate plain has an xcoordinate (the abscissa) and a ycoordinate (the ordinate).
More information21 Power and Radical Functions
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 35. Evaluate the function for several xvalues in
More informationObjectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function
SECTIONS 3.5: Rational Functions Objectives Graph and Analyze Rational Functions Find the Domain, Asymptotes, Holes, and Intercepts of a Rational Function I. Rational Functions A rational function is a
More informationObjectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using
Ch 13  RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using trigonometric
More informationMaths Year 11 Mock Revision list
Maths Year 11 Mock Revision list F = Foundation Tier = Foundation and igher Tier = igher Tier Number Tier Topic know and use the word integer and the equality and inequality symbols use fractions, decimals
More informationMath 1330 Section : Rational Functions Definition: A rational function is a function that can be written in the form f ( x ), where
2.3: Rational Functions P( x ) Definition: A rational function is a function that can be written in the form f ( x ), where Q( x ) and Q are polynomials, consists of all real numbers x such that You will
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 informationSection a) f(x3)+4 = (x 3) the (3) in the parenthesis moves right 3, the +4 moves up 4
Section 4.3 1a) f(x3)+4 = (x 3) 2 + 4 the (3) in the parenthesis moves right 3, the +4 moves up 4 Answer 1a: f(x3)+4 = (x 3) 2 + 4 The graph has the same shape as f(x) = x 2, except it is shifted right
More informationTrigonometric Functions of Any Angle
Trigonometric Functions of Any Angle MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: evaluate trigonometric functions of any angle,
More informationUnit 2: Function Transformation Chapter 1. Basic Transformations Reflections Inverses
Unit 2: Function Transformation Chapter 1 Basic Transformations Reflections Inverses Section 1.1: Horizontal and Vertical Transformations A transformation of a function alters the equation and any combination
More informationPreCalculus Chapter 9 Practice Test Name:
This ellipse has foci 0,, and therefore has a vertical major axis. The standard form for an ellipse with a vertical major axis is: 1 Note: graphs of conic sections for problems 1 to 1 were made with the
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 informationMultiplying and Dividing Rational Expressions
Page 1 of 14 Multiplying and Dividing Rational Expressions Attendance Problems. Simplify each expression. Assume all variables are nonzero. x 6 y 2 1. x 5 x 2 2. y 3 y 3 3. 4. x 2 y 5 Factor each expression.
More informationQuiz 6 Practice Problems
Quiz 6 Practice Problems Practice problems are similar, both in difficulty and in scope, to the type of problems you will see on the quiz. Problems marked with a are for your entertainment and are not
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 xvalues that make the equation defined Defined: There is no division by zero or square roots of negative numbers
More informationChapter 4. Trigonometric Functions. 4.6 Graphs of Other. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 4 Trigonometric Functions 4.6 Graphs of Other Trigonometric Functions Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: Understand the graph of y = tan x. Graph variations of y =
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 information2.7 Graphing Tangent, Cotangent, Secant, and
www.ck12.org Chapter 2. Graphing Trigonometric Functions 2.7 Graphing Tangent, Cotangent, Secant, and Cosecant Learning Objectives Apply transformations to the remaining four trigonometric functions. Identify
More informationChapter 5. Radicals. Lesson 1: More Exponent Practice. Lesson 2: Square Root Functions. Lesson 3: Solving Radical Equations
Chapter 5 Radicals Lesson 1: More Exponent Practice Lesson 2: Square Root Functions Lesson 3: Solving Radical Equations Lesson 4: Simplifying Radicals Lesson 5: Simplifying Cube Roots This assignment is
More informationMath 301 Sample Test Questions
Math 301 Sample Test Questions Instructions: This sample test is designed to give the student some prior indication of what the course content for Math 301 is like It is to be used to help the student
More informationMAT 115: Precalculus Mathematics Constructing Graphs of Trigonometric Functions Involving Transformations by Hand. Overview
MAT 115: Precalculus Mathematics Constructing Graphs of Trigonometric Functions Involving Transformations by Hand Overview Below are the guidelines for constructing a graph of a trigonometric function
More informationMATH 1113 Exam 1 Review. Fall 2017
MATH 1113 Exam 1 Review Fall 2017 Topics Covered Section 1.1: Rectangular Coordinate System Section 1.2: Circles Section 1.3: Functions and Relations Section 1.4: Linear Equations in Two Variables and
More informationTest Name: Chapter 3 Review
Test Name: Chapter 3 Review 1. For the following equation, determine the values of the missing entries. If needed, write your answer as a fraction reduced to lowest terms. 10x  8y = 18 Note: Each column
More information3.7 Rational Functions. Copyright Cengage Learning. All rights reserved.
