Outline. 1 The Role of Functions. 2 Polynomial Functions. 3 Power Functions. 4 Rational Functions. 5 Exponential & Logarithmic Functions
|
|
- Maria Gregory
- 5 years ago
- Views:
Transcription
1 Outline MS11: IT Mathematics Functions Catalogue of Essential Functions John Carroll School of Mathematical Sciences Dublin City University 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions 6 The Role of Functions The Role of Functions Mathematical Models Outline Mathematical Model 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions 6 Functions (/4) MS11: IT Mathematics John Carroll 3 / 58 A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life epectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. Functions (/4) MS11: IT Mathematics John Carroll 4 / 58
2 The Role of Functions Mathematical Models Mathematical Model Outline 1 The Role of Functions A mathematical model is never a completely accurate representation of a physical situation it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions 6 Functions (/4) MS11: IT Mathematics John Carroll 5 / 58 Functions (/4) MS11: IT Mathematics John Carroll 6 / 58 Coordinate Geometry & Lines Coordinate Geometry & Lines (Cont d) Any point in the plane can be located by a unique ordered pair of numbers as follows: Draw lines through perpendicular to the - and y-aes. These lines intersect the aes in points with coordinates a and b as shown above. Functions (/4) MS11: IT Mathematics John Carroll 7 / 58 Then the point P is assigned the ordered pair (a, b). The first number a is called the -coordinate of P; the second number b is called the y-coordinate of P. We say that P is the point with coordinates (a, b), and we denote the point by the symbol P(a, b). Several points are labeled with their coordinates in the second plot above. Functions (/4) MS11: IT Mathematics John Carroll 8 / 58
3 Regions in the Plane Constant Functions: Slope m = 0 y y =.5 y = 1 y = 0 Formally {(, y) 0} {(, y) y = 1} {(, y) y < 1} Functions (/4) MS11: IT Mathematics John Carroll 9 / 58 y = 1. y = Functions (/4) MS11: IT Mathematics John Carroll 10 / 58 Linear Functions Linear Functions A function of the form y = f () = m + c, where m and c are constants, is called a linear function. y = 3 y y = y = y = When we say that y is a linear function of, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for the function as y = 0 y = f () = m + c where m is the slope of the line and c is the y-intercept. A characteristic feature of linear functions is that they grow at a constant rate. Case 1: c = 0 Functions (/4) MS11: IT Mathematics John Carroll 11 / 58 Functions (/4) MS11: IT Mathematics John Carroll 1 / 58
4 Slope of a Straight Line Slope of a Straight Line (Cont d) The slope of a nonvertical line which passes through the points P 1 ( 1, y 1 ) and P (, y ) is m = y = y y 1 1 The slope of a vertical line is not defined. Functions (/4) MS11: IT Mathematics John Carroll 13 / 58 Functions (/4) MS11: IT Mathematics John Carroll 14 / 58 Straight Line: Equation & Inequality Slope as the Rate of Change Graph of equation 3 5y = 15 Graph of inequality + y > 5 Consider the graph of the linear function y = f () = 3 You will notice that whenever increases by 1, the value of y increases by 3. So y increases three times as fast as. Thus the slope of the graph, namely 3, can be interpreted as the rate of change of y with respect to. Functions (/4) MS11: IT Mathematics John Carroll 15 / 58 Functions (/4) MS11: IT Mathematics John Carroll 16 / 58
5 Polynomials Polynomials Polynomials Polynomial of Degree : Quadratic Function Definition A function is called a polynomial if P() = a n n + a n 1 n a + a 1 + a 0 where n is a nonnegative integer and the numbers a 0, a 1,..., a n are constants called the coefficients of the polynomial. The domain of any polynomial R = (, + ). If the leading coefficient a n 0, then the degree of the polynomial is n. For eample, the function is a polynomial of degree 6. P() = Functions (/4) MS11: IT Mathematics John Carroll 17 / 58 A polynomial of degree is of the form and is called a quadratic function. P() = a + b + c Its graph is always a parabola obtained by shifting the parabola y = a. The parabola opens upward if a > 0 and downward if a < 0. Functions (/4) MS11: IT Mathematics John Carroll 18 / 58 Quadratic Functions Quadratic Functions Quadratic functions Where is ( 3)( 5) = > 0? f () = a + b + c 3 The roots of this function are the solutions to the quadratic equation Questions to ask ( 3)( 5) = 0 Where is f () = ( 3)( 5) positive? Where is f () = ( a)( b), when a < b, positive? Where is g() = b = ( b)( + b), when 0 < b, positive? Where is h() = b = (b )(b + ) = g(), when 0 < b, positive? So i.e. = 3 and = 5. f () > 0 < 3 or > 5, f () < 0 3 < < 5. Functions (/4) MS11: IT Mathematics John Carroll 19 / 58 Functions (/4) MS11: IT Mathematics John Carroll 0 / 58
