2.1 Increasing, Decreasing, and Piecewise Functions; Applications


 Beverley Martin
 7 years ago
 Views:
Transcription
1 2.1 Increasing, Decreasing, and Piecewise Functions; Applications Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima. Given an application, find a function that models the application; find the domain of the function and function values, and then graph the function. Graph functions defined piecewise.
2 Increasing, Decreasing, and Constant Functions On a given interval, if the graph of a function rises from left to right, it is said to be increasing on that interval. If the graph drops from left to right, it is said to be decreasing. If the function values stay the same from left to right, the function is said to be constant. Slide 2.12
3 Definitions A function f is said to be increasing on an open interval I, if for all a and b in that interval, a < b implies f(a) < f(b). Slide 2.13
4 Definitions continued A function f is said to be decreasing on an open interval I, if for all a and b in that interval, a < b implies f(a) > f(b). Slide 2.14
5 Definitions continued A function f is said to be constant on an open interval I, if for all a and b in that interval, f(a) = f(b). Slide 2.15
6 Relative Maximum and Minimum Values Suppose that f is a function for which f(c) exists for some c in the domain of f. Then: f(c) is a relative maximum if there exists an open interval I containing c such that f(c) > f(x), for all x in I where x c; and f(c) is a relative minimum if there exists an open interval I containing c such that f(c) < f(x), for all x in I where x c. Slide 2.16
7 Relative Maximum and Minimum Values y Relative maximum f Relative minimum c 1 c 2 c 3 x Slide 2.17
8 Applications of Functions Many realworld situations can be modeled by functions. Example A man plans to enclose a rectangular area using 80 yards of fencing. If the area is w yards wide, express the enclosed area as a function of w. Solution We want area as a function of w. Since the area is rectangular, we have A = lw. We know that the perimeter, 2 lengths and 2 widths, is 80 yds, so we have 40 yds for one length and one width. If the width is w, then the length, l, can be given by l = 40 w. Now A(w) = (40 w)w = 40w w 2. Slide 2.18
9 Functions Defined Piecewise Some functions are defined piecewise using different output formulas for different parts of the domain. For the function defined as: find f(3), f(1), and f(5). x 2, for x 0, f ( x) 4, for 0 x 2, x 1, for x 2, Since 3 0, use f(x) = x 2 : f( 3) = ( 3) 2 = 9. Since 0 < 1 2, use f(x) = 4: f(1) = 4. Since 5 > 2 use f(x) = x 1: f(5) = 5 1 = 4. Slide 2.19
10 Functions Defined Piecewise Graph the function defined as: 3 for x 0 f x x x x 1 for x ( ) 3 for 0 2 a) We graph f(x) = 3 only for inputs x less than or equal to 0. f(x) = 3, for x 0 x f ( x) 1for x 2 2 f(x) = 3 + x 2, for 0 < x 2 b) We graph f(x) = 3 + x 2 only for inputs x greater than 0 and x less than or equal to 2. c) We graph f(x) = 1 2 only for inputs x greater than 2. Slide
11 Greatest Integer Function = the greatest integer less than or equal to x. The greatest integer function pairs the input with the greatest integer less than or equal to that input Slide
12 2.2 The Algebra of Functions Find the sum, the difference, the product, and the quotient of two functions, and determine the domains of the resulting functions. Find the difference quotient for a function.
13 Sums, Differences, Products, and Quotients of Functions If f and g are functions and x is in the domain of each function, then ( f g)( x) f ( x) g( x) ( f g)( x) f ( x) g( x) ( fg)( x) f ( x) g( x) ( f / g)( x) f ( x) / g( x), provided g( x) 0 Slide
14 Example Given that f(x) = x + 2 and g(x) = 2x + 5, find each of the following. a) (f + g)(x) b) (f + g)(5) Solution: a) ( f g)( x) f ( x) g( x) x 2 2x 5 3x 7 b) We can find (f + g)(5) provided 5 is in the domain of each function. This is true. f(5) = = 7 g(5) = 2(5) + 5 = 15 (f + g)(5) = f(5) + g(5) = = 22 or (f + g)(5) = 3(5) + 7 = 22 Slide
15 Another Example Given that f(x) = x and g(x) = x 3, find each of the following. a) The domain of f + g, f g, fg, and f/g b) (f g)(x) c) (f/g)(x) Solution: a) The domain of f is the set of all real numbers. The domain of g is also the set of all real numbers. The domains of f + g, f g, and fg are the set of numbers in the intersection of the domains that is, the set of numbers in both domains, or all real numbers. For f/g, we must exclude 3, since g(3) = 0. Slide
16 Another Example continued b) (f g)(x) = f(x) g(x) = (x 2 + 2) (x 3) = x 2 x + 5 c) (f/g)(x) = ( f / g)( x) f( x) gx ( ) 2 x 2 x 3 Remember to add the stipulation that x 3, since 3 is not in the domain of (f/g)(x). Slide
17 Difference Quotient The ratio below is called the difference quotient, or average rate of change. f ( x h) f ( x) h Slide
18 Example For the function f given by f(x) = 5x 1, find the difference quotient f ( x h) f ( x). h Solution: We first find f(x + h): f ( x h) 5( x h) 1 5x 5h 1 Slide
19 Example continued f ( x h) f ( x) h 5x 5h 1 (5x 1) h 5h h 5 Slide
20 Another Example For the function f given by f(x) = x 2 + 2x 3, find the difference quotient. Solution: We first find f(x + h): f ( x h) 2 ( x h) 2( x h) x xh h x h Slide
21 Example continued f ( x h) f ( x) h x 2xh h 2x 2h 3 ( x 2x 3) h x 2xh h 2x 2h 3 x 2x 3 h 2 2xh h 2h h h(2x h 2) 2x h 2 h Slide
22 2.3 The Composition of Functions Find the composition of two functions and the domain of the composition. Decompose a function as a composition of two functions.
