MA123, Chapter 1: Equations, functions and graphs (pp. 1-15)
|
|
- Dale Matthews
- 6 years ago
- Views:
Transcription
1 MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand function notation. Find inverse functions. What is a graph? Sketch the graph of an equation in two variables. Find the equation of a line and a parabola. Find the intersection point(s) of two graphs. Assignments: Assignment 00 Assignment 01 Equations and solution(s) to equations: One wa in which humanit increases its understanding of the universe is b discovering relationships between various objects, concepts, quantities, and so on. Our understanding of a relationship between two quantities is sharpest when this relationship can be completel quantified and epressed in an equation. Roughl speaking, an equation is a statement that two mathematical epressions are equal. For instance, = 5 is an equation relating and. A set of numbers that can be substituted for the variables in an equation so that the equalit is true is a solution for the equation. A solution is said to satisf the equation. Eample 1: Is = 1 and = 2 a solution for the equation 2 + = 2? What about = 1 and = 1? Man problems in the sciences, economics, finance, medicine and numerous other fields can be formulated into algebraic terms b identifing variables epressing unknown quantities and b setting up appropriate equations relating these variables. Eample 2: Suppose a fuel miture is 4% ethanol and 96% gasoline. How much ethanol (in gallons) must ou add to one gallon of fuel so that the new fuel miture is 10% ethanol? 1
2 Equations into functions: An equation in two (or more) variables can sometimes be solved in terms of one of the variables. This tpe of equation is closel related to the notion of a function. Eample 3: Solve the equation = 7 for in terms of. Observe that in the equation = 7 3, the epression on the right-hand side can be viewed as a recipe that associates to an given value of precisel one corresponding value for. Definition of function: A function f is a rule that assigns to each element in a set A eactl one element, called f(), in a set B. The set A is called the domain of f whereas the set B is called the codomain of f; f() is called the value of f at, or the image of under f. The range of f is the set of all possible values of f() as varies throughout the domain: range of f = {f() A}. input f Machine diagram of f f() output A a Arrow diagram of f b f f() B f(a) = f(b) range of f Evaluating a function: The smbol that represents an arbitrar number in the domain of a function f is called an independent variable. The smbol that represents a number in the range of f is called a dependent variable. In the definition of a function the independent variable plas the role of a placeholder. For eample, the function f() = can be thought of as f( ) = To evaluate f at a number (epression), we substitute the number (epression) for the placeholder. Note: If f is a function of, then = f() is a special kind of equation, in which the variable appears alone on the left side of the equal sign and the epression on the right side of the equal sign involves onl the other variable. Conversel, when we have this special kind of equation, such as = e , it is common to think of the right hand side as defining a function f(), and of the equation as being simpl = f(). Eample 4: Find the domain of the following functions: f() = 3 g() = h() =
3 Eample 5: If f() = 6 + 4, write an epression for: [ ] [ ] f(1 + h) f(1) f(1 + h) + f(1). Eample 6: If P() = and we rewrite P() in the form what are the values of A and B? P() = A + B( 1) + C( 1)( 2) + D( 1)( 2)( 3), Eample 7: If we rewrite the function f() = 3 ( 1)( 2) in the form: what are the values of A, B, and C? f() = A + B 1 + C 2, Inverse of a function: Recall that two functions f() and g() are said to be inverse of each other if f(g()) = and g(f()) =. Eample 8: If h(t) = 3t + 7, find a function g(t) such that h(g(t)) = t. 3
4 Cartesian plane and the graph of a function: Points in a plane can be identified with ordered pairs of numbers to form the coordinate plane. To do this, we draw two perpendicular oriented lines (one horizontal and the other vertical) that intersect at 0 on each line. The horizontal line with positive direction to the right is called the -ais; the other line with positive direction upward is called the -ais. The point of intersection of the two aes is the origin O. The two aes divide the plane into four quadrants, labeled I, II, III, and IV. The coordinate plane is also called Cartesian plane in honor of the French mathematician/philosopher René Descartes ( ). An point P in the coordinate plane can be located b a unique ordered pair of numbers (a, b) as shown in the picture. The first number a is called the -coordinate of P; the second number b is called the -coordinate of P. II b O P(a,b) a I III IV Graphing functions: If f is a function with domain A, then the graph of f is the set of ordered pairs f(6) f() (6,f(6)) (,f()) graph of f = {(,f()) A}. f(2) (2,f(2)) In other words, the graph of f is the set of all points (,) such that = f(); that is, the graph of f is the graph of the equation = f() Obtaining information from the graph of a function: The values of a function are represented b the height of its graph above the -ais. So, we can read off the values of a function from its graph. In addition, the graph of a function helps us picture the domain and range of the function on the -ais and -ais as shown in the picture: 0 Range = f() Domain The graph of a function is a curve in the plane. But the question arises: Which curves in the -plane are graphs of functions? The vertical line test: A curve in the coordinate plane is the graph of a function if and onl if no vertical line intersects the curve more than once. Graph of a function Not a graph of a function 4