6 Quadratic Functions Quadratic Functions Quadratic functions Some Answers The quadratic function f () = ( a)( b), where a < b, is zero when = a and = b; positive if < a or > b; negative when a < < b. The quadratic function g() = b = ( b)( + b), where 0 < b, is zero when = ±b; positive if < b or > b; negative when b < < b. The quadratic function h() = b = (b )(b + ) = g(), where 0 < b, is zero when = ±b; positive if b < < b. negative when < b or > b; of Degree n A polynomial of degree 3 is of the form P() = a 3 + b + c + d, a 0 and is called a cubic function. Functions (/4) MS11: IT Mathematics John Carroll 1 / 58 Functions (/4) MS11: IT Mathematics John Carroll / 58 More General Polynomials More General Polynomials Polynomials: Recap Eamples of Polynomials A function f is a polynomial if f () = ( + 1)( 1) 1 f () = ( ) 4 ( + 1) 3 ( 1) 5 f () = a n n + a n 1 n a 1 + a o where n is a non negative integer, called the degree of the polynomial, and a 0, a 1,..., a n are real constants, called coefficients, with a n Functions (/4) MS11: IT Mathematics John Carroll 3 / 58 Functions (/4) MS11: IT Mathematics John Carroll 4 / 58
7 More General Polynomials More General Polynomials Polynomials Eamples of Polynomials Properties All polynomials have domain R = (, ). - y = m + b (with m 0) are polynomials of degree 1. Quadratic functions y = a + b + c (with a 0) are polynomials of degree. Cubic functions y = a 3 + b + c + d (with a 0) are polynomials of degree 3. If n is even, then f has even degree and if n is odd, then f has odd degree. A ball is thrown into the air with an initial velocity of 160ft/sec. It reaches a height, h, after t seconds: h(t) = 160t 16t ft. The supply and demand functions for a commodity are where P is the price per unit. P = Q S + 9Q S P = Q D 1Q D + 75 Functions (/4) MS11: IT Mathematics John Carroll 5 / 58 Functions (/4) MS11: IT Mathematics John Carroll 6 / 58 More General Polynomials More General Polynomials Evaluating Polynomials Graph of p() = Let p() = p is a polynomial of degree 3 (a cubic). We can, for eample, evaluate at = 0 to find p(0) = 7 (0) 3 1 (0) 8 (0) + 8 = 8 10 f () = While evaluating at = gives p() = 7 () 3 1 () 8 () + 8 = 0 In this case, = is called a root of the polynomial p. Note that p() = ( )(7 + 4). Functions (/4) MS11: IT Mathematics John Carroll 7 / 58 Functions (/4) MS11: IT Mathematics John Carroll 8 / 58
8 More General Polynomials Power Functions Roots of a Polynomial Outline In general, given any function f, we call a solution of the equation a root or zero of the function f. f () = 0 If = a is a root of a polynomial p, then p() = ( a)q(), for some polynomial q. We say that ( a) is a factor of p. A root = a is called a repeated root of the polynomial p, if p() = ( a) m s(), for some m and some polynomial s(). If p is a polynomial of degree n, then p has at most n real roots. 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions 6 Functions (/4) MS11: IT Mathematics John Carroll 9 / 58 Functions (/4) MS11: IT Mathematics John Carroll 30 / 58 Power Functions Types of Power Functions Power Functions Types of Power Functions Power Functions Families of Power Functions A function of the form y = f () = n, where n is a constant, is called a power function. The graphs for n = 1,, 3, 4, 5 are displayed below: y y = y y = y y = 3 y y = 4 y y = 5 Functions (/4) MS11: IT Mathematics John Carroll 31 / 58 The general shape of the graph of f () = n depends on whether n is even or odd. Note that, as n increases, the graph of y = n becomes flatter near = 0 and steeper when 1. Functions (/4) MS11: IT Mathematics John Carroll 3 / 58
9 Power Functions Types of Power Functions Power Functions Types of Power Functions Power Functions: 1 n, n a positive integer Power Functions: Two Further Eamples 4 f () = 1 4 f () = Note that the domain of is [0, ) whose graph is the upper half of the parabola = y. The domain of 3 is R (every real number has a cube root). Functions (/4) MS11: IT Mathematics John Carroll 33 / 58 Functions (/4) MS11: IT Mathematics John Carroll 34 / 58 Rational Functions Rational Functions Definition Outline Rational Functions 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions A rational function is a quotient or ratio of two polynomial functions: f () = p() q() where p() and q() are polynomials. The domain of a rational function is the set of all for which q() 0. 6 Functions (/4) MS11: IT Mathematics John Carroll 35 / 58 Functions (/4) MS11: IT Mathematics John Carroll 36 / 58