3.7 Rational Functions Copyright Cengage Learning. All rights reserved. Objectives Rational Functions and Asymptotes Transformations of y = 1/x Asymptotes of Rational Functions Graphing Rational Functions
More informationChapter Three Chapter Three
Chapter Three Chapter Three 90 CHAPTER THREE ConcepTests for Section.. If f () = g (), then f() = g(). (a) True (b) False (b). If f () = g (), then f() = g() + C, where C is some constant. You might point
More informationProof of Identities TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TINspire Navigator System
Math Objectives Students will be able to interpret reciprocal, negative angle, cofunction, and Pythagorean identities in terms of the graphs of the trigonometric functions involved Students will be able
More informationVertical and Horizontal Translations
SECTION 4.3 Vertical and Horizontal Translations Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the vertical translation of a sine or cosine function. Find the horizontal
More information7 Fractions. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability
7 Fractions GRADE 7 FRACTIONS continue to develop proficiency by using fractions in mental strategies and in selecting and justifying use; develop proficiency in adding and subtracting simple fractions;
More information0.4 Family of Functions/Equations
0.4 Family of Functions/Equations By a family of functions, we are referring to a function definition such as f(x) = mx + 2 for m = 2, 1, 1, 0, 1, 1, 2. 2 2 This says, work with all the functions obtained
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II 3 rd Nine Weeks, 20162017 1 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationCollege Algebra. Fifth Edition. James Stewart Lothar Redlin Saleem Watson
College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson 4 Polynomial and Rational Functions 4.6 Rational Functions Rational Functions A rational function is a function of the form Px (
More informationd f(g(t), h(t)) = x dt + f ( y dt = 0. Notice that we can rewrite the relationship on the left hand side of the equality using the dot product: ( f
Gradients and the Directional Derivative In 14.3, we discussed the partial derivatives f f and, which tell us the rate of change of the x y height of the surface defined by f in the x direction and the
More informationSection 5.3 Graphs of the Cosecant and Secant Functions 1
Section 5.3 Graphs of the Cosecant, Secant, Tangent, and Cotangent Functions The Cosecant Graph RECALL: 1 csc x so where sin x 0, csc x has an asymptote. sin x To graph y Acsc( Bx C) D, first graph THE
More information1. GRAPHS OF THE SINE AND COSINE FUNCTIONS
GRAPHS OF THE CIRCULAR FUNCTIONS 1. GRAPHS OF THE SINE AND COSINE FUNCTIONS PERIODIC FUNCTION A period function is a function f such that f ( x) f ( x np) for every real numer x in the domain of f every
More informationNelson Functions 11 Errata
Nelson Functions 11 Errata 1: Introduction to Functions Location Question Correct Answer Getting Started 6a Graph is correct but vertex and axis of symmetry are not labelled. Add blue point labelled (in
More informationGraphing Calculator Tutorial
Graphing Calculator Tutorial This tutorial is designed as an interactive activity. The best way to learn the calculator functions will be to work the examples on your own calculator as you read the tutorial.