5 Lines and parabolas: The simplest tpes of functions are linear and quadratic functions. A linear function is a function f of the form f() = m + b, where m and b are real numbers. The graph of the equation = m + b is a (non-vertical) line in the -plane. The numbers m and b are called the slope and -intercept, respectivel. A quadratic function is a function f of the form f() = a 2 + b + c, where a,b, and c are real numbers and a 0. The graph of an quadratic function is a parabola; it can be obtained from the graph of f() = 2 b using shifting, reflecting and stretching transformations. Indeed, b completing the square a quadratic function f() = a 2 + b + c can be epressed in the standard form k Verte (h, k) (Minimum) k (Maimum) Verte (h, k) f() = a( h) 2 + k. h h The graph of f is a parabola with verte (h,k); the parabola opens upward if a > 0, or downward if a < 0. f() = a( h) 2 + k, a > 0 f() = a( h) 2 + k, a < 0 Eample 9: If the equation of the line through the points (3,4) and ( 1,6) is written as = A + B( + 1), what are the values of A and B? Eample 10: The parabola = intersects the -ais at the two points P and Q. What is the distance from P to Q? If we rewrite the inequalit < 0 in the form A < < B, what are the values of A and B? 5
6 Intersection points: The graphs of two equations intersect at a point if and onl if that point is a solution for both equations. Eample 11: Find the point(s) of intersection between the graph of the equation = 36 and the line with equation = 2; = 1. Eample 12: Find all points where the graph of ( 1) 2 2 = crosses the -ais; the -ais. Eample 13: The area of a right triangle is 7. The sum of the lengths of the two sides adjacent to the right angle of the triangle is 11. What is the length of the hpotenuse of the triangle? Eample 14: What is the smallest root of the polnomial Q() = ? 6
MA123, Chapter 1: Equations, functions, and graphs (pp. 1-15, Gootman)
MA123, Chapter 1: Equations, functions, and graphs (pp. 1-15, Gootman) Chapter Goals: Solve an equation for one variable in terms of another. What is a function? Find inverse functions. What is a graph?
More informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More informationMath RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus
Math 0-03-RE - Calculus I Functions Page of 0 Definition of a function f() : Topics of Functions used in Calculus A function = f() is a relation between variables and such that for ever value onl one value.
More information= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationGlossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
More information4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
More informationMath 3201 UNIT 5: Polynomial Functions NOTES. Characteristics of Graphs and Equations of Polynomials Functions
1 Math 301 UNIT 5: Polnomial Functions NOTES Section 5.1 and 5.: Characteristics of Graphs and Equations of Polnomials Functions What is a polnomial function? Polnomial Function: - A function that contains
More informationLesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
More informationRELATIONS AND FUNCTIONS through
RELATIONS AND FUNCTIONS 11.1.2 through 11.1. Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More information1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10
CNTENTS Algebra Chapter Chapter Chapter Eponents and logarithms. Laws of eponents. Conversion between eponents and logarithms 6. Logarithm laws 8. Eponential and logarithmic equations 0 Sequences and series.
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationSection 4.1 Increasing and Decreasing Functions
Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
More informationCHAPTER 3 Graphs and Functions
CHAPTER Graphs and Functions Section. The Rectangular Coordinate Sstem............ Section. Graphs of Equations..................... 7 Section. Slope and Graphs of Linear Equations........... 7 Section.
More informationWriting Quadratic Functions in Standard Form
Chapter Summar Ke Terms standard form (general form) of a quadratic function (.1) parabola (.1) leading coefficient (.) second differences (.) vertical motion model (.3) zeros (.3) interval (.3) open interval
More information3 Polynomial and Rational Functions
3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental
More informationUNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
More informationAlgebra 2 Semester Exam Review
Algebra Semester Eam Review 7 Graph the numbers,,,, and 0 on a number line Identif the propert shown rs rs r when r and s Evaluate What is the value of k k when k? Simplif the epression 7 7 Solve the equation
More informationReady To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions
Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte
More information10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities.
Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + =
More informationSample. Sample. Sample. Sample (1,2) (-1,1) (3,-1) (-3,-5) Sample (1,2) (-1,1) (3,-1) (-3,-5) Sample. (x, y) Domain: {-3, -1, 1, 3} (1,2) (-1,1)
(-1,1) (1,2) Algebra 2 HS Mathematics Unit: 02 Lesson: 01 (3,-1) (-3,-5) Range: {-5, 1, 2, -1} (-1,1) (-3,-5) (1,2) (3,-1) (-1,1) (-3,-5) (1,2) (3,-1) Domain: {-3, -1, 1, 3} (1,2) (-1,1) (3,-1) (-3,-5)
More informationa [A] +Algebra 2/Trig Final Exam Review Fall Semester x [E] None of these [C] 512 [A] [B] 1) Simplify: [D] x z [E] None of these 2) Simplify: [A]
) Simplif: z z z 6 6 z 6 z 6 ) Simplif: 9 9 0 ) Simplif: a a a 0 a a ) Simplif: 0 0 ) Simplif: 9 9 6) Evaluate: / 6 6 6 ) Rationalize: ) Rationalize: 6 6 0 6 9) Which of the following are polnomials? None
More informationName: Richard Montgomery High School Department of Mathematics. Summer Math Packet. for students entering. Algebra 2/Trig*
Name: Richard Montgomer High School Department of Mathematics Summer Math Packet for students entering Algebra 2/Trig* For the following courses: AAF, Honors Algebra 2, Algebra 2 (Please go the RM website
More information4Cubic. polynomials UNCORRECTED PAGE PROOFS
4Cubic polnomials 4.1 Kick off with CAS 4. Polnomials 4.3 The remainder and factor theorems 4.4 Graphs of cubic polnomials 4.5 Equations of cubic polnomials 4.6 Cubic models and applications 4.7 Review
More informationMathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.
Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of
More informationAlgebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.
Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph
More informationMath 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.
Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the
More informationQUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta
QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape
More informationRev Name Date. Solve each of the following equations for y by isolating the square and using the square root property.
Rev 8-8-3 Name Date TI-8 GC 3 Using GC to Graph Parabolae that are Not Functions of Objectives: Recall the square root propert Practice solving a quadratic equation f Graph the two parts of a hizontal
More informationUnit 10 - Graphing Quadratic Functions
Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif
More informationChapter 18 Quadratic Function 2
Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are
More informationUNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives
Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.
More informationAPPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I
APPENDIXES A Numbers, Inequalities, and Absolute Values B Coordinate Geometr and Lines C Graphs of Second-Degree Equations D Trigonometr E F Sigma Notation Proofs of Theorems G The Logarithm Defined as
More informationMath Intermediate Algebra
Math 095 - Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and
More informationCoordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
More informationDerivatives 2: The Derivative at a Point
Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned
More informationQUADRATIC FUNCTION REVIEW
Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important
More informationMAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function
MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,
More informationMATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED
FOM 11 T GRAPHING LINEAR INEQUALITIES & SET NOTATION - 1 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) INEQUALITY = a mathematical statement that contains one of these four inequalit signs: ,.
More informationCollege Algebra Final, 7/2/10
NAME College Algebra Final, 7//10 1. Factor the polnomial p() = 3 5 13 4 + 13 3 + 9 16 + 4 completel, then sketch a graph of it. Make sure to plot the - and -intercepts. (10 points) Solution: B the rational
More informationSection 2.5: Graphs of Functions
Section.5: Graphs of Functions Objectives Upon completion of this lesson, ou will be able to: Sketch the graph of a piecewise function containing an of the librar functions. o Polnomial functions of degree
More informationy = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is
Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (,
More informationThe standard form of the equation of a circle is based on the distance formula. The distance formula, in turn, is based on the Pythagorean Theorem.
Unit, Lesson. Deriving the Equation of a Circle The graph of an equation in and is the set of all points (, ) in a coordinate plane that satisf the equation. Some equations have graphs with precise geometric
More informationQuadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.
Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to
More informationUnit 2 Notes Packet on Quadratic Functions and Factoring
Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a
More informationUsing Intercept Form
8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of
More information8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.
8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,
More informationc) Words: The cost of a taxicab is $2.00 for the first 1/4 of a mile and $1.00 for each additional 1/8 of a mile.