10 Rational Functions Eamples Rational Functions Eamples Eample of a Rational Function Rational Function with a Discontinuity The function f () = f () = 1 4 is a rational function with domain all real numbers. 4 Functions (/4) MS11: IT Mathematics John Carroll 37 / 58 Functions (/4) MS11: IT Mathematics John Carroll 38 / 58 Rational Functions Eamples Rational Functions Eamples Rational Function with a Discontinuity Rational Function with a Discontinuity The function 4 f () = is a rational function with domain all real numbers ecept when the denominator is zero: = 0 = 5 8. { So the domain is R\ (, 5 8 ) ( 58, ). 5 8 } = f () = Functions (/4) MS11: IT Mathematics John Carroll 39 / 58 Functions (/4) MS11: IT Mathematics John Carroll 40 / 58
11 Eponential & Logarithmic Functions Eponential & Logarithmic Functions Eponential Functions Outline Eponential Functions 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions Functions of the form y = a, where a > 0 are called eponential functions, a is called the base. All eponential functions have domain R. If a 1, then the range of y = a is (0, ). 6 Functions (/4) MS11: IT Mathematics John Carroll 41 / 58 Functions (/4) MS11: IT Mathematics John Carroll 4 / 58 Eponential & Logarithmic Functions Eponential Functions Eponential & Logarithmic Functions Eponential Functions Eponential Function: f () = a Eponential Functions: Family Members In both cases, the domain is (, ), and the range is (0, ) Functions (/4) MS11: IT Mathematics John Carroll 43 / 58 Functions (/4) MS11: IT Mathematics John Carroll 44 / 58
12 Eponential & Logarithmic Functions Eponential Functions Eponential & Logarithmic Functions Eponential Functions The 3 kinds of Eponential Functions y = a Eponential Functions Application Suppose that e10, 000 euro is invested in an account with an interest rate of % per annum. The amount of money in the account after t years is P(t) = 10, 000(1.0) t. Functions (/4) MS11: IT Mathematics John Carroll 45 / 58 Functions (/4) MS11: IT Mathematics John Carroll 46 / 58 Eponential & Logarithmic Functions Logarithmic Functions Eponential & Logarithmic Functions Logarithmic Functions Logarithmic Functions Logarithmic Function: f () = log a Logarithmic functions are functions of the form y = log a where the base a 1 is a positive constant, are the inverse functions of the eponential functions. All logarithmic functions have domain (0, ) and range R. The domain is (0, ), and the range is (, ) Functions (/4) MS11: IT Mathematics John Carroll 47 / 58 Functions (/4) MS11: IT Mathematics John Carroll 48 / 58
13 Eponential & Logarithmic Functions Logarithmic Functions Eponential & Logarithmic Functions Logarithmic Functions Eponential & Logarithmic Functions a > 1 Logarithmic Functions Base a Functions (/4) MS11: IT Mathematics John Carroll 49 / 58 Functions (/4) MS11: IT Mathematics John Carroll 50 / 58 Eponential & Logarithmic Functions Logarithmic Functions Eponential & Logarithmic Functions Logarithmic Functions Natural Logarithmic Function y = ln Logarithmic Function y = ln( ) 1 Functions (/4) MS11: IT Mathematics John Carroll 51 / 58 Functions (/4) MS11: IT Mathematics John Carroll 5 / 58
14 Definition Outline 1 The Role of Functions Piecewise-defined functions are functions which are defined by different formulae in different parts of their domains. Simple Eample f () = 3 3 Power Functions 4 4 Rational Functions f () = f () = 5 Eponential & Logarithmic Functions Functions (/4) MS11: IT Mathematics John Carroll 53 / 58 4 Functions (/4) MS11: IT Mathematics John Carroll 54 / 58 Eamples Eamples Another eample of a function which is described by using different formulae on different parts of its domain is the absolute value function: {, if 0 f () = =, if < 0 Graphing The function, if < 0 f () =, if 0 1 1, if > 1. is defined on the real line but has values given by different formulas depending on the position of. y y = y = y = 1 0 y = 1 y = 1 Domain: (, ) Range: [0, ) Functions (/4) MS11: IT Mathematics John Carroll 55 / 58 Functions (/4) MS11: IT Mathematics John Carroll 56 / 58