More informationLearning Packet. Lesson 6 Exponents and Rational Functions THIS BOX FOR INSTRUCTOR GRADING USE ONLY
Learning Packet Student Name Due Date Class Time/Day Submission Date THIS BOX FOR INSTRUCTOR GRADING USE ONLY MiniLesson is complete and information presented is as found on media links (0 5 pts) Comments:
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 information5.2 Verifying Trigonometric Identities
360 Chapter 5 Analytic Trigonometry 5. Verifying Trigonometric Identities Introduction In this section, you will study techniques for verifying trigonometric identities. In the next section, you will study
More informationUnit 7: Trigonometry Part 1
100 Unit 7: Trigonometry Part 1 Right Triangle Trigonometry Hypotenuse a) Sine sin( α ) = d) Cosecant csc( α ) = α Adjacent Opposite b) Cosine cos( α ) = e) Secant sec( α ) = c) Tangent f) Cotangent tan(
More informationGraphs and transformations 4G
Graphs and transformations 4G a f(x + ) is a translation by one unit to the left. d A (0, ), B ( ),0, C (, 4), D (, 0) A (, ), B (0, 0), C (, 4), D (5, 0) e f(x) is a stretch with scale factor b f(x) 4
More informationSect Graphing Techniques: Transformations
Sect.  Graphing Techniques: Transformations Recall the general shapes of each of the following basic functions and their properties: Identity Function Square Function f(x) = x f(x) = x        
More informationb) develop mathematical thinking and problem solving ability.
Submission for PreCalculus MATH 20095 1. Course s instructional goals and objectives: The purpose of this course is to a) develop conceptual understanding and fluency with algebraic and transcendental
More informationChapter 1 Notes, Calculus I with Precalculus 3e Larson/Edwards
Contents 1.1 Functions.............................................. 2 1.2 Analzing Graphs of Functions.................................. 5 1.3 Shifting and Reflecting Graphs..................................
More informationo Checkpoint Plot the points in the same coordinate plane.
The ~ al Coordinate Plane Plot points in a coordinate plane. VOCABULARY Coordinate plane Origin xaxls axis Ordered pair xcoordinate coordinate Quadrant Scatter plot ~. Lesson 4.. Algebra. Concepts
More informationUnit 4 Graphs of Trigonometric Functions  Classwork
Unit Graphs of Trigonometric Functions  Classwork For each of the angles below, calculate the values of sin x and cos x ( decimal places) on the chart and graph the points on the graph below. x 0 o 30
More informationInverse Trigonometric Functions:
Inverse Trigonometric Functions: Trigonometric functions can be useful models for many real life phenomena. Average monthly temperatures are periodic in nature and can be modeled by sine and/or cosine
More informationHigher. The Wave Equation. The Wave Equation 146
Higher Mathematics UNIT OUTCOME 4 The Wave Equation Contents The Wave Equation 146 1 Expressing pcosx + qsinx in the form kcos(x a 146 Expressing pcosx + qsinx in other forms 147 Multiple Angles 148 4
More informationTrigonometric Integrals
Most trigonometric integrals can be solved by using trigonometric identities or by following a strategy based on the form of the integrand. There are some that are not so easy! Basic Trig Identities and
More informationLesson 11 Rational Functions
Lesson 11 Rational Functions In this lesson, you will embark on a study of rational functions. These may be unlike any function you have ever seen. Rational functions look different because they are in
More information1.6 Applying Trig Functions to Angles of Rotation
wwwck1org Chapter 1 Right Triangles and an Introduction to Trigonometry 16 Applying Trig Functions to Angles of Rotation Learning Objectives Find the values of the six trigonometric functions for angles
More informationUnit 4 Trigonometry. Study Notes 1 Right Triangle Trigonometry (Section 8.1)
Unit 4 Trigonometr Stud Notes 1 Right Triangle Trigonometr (Section 8.1) Objective: Evaluate trigonometric functions of acute angles. Use a calculator to evaluate trigonometric functions. Use trigonometric