Functions Definition: A function f, defined from a set A to a set B, is a rule that associates with each element of the set A one, and onl one, element of the set B. Eamples: a) Graphs: b) Tables: 0 50
More informationAlgebra 2 Honors Summer Packet 2018
Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear
More informationVertex. March 23, Ch 9 Guided Notes.notebook
March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function
More informationEOC Review. Algebra I
EOC Review Algebra I Order of Operations PEMDAS Parentheses, Eponents, Multiplication/Division, Add/Subtract from left to right. A. Simplif each epression using appropriate Order of Operations.. 5 6 +.
More informationVertex form of a quadratic equation
Verte form of a quadratic equation Nikos Apostolakis Spring 017 Recall 1. Last time we looked at the graphs of quadratic equations in two variables. The upshot was that the graph of the equation: k = a(
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationReady To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities
A Read To Go n? Skills Intervention -1 Solving Linear Equations and Inequalities Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular equation solution of an equation linear
More informationTable of Contents. Module 1
Table of Contents Module 1 11 Order of Operations 16 Signed Numbers 1 Factorization of Integers 17 Further Signed Numbers 13 Fractions 18 Power Laws 14 Fractions and Decimals 19 Introduction to Algebra
More informationBRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH 05 Review Sheet
BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH 05 Review Sheet Go to http://www.cun.edu/testing for more information on the CUNY Elementar Algebra
More informationHCC-SE MATH DEPT. 1 Revised Fall 2008
FINAL EXAM REVIEW ITEMS Math : College Algebra Find the -intercepts and an -intercepts. ) f() = + 7-0 ) = Name ) Select the equation that describes the graph. Solve the equation and epress the solution
More informationGrade 12 Mathematics. unimaths.co.za. Revision Questions. (Including Solutions)
Grade 12 Mathematics Revision Questions (Including Solutions) unimaths.co.za Get read for universit mathematics b downloading free lessons taken from Unimaths Intro Workbook. Visit unimaths.co.za for more
More informationMAT 116 Final Exam Review
MAT 116 Final Eam Review 1. Determine the equation for the line described, writing the answer in slope-intercept form. The line is parallel to = 8 and passes through ( 1,9). The line is perpendicular to
More informationFirst Semester Final Review NON-Graphing Calculator
Algebra First Semester Final Review NON-Graphing Calculator Name:. 1. Find the slope of the line passing through the points ( 5, ) and ( 3, 7).. Find the slope-intercept equation of the line passing through
More informationLinear Relationships
Linear Relationships Curriculum Read www.mathletics.com Basics Page questions. Draw the following lines on the provided aes: a Line with -intercept and -intercept -. The -intercept is ( 0and, ) the -intercept
More information2.3 Quadratic Functions
88 Linear and Quadratic Functions. 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:
More informationPower Functions. A polynomial expression is an expression of the form a n. x n 2... a 3. ,..., a n. , a 1. A polynomial function has the form f(x) a n
1.1 Power Functions A rock that is tossed into the water of a calm lake creates ripples that move outward in a circular pattern. The area, A, spanned b the ripples can be modelled b the function A(r) πr,
More informationZETA MATHS. Higher Mathematics Revision Checklist
ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions
More informationf(x) = 2x 2 + 2x - 4
4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100
Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 5 1 Quadratics 5 The Discriminant 54 Completing the Square 55 4 Sketching Parabolas 57 5 Determining the Equation
More informationUnit 2: Functions and Graphs
AMHS Precalculus - Unit 16 Unit : Functions and Graphs Functions A function is a rule that assigns each element in the domain to eactly one element in the range. The domain is the set of all possible inputs
More informationSolve Quadratics Using the Formula
Clip 6 Solve Quadratics Using the Formula a + b + c = 0, = b± b 4 ac a ) Solve the equation + 4 + = 0 Give our answers correct to decimal places. ) Solve the equation + 8 + 6 = 0 ) Solve the equation =
More informationFunctions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
More informationUNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS
Answer Ke Name: Date: UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Part I Questions. For the quadratic function shown below, the coordinates of its verte are, (), 7 6,, 6 The verte is
More informationChapter 1 Coordinates, points and lines
Cambridge Universit Press 978--36-6000-7 Cambridge International AS and A Level Mathematics: Pure Mathematics Coursebook Hugh Neill, Douglas Quadling, Julian Gilbe Ecerpt Chapter Coordinates, points and
More informationNational 5 Mathematics
St Andrew s Academ Mathematics Department National 5 Mathematics UNIT 4 ASSESSMENT PREPARATION St Andrew's Academ Maths Dept 016-17 1 Practice Unit Assessment 4A for National 5 1. Simplif, giving our answer
More information20.2 Connecting Intercepts and Linear Factors
Name Class Date 20.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and
More informationAnswer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE
The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test
More informationLesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas
Unit 4 Polnomial/Rational Functions Quadratic Functions (Chap 0.3) William (Bill) Finch Lesson Goals When ou have completed this lesson ou will: Graph and analze the graphs of quadratic functions. Solve
More information1.3. Absolute Value and Piecewise-Defined Functions Absolutely Piece-ful. My Notes ACTIVITY
Absolute Value and Piecewise-Defined Functions Absolutel Piece-ful SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Quickwrite. Graph both = - for < 3 and = - + 7 for
More informationChapter 1 Graph of Functions
Graph of Functions Chapter Graph of Functions. Rectangular Coordinate Sstem and Plotting points The Coordinate Plane Quadrant II Quadrant I (0,0) Quadrant III Quadrant IV Figure. The aes divide the plane
More informationName Date. and y = 5.
Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five
More informationFair Game Review. Chapter 2. and y = 5. Evaluate the expression when x = xy 2. 4x. Evaluate the expression when a = 9 and b = 4.
Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five
More informationNorthwest High School s Algebra 2/Honors Algebra 2
Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar
More informationPRINCIPLES OF MATHEMATICS 11 Chapter 2 Quadratic Functions Lesson 1 Graphs of Quadratic Functions (2.1) where a, b, and c are constants and a 0
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 1 Graphs of Quadratic Functions (.1) Date A. QUADRATIC FUNCTIONS A quadratic function is an equation that can be written in the following
More informationPure Core 1. Revision Notes
Pure Core Revision Notes Ma 06 Pure Core Algebra... Indices... Rules of indices... Surds... 4 Simplifing surds... 4 Rationalising the denominator... 4 Quadratic functions... 5 Completing the square....
More information10.2 INTRODUCTION TO CONICS: PARABOLAS
Section 0.2 Introduction to Conics: Parabolas 733 0.2 INTRODUCTION TO CONICS: PARABOLAS What ou should learn Recognize a conic as the intersection of a plane a double-napped cone. Write equations of parabolas
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationChapter 2 Polynomial, Power, and Rational Functions
Section. Linear and Quadratic Functions and Modeling 6 Chapter Polnomial, Power, and Rational Functions Section. Linear and Quadratic Functions and Modeling Eploration. $000 per ear.. The equation will
More informationGlossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
More informationAlgebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3
1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 Page 1 of 0 11 Practice Questions 6 1 5. Which
More informationKEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
More informationChapter 8 Vocabulary Check
28 CHAPTER 8 Quadratic Equations and Functions d. What is the level of methane emissions for that ear? (Use our rounded answer from part (c).) (Round this answer to 2 decimals places.) Use a graphing calculator
More information(2.5) 1. Solve the following compound inequality and graph the solution set.
Intermediate Algebra Practice Final Math 0 (7 th ed.) (Ch. -) (.5). Solve the following compound inequalit and graph the solution set. 0 and and > or or (.7). Solve the following absolute value inequalities.
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationUnit 3 Notes Mathematical Methods
Unit 3 Notes Mathematical Methods Foundational Knowledge Created b Triumph Tutoring Copright info Copright Triumph Tutoring 07 Triumph Tutoring Pt Ltd ABN 60 607 0 507 First published in 07 All rights
More informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationSample Problems For Grade 9 Mathematics. Grade. 1. If x 3
Sample roblems For 9 Mathematics DIRECTIONS: This section provides sample mathematics problems for the 9 test forms. These problems are based on material included in the New York Cit curriculum for 8.
More informationLearning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1
College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL-) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer,
More informationModule 3, Section 4 Analytic Geometry II
Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related
More informationLinear Programming. Maximize the function. P = Ax + By + C. subject to the constraints. a 1 x + b 1 y < c 1 a 2 x + b 2 y < c 2
Linear Programming Man real world problems require the optimization of some function subject to a collection of constraints. Note: Think of optimizing as maimizing or minimizing for MATH1010. For eample,
More informationSection 3.3 Graphs of Polynomial Functions
3.3 Graphs of Polynomial Functions 179 Section 3.3 Graphs of Polynomial Functions In the previous section we eplored the short run behavior of quadratics, a special case of polynomials. In this section
More information