15 Eamples Eamples The Greatest Integer function is the function whose value at is the greatest integer less than or equal to it is also called the integer floor function and is denoted. For eample 3. = 3 = 3 = 3.99, 3.7 = 4 = y y = Domain: (, ) Range: Z Functions (/4) MS11: IT Mathematics John Carroll 57 / 58 The Least Integer function is the function whose value at is the smallest integer greater than or equal to it is also called the integer ceiling function and is denoted. For eample 3. = 4 = 4 = , 3.7 = 3 = y y = Domain: (, ) Range: Z Functions (/4) MS11: IT Mathematics John Carroll 58 / 58
FUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS In this section, we will learn about: The purpose of mathematical models. MATHEMATICAL MODELS A mathematical
More information1.2 Mathematical Models: A Catalog of Essential Functions
1.2 Mathematical Models: A Catalog of Essential Functions A is a mathematical description of a real-world phenomenon. e.g., the size of a population, the demand for a product, the speed of a falling object,
More informationMath 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have
Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember
More informationMAC Module 7 Additional Equations and Inequalities. Rev.S08
MAC 1105 Module 7 Additional Equations and Inequalities Learning Objectives Upon completing this module, you should be able to: 1. Use properties of rational exponents (rational powers). 2. Understand
More informationMAC Learning Objectives. Module 7 Additional Equations and Inequalities. Let s Review Some Properties of Rational Exponents
MAC 1105 Module 7 Additional Equations and Inequalities Learning Objectives Upon completing this module, you should be able to: 1. Use properties of rational exponents (rational powers). 2. Understand
More informationChapter 2 Analysis of Graphs of Functions
Chapter Analysis of Graphs of Functions Chapter Analysis of Graphs of Functions Covered in this Chapter:.1 Graphs of Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry..
More information1 Functions and Graphs
1 Functions and Graphs 1.1 Functions Cartesian Coordinate System A Cartesian or rectangular coordinate system is formed by the intersection of a horizontal real number line, usually called the x axis,
More informationQuestion 1. Find the coordinates of the y-intercept for. f) None of the above. Question 2. Find the slope of the line:
of 4 4/4/017 8:44 AM Question 1 Find the coordinates of the y-intercept for. Question Find the slope of the line: of 4 4/4/017 8:44 AM Question 3 Solve the following equation for x : Question 4 Paul has
More information9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater.
Chapter 9 Section 5 9.5 Polynomial and Rational Inequalities Objectives 1 3 Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Solve rational inequalities. Objective 1
More information3.1 Power Functions & Polynomial Functions
3.1 Power Functions & Polynomial Functions A power function is a function that can be represented in the form f() = p, where the base is a variable and the eponent, p, is a number. The Effect of the Power
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 informationMath 120 Handouts. Functions Worksheet I (will be provided in class) Point Slope Equation of the Line 5. Functions Worksheet III 17
Math 0 Handouts HW # (will be provided to class) Lines: Concepts from Previous Classes (emailed to the class) Parabola Plots # (will be provided in class) Functions Worksheet I (will be provided in class)
More informationIntermediate Algebra 100A Final Exam Review Fall 2007
1 Basic Concepts 1. Sets and Other Basic Concepts Words/Concepts to Know: roster form, set builder notation, union, intersection, real numbers, natural numbers, whole numbers, integers, rational numbers,
More informationCalculus I Practice Test Problems for Chapter 2 Page 1 of 7
Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual
More informationLecture Notes Basic Functions and their Properties page 1
Lecture Notes Basic Functions and their Properties page De nition: A function f is (or injective) if for all a and b in its domain, if a = b, then f (a) = f (b). Alternative de nition: A function f is
More informationPre-Calculus Midterm Practice Test (Units 1 through 3)
Name: Date: Period: Pre-Calculus Midterm Practice Test (Units 1 through 3) Learning Target 1A I can describe a set of numbers in a variety of ways. 1. Write the following inequalities in interval notation.