More informationQuadratic Functions CHAPTER. 1.1 Lots and Projectiles Introduction to Quadratic Functions p. 31
CHAPTER Quadratic Functions Arches are used to support the weight of walls and ceilings in buildings. Arches were first used in architecture by the Mesopotamians over 4000 years ago. Later, the Romans
More informationUnit 2 Intro to Angles and Trigonometry
HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 1 Unit 2 Intro to Angles and Trigonometry This is a BASIC CALCULATORS ONLY unit. (2) Definition of an Angle (3) Angle Measurements & Notation (4) Conversions of
More informationPractice Test (page 391) 1. For each line, count squares on the grid to determine the rise and the run. Use slope = rise
Practice Test (page 91) 1. For each line, count squares on the grid to determine the rise and the. Use slope = rise 4 Slope of AB =, or 6 Slope of CD = 6 9, or Slope of EF = 6, or 4 Slope of GH = 6 4,
More informationPolynomial and Rational Functions
Chapter 3 Polynomial and Rational Functions Review sections as needed from Chapter 0, Basic Techniques, page 8. Refer to page 187 for an example of the work required on paper for all graded homework unless
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 information8B.2: Graphs of Cosecant and Secant
Opp. Name: Date: Period: 8B.: Graphs of Cosecant and Secant Or final two trigonometric functions to graph are cosecant and secant. Remember that So, we predict that there is a close relationship between
More informationGCSE Higher Revision List
GCSE Higher Revision List Level 8/9 Topics I can work with exponential growth and decay on the calculator. I can convert a recurring decimal to a fraction. I can simplify expressions involving powers or
More informationPolynomial And Rational Functions. Copyright Cengage Learning. All rights reserved.
Polynomial And Rational Functions Copyright Cengage Learning. All rights reserved. 3.7 Rational Functions Copyright Cengage Learning. All rights reserved. Objectives Rational Functions and Asymptotes Transformations
More informationMath 144 Activity #2 Right Triangle Trig and the Unit Circle
1 p 1 Right Triangle Trigonometry Math 1 Activity #2 Right Triangle Trig and the Unit Circle We use right triangles to study trigonometry. In right triangles, we have found many relationships between the
More informationCh. 8.7 Graphs of Rational Functions Learning Intentions: Identify characteristics of the graph of a rational function from its equation.
Ch. 8.7 Graphs of Rational Functions Learning Intentions: Identify characteristics of the graph of a rational function from its equation. Learn to write the equation of a rational function from its graph.
More informationTexas Instruments TI83, TI83 Plus, TI84 Plus Graphics Calculator
Part II: Texas Instruments TI83, TI83 Plus, TI84 Plus Graphics Calculator II.1 Getting started with the TI83, TI83 Plus, TI84 Plus Note: All keystroke sequences given for the TI83 are applicable
More informationTrigonometry Curriculum Guide Scranton School District Scranton, PA
Trigonometry Scranton School District Scranton, PA Trigonometry Prerequisite: Algebra II, Geometry, Algebra I Intended Audience: This course is designed for the student who has successfully completed Algebra
More informationAlgebra II Chapter 6: Rational Exponents and Radical Functions
Algebra II Chapter 6: Rational Exponents and Radical Functions Chapter 6 Lesson 1 Evaluate nth Roots and Use Rational Exponents Vocabulary 1 Example 1: Find nth Roots Note: and Example 2: Evaluate Expressions
More information5.2. The Sine Function and the Cosine Function. Investigate A
5.2 The Sine Function and the Cosine Function What do an oceanographer, a stock analyst, an audio engineer, and a musician playing electronic instruments have in common? They all deal with periodic patterns.
More informationTriangles. Leg = s. Hypotenuse = s 2
Honors Geometry Second Semester Final Review This review is designed to give the student a BASIC outline of what needs to be reviewed for the second semester final exam in Honors Geometry. It is up to
More information