More informationName: Period: SM Starter on Reading Quadratic Graph. This graph and equation represent the path of an object being thrown.
SM Name: Period: 7.5 Starter on Reading Quadratic Graph This graph and equation represent the path of an object being thrown. 1. What is the -ais measuring?. What is the y-ais measuring? 3. What are the
More informationGraphing Review Part 1: Circles, Ellipses and Lines
Graphing Review Part : Circles, Ellipses and Lines Definition The graph of an equation is the set of ordered pairs, (, y), that satisfy the equation We can represent the graph of a function by sketching
More informationFinal Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14
Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)
More informationSect Polynomial and Rational Inequalities
158 Sect 10.2 - Polynomial and Rational Inequalities Concept #1 Solving Inequalities Graphically Definition A Quadratic Inequality is an inequality that can be written in one of the following forms: ax
More informationNONLINEAR FUNCTIONS A. Absolute Value Exercises: 2. We need to scale the graph of Qx ( )
NONLINEAR FUNCTIONS A. Absolute Value Eercises:. We need to scale the graph of Q ( ) f ( ) =. The graph is given below. = by the factor of to get the graph of 9 - - - - -. We need to scale the graph of
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 6x + 4
Math1420 Review Comprehesive Final Assessment Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Add or subtract as indicated. x + 5 1) x2
More informationThe Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function
8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line
More informationfunction independent dependent domain range graph of the function The Vertical Line Test
Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding
More informationAlgebra 2A Unit 1 Week 1 Day Activity Unit 1 Week 2 Day Activity Unit 1 Week 3 Day Activity Unit 2 Week 1 Day Activity
Algebra 2A Unit 1 Week 1 1 Pretest Unit 1 2 Evaluating Rational Expressions 3 Restrictions on Rational Expressions 4 Equivalent Forms of Rational Expressions 5 Simplifying Rational Expressions Unit 1 Week
More informationMATH Spring 2010 Topics per Section
MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line
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 informationChapter 1. Functions 1.1. Functions and Their Graphs
1.1 Functions and Their Graphs 1 Chapter 1. Functions 1.1. Functions and Their Graphs Note. We start by assuming that you are familiar with the idea of a set and the set theoretic symbol ( an element of
More information1/100 Range: 1/10 1/ 2. 1) Constant: choose a value for the constant that can be graphed on the coordinate grid below.
Name 1) Constant: choose a value or the constant that can be graphed on the coordinate grid below a y Toolkit Functions Lab Worksheet thru inverse trig ) Identity: y ) Reciprocal: 1 ( ) y / 1/ 1/1 1/ 1
More informationAlgebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
More informationFinal Exam Review Sheet Algebra for Calculus Fall Find each of the following:
Final Eam Review Sheet Algebra for Calculus Fall 007 Find the distance between each pair of points A) (,7) and (,) B), and, 5 5 Find the midpoint of the segment with endpoints (,) and (,) Find each of
More informationMath 120 Handouts. Functions Worksheet I (will be provided in class) Point Slope Equation of the Line 3. Functions Worksheet III 15
Math 0 Handouts HW # (will be provided to class) Lines: Concepts from Previous Classes (emailed to the class) Parabola Plots # (will be provided in class) Functions Worksheet I (will be provided in class)
More informationIntroduction to Rational Functions
Introduction to Rational Functions The net class of functions that we will investigate is the rational functions. We will eplore the following ideas: Definition of rational function. The basic (untransformed)
More information3 Inequalities Absolute Values Inequalities and Intervals... 4
Contents 1 Real Numbers, Exponents, and Radicals 2 1.1 Rationalizing the Denominator................................... 2 1.2 Factoring Polynomials........................................ 2 1.3 Algebraic
More informationAll quadratic functions have graphs that are U -shaped and are called parabolas. Let s look at some parabolas
Chapter Three: Polynomial and Rational Functions 3.1: Quadratic Functions Definition: Let a, b, and c be real numbers with a 0. The function f (x) = ax 2 + bx + c is called a quadratic function. All quadratic
More information3 Inequalities Absolute Values Inequalities and Intervals... 18
Contents 1 Real Numbers, Exponents, and Radicals 1.1 Rationalizing the Denominator................................... 1. Factoring Polynomials........................................ 1. Algebraic and Fractional
More informationFinal Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i
Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add
More informationHonors Calculus Summer Preparation 2018
Honors Calculus Summer Preparation 08 Name: ARCHBISHOP CURLEY HIGH SCHOOL Honors Calculus Summer Preparation 08 Honors Calculus Summer Work and List of Topical Understandings In order to be a successful
More informationFinal Exam Study Guide Mathematical Thinking, Fall 2003
Final Exam Study Guide Mathematical Thinking, Fall 2003 Chapter R Chapter R contains a lot of basic definitions and notations that are used throughout the rest of the book. Most of you are probably comfortable
More information1 Numbers, Sets, Algebraic Expressions
AAU - Business Mathematics I Lecture #1, February 27, 2010 1 Numbers, Sets, Algebraic Expressions 1.1 Constants, Variables, and Sets A constant is something that does not change, over time or otherwise:
More informationImportant Math 125 Definitions/Formulas/Properties
Exponent Rules (Chapter 3) Important Math 125 Definitions/Formulas/Properties Let m & n be integers and a & b real numbers. Product Property Quotient Property Power to a Power Product to a Power Quotient
More informationFunctions and Their Graphs
Functions and Their Graphs DEFINITION Function A function from a set D to a set Y is a rule that assigns a unique (single) element ƒ(x) Y to each element x D. A symbolic way to say y is a function of x
More informationAlgebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
More informationMATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline
MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 1-75 odd) A. Simplify fractions. B. Change mixed numbers
More informationSection 2.7 Notes Name: Date: Polynomial and Rational Inequalities
Section.7 Notes Name: Date: Precalculus Polynomial and Rational Inequalities At the beginning of this unit we solved quadratic inequalities by using an analysis of the graph of the parabola combined with
More informationChapter 1 Functions and Models
Chapter 1 Functions and Models 1.2 Mathematical Models: A catalog of Essential Functions A mathematical model is a mathematical description of a real world situations such as the size of a population,
More informationCheck boxes of Edited Copy of Sp Topics (was 261-pilot)
Check boxes of Edited Copy of 10023 Sp 11 253 Topics (was 261-pilot) Intermediate Algebra (2011), 3rd Ed. [open all close all] R-Review of Basic Algebraic Concepts Section R.2 Ordering integers Plotting
More informationThe Quadratic Formula. ax 2 bx c 0 where a 0. Deriving the Quadratic Formula. Isolate the constant on the right side of the equation.
SECTION 11.2 11.2 The Quadratic Formula 11.2 OBJECTIVES 1. Solve quadratic equations by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation by using the discriminant
More informationCollege Algebra and College Algebra with Review Final Review
The final exam comprises 30 questions. Each of the 20 multiple choice questions is worth 3 points and each of the 10 open-ended questions is worth 4 points. Instructions for the Actual Final Exam: Work
More informationBeginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions
1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:
More informationPart I: Multiple Choice Questions
Name: Part I: Multiple Choice Questions. What is the slope of the line y=3 A) 0 B) -3 ) C) 3 D) Undefined. What is the slope of the line perpendicular to the line x + 4y = 3 A) -/ B) / ) C) - D) 3. Find
More informationBasic Math Formulas. Unit circle. and. Arithmetic operations (ab means a b) Powers and roots. a(b + c)= ab + ac
Basic Math Formulas Arithmetic operations (ab means ab) Powers and roots a(b + c)= ab + ac a+b c = a b c + c a b + c d = ad+bc bd a b = a c d b d c a c = ac b d bd a b = a+b ( a ) b = ab (y) a = a y a
More informationHarbor Creek School District. Algebra II Advanced. Concepts Timeframe Skills Assessment Standards Linear Equations Inequalities
Algebra II Advanced and Graphing and Solving Linear Linear Absolute Value Relation vs. Standard Forms of Linear Slope Parallel & Perpendicular Lines Scatterplot & Linear Regression Graphing linear Absolute
More informationCalculus I. 1. Limits and Continuity
2301107 Calculus I 1. Limits and Continuity Outline 1.1. Limits 1.1.1 Motivation:Tangent 1.1.2 Limit of a function 1.1.3 Limit laws 1.1.4 Mathematical definition of a it 1.1.5 Infinite it 1.1. Continuity
More informationFor all questions, answer choice E. NOTA" means none of the above answers is correct.
For all questions, answer choice " means none of the above answers is correct. 1. The sum of the integers 1 through n can be modeled by a quadratic polynomial. What is the product of the non-zero coefficients
More informationTitle: Jul 7 8:49 PM (1 of 18)
Title: Jul 7 8:49 PM (1 of 18) Let's review some of the terms used to describe functions. domain: continuous: bounded: even: the set of values from which the input values, x values, are taken every point
More informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types
More informationGraphing Calculator Computations 2
Graphing Calculator Computations A) Write the graphing calculator notation and B) Evaluate each epression. 4 1) 15 43 8 e) 15 - -4 * 3^ + 8 ^ 4/ - 1) ) 5 ) 8 3 3) 3 4 1 8 3) 7 9 4) 1 3 5 4) 5) 5 5) 6)
More informationFinding Slope. Find the slopes of the lines passing through the following points. rise run
Finding Slope Find the slopes of the lines passing through the following points. y y1 Formula for slope: m 1 m rise run Find the slopes of the lines passing through the following points. E #1: (7,0) and
More informationMath 141. Chapter 1: Equations and Inequalities. Notes. Name...
Math 141 Chapter 1: Equations and Inequalities Notes Name... 1 1.1 The Coordinate Plane Graphing Regions in the Coordinate Plane 1. Example: Sketch the region given by the set (a) {(x, y) x = 3} (b) {(x,
More informationCopyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 2/10
Prep for Calculus This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (281 topics + 125 additional topics) Real
More informationDirections: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.
AP Calculus AB SUMMER ASSIGNMENT Multiple Choice Section Directions: Please read questions carefully It is recommended that you do the Short Answer Section prior to doing the Multiple Choice Show all work
More informationPrecalculus Notes: Functions. Skill: Solve problems using the algebra of functions.
Skill: Solve problems using the algebra of functions. Modeling a Function: Numerical (data table) Algebraic (equation) Graphical Using Numerical Values: Look for a common difference. If the first difference
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationNote: The zero function f(x) = 0 is a polynomial function. It has no degree and no leading coefficient. Sep 15 2:51 PM
2.1 Linear and Quadratic Name: Functions and Modeling Objective: Students will be able to recognize and graph linear and quadratic functions, and use these functions to model situations and solve problems.
More informationINTERNET MAT 117. Solution for the Review Problems. (1) Let us consider the circle with equation. x 2 + 2x + y 2 + 3y = 3 4. (x + 1) 2 + (y + 3 2
INTERNET MAT 117 Solution for the Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (i) Group
More informationChapter 3: Inequalities, Lines and Circles, Introduction to Functions
QUIZ AND TEST INFORMATION: The material in this chapter is on Quiz 3 and Exam 2. You should complete at least one attempt of Quiz 3 before taking Exam 2. This material is also on the final exam and used
More information3.1 Functions. We will deal with functions for which both domain and the range are the set (or subset) of real numbers
3.1 Functions A relation is a set of ordered pairs (, y). Eample: The set {(1,a), (1, b), (,b), (3,c), (3, a), (4,a)} is a relation A function is a relation (so, it is the set of ordered pairs) that does
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More informationAccuplacer College Level Math Study Guide
Testing Center Student Success Center Accuplacer Study Guide The following sample questions are similar to the format and content of questions on the Accuplacer College Level Math test. Reviewing these
More informationExample 1: What do you know about the graph of the function
Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What
More informationFunctions of One Variable
Functions of One Variable Mathematical Economics Vilen Lipatov Fall 2014 Outline Functions of one real variable Graphs Linear functions Polynomials, powers and exponentials Reading: Sydsaeter and Hammond,
More informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More informationSect 2.6 Graphs of Basic Functions
Sect. Graphs of Basic Functions Objective : Understanding Continuity. Continuity is an extremely important idea in mathematics. When we say that a function is continuous, it means that its graph has no
More informationSKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution.
SKILL BUILDER TEN Graphs of Linear Equations with Two Variables A first degree equation is called a linear equation, since its graph is a straight line. In a linear equation, each term is a constant or
More informationSTANDARD FORM is a QUADRATIC FUNCTION and its graph is a PARABOLA. The domain of a quadratic function is the set of all real numbers.
EXERCISE 2-3 Things to remember: 1. QUADRATIC FUNCTION If a, b, and c are real numbers with a 0, then the function f() = a 2 + b + c STANDARD FORM is a QUADRATIC FUNCTION and its graph is a PARABOLA. The
More informationReference Material /Formulas for Pre-Calculus CP/ H Summer Packet
Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step
More information3x 2. x ))))) and sketch the graph, labelling everything.
Fall 2006 Practice Math 102 Final Eam 1 1. Sketch the graph of f() =. What is the domain of f? [0, ) Use transformations to sketch the graph of g() = 2. What is the domain of g? 1 1 2. a. Given f() = )))))
More informationCME Project, Algebra Correlated to: Michigan High School Content Expectations, 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
More informationChapter 2 Polynomial and Rational Functions
SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear
More informationUNLV University of Nevada, Las Vegas
UNLV University of Nevada, Las Vegas The Department of Mathematical Sciences Information Regarding Math 14 Final Exam Revised.8.018 While all material covered in the syllabus is essential for success in
More informationBasic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
More informationSummer Packet for Students Taking Introduction to Calculus in the Fall
Summer Packet for Students Taking Introduction to Calculus in the Fall Algebra 2 Topics Needed for Introduction to Calculus Need to know: à Solve Equations Linear Quadratic Absolute Value Polynomial Rational
More informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
More informationAdvanced Algebra Scope and Sequence First Semester. Second Semester
Last update: April 03 Advanced Algebra Scope and Sequence 03-4 First Semester Unit Name Unit : Review of Basic Concepts and Polynomials Unit : Rational and Radical Epressions Sections in Book 0308 SLOs
More informationMTH4100 Calculus I. Lecture notes for Week 2. Thomas Calculus, Sections 1.3 to 1.5. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 2 Thomas Calculus, Sections 1.3 to 1.5 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Reading Assignment: read Thomas
More informationAlgebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:
More information1 Chapter 1: Graphs, Functions, and Models
1 Chapter 1: Graphs, Functions, and Models 1.1 Introduction to Graphing 1.1.1 Know how to graph an equation Eample 1. Create a table of values and graph the equation y = 1. f() 6 1 0 1 f() 3 0 1 0 3 4
More informationSummer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
More informationMAT135 Review for Test 4 Dugopolski Sections 7.5, 7.6, 8.1, 8.2, 8.3, 8.4
Sections 7.5, 7.6, 8.1, 8., 8., 8.4 1. Use the discriminant to determine the number and type(s) of solutions for 4x 8x 4 0. One real solution B. One complex solution Two real solutions Two complex solutions.
More informationMORE CURVE SKETCHING
Mathematics Revision Guides More Curve Sketching Page of 3 MK HOME TUITION Mathematics Revision Guides Level: AS / A Level MEI OCR MEI: C4 MORE CURVE SKETCHING Version : 5 Date: 05--007 Mathematics Revision
More information12.1 The Extrema of a Function
. The Etrema of a Function Question : What is the difference between a relative etremum and an absolute etremum? Question : What is a critical point of a function? Question : How do you find the relative
More informationMock Final Exam Name. Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) A) {- 30} B) {- 6} C) {30} D) {- 28}
Mock Final Exam Name Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) 1) A) {- 30} B) {- 6} C) {30} D) {- 28} First, write the value(s) that make the denominator(s) zero. Then solve the
More informationA Library of Functions
LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us
More informationOBJECTIVES UNIT 1. Lesson 1.0
OBJECTIVES UNIT 1 Lesson 1.0 1. Define "set," "element," "finite set," and "infinite set," "empty set," and "null set" and give two examples of each term. 2. Define "subset," "universal set," and "disjoint
More informationRational Functions 4.5
Math 4 Pre-Calculus Name Date Rational Function Rational Functions 4.5 g ( ) A function is a rational function if f ( ), where g ( ) and ( ) h ( ) h are polynomials. Vertical asymptotes occur at -values
More informationPrep for College Algebra
Prep for College Algebra This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (219 topics + 85 additional topics)
More informationAlgebra I Quadratics Practice Questions
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 From CCSD CSE S Page 1 of 6 1 5. Which is equivalent
More information