23 Composition of Functions Definition: The composite function f g, the composition of f and g, is defined as ( f g)( x) f ( g( x)), where x is in the domain of g and g( x) is in the domain of f. Slide
24 Example Given that f(x) = 3x 1 and g(x) = x 2 + x 3, find: a) ( f g)( x) b) ( g f )( x) a) 2 ( f g)( x) f ( g( x)) f ( x x 3) 2 3( x x 3) 1 2 3x 3x x 3x 10 Slide
25 Example Given that f(x) = 3x 1 and g(x) = x 2 + x 3, find: a) ( f g)( x) b) ( g f )( x) b) ( g f )( x) g( f ( x)) g( 3x 1) 2 ( x ) ( x ) 3 2 9x 6x 1 3x x 3x 3 Slide
26 Example Given that f(x) = 3x 1 and g(x) = x 2 + x 3, find: a) ( f g)(2) b) ( g f)(2) a) ( f g)(2) f ( g(2)) f 2 ( 2 2 3) f ( 3) 3( 3) 1 8 Slide
27 Example Given that f(x) = 3x 1 and g(x) = x 2 + x 3, find: a) ( f g)(2) b) ( g f)(2) b) ( g f )(2) g( f (2)) g( 3(2) 1) 2 ( 5) ( 5) 3 27 Slide
28 Example 4 Given f ( x) x and g( x) 2x 3, find the domain of ( f g)( x). Solution: f(x) is not defined for negative radicands. Since the inputs of f g are the outputs of g, the domain of f g consists of all the values in the domain of g for which g(x) is nonnegative. gx ( ) 0 2x 3 0 x 3/ 2 The domain is { x x 3/ 2}, or [ 3/ 2, ). Slide
29 Decomposing a Function as a Composition In calculus, one needs to recognize how a function can be expressed as the composition of two functions. This can be thought of as decomposing the function. Slide
30 Example If h(x) = (3x 1) 4, find f(x) and g(x) such that h( x) ( f g)( x). Solution: The function h(x) raises (3x 1) to the fourth power. Two functions that can be used for the composition are: h( x) ( f g)( x) f ( g( x)) f (3x 1) 3x 1 4 f(x) = x 4 and g(x) = 3x 1. Slide
31 2.4 Symmetry and Transformations Determine whether a graph is symmetric with respect to the xaxis, the yaxis, and the origin. Determine whether a function is even, odd, or neither even nor odd. Given the graph of a function, graph its transformation under translations, reflections, stretchings, and shrinkings.
32 Symmetry Algebraic Tests of Symmetry xaxis: If replacing y with y produces an equivalent equation, then the graph is symmetric with respect to the xaxis. yaxis: If replacing x with x produces an equivalent equation, then the graph is symmetric with respect to the yaxis. Origin: If replacing x with x and y with y produces an equivalent equation, then the graph is symmetric with respect to the origin. Slide
33 Example Test x = y for symmetry with respect to the xaxis, the yaxis, and the origin. xaxis: We replace y with y: x x 2 ( y) 2 y 2 2 The resulting equation is equivalent to the original so the graph is symmetric with respect to the xaxis. Slide
34 Example continued Test x = y for symmetry with respect to the xaxis, the yaxis, and the origin. yaxis: We replace x with x: x y x y The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the yaxis. Slide
35 Example continued Origin: We replace x with x and y with y: x x x y 2 2 ( y) 2 y 2 The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the origin. 2 2 Slide
36 Even and Odd Functions If the graph of a function f is symmetric with respect to the yaxis, we say that it is an even function. That is, for each x in the domain of f, f(x) = f( x). If the graph of a function f is symmetric with respect to the origin, we say that it is an odd function. That is, for each x in the domain of f, f( x) = f(x). Slide
37 Example Determine whether the function is even, odd, or neither. h( x) x 4x h( x) ( x) 4( x) x 4 2 4x 4 2 y = x 4 4x 2 We see that h(x) = h( x). Thus, h is even. Slide
38 Example Determine whether the function is even, odd, or neither. h( x) x 4x h( x) ( x 4 x ) x 4x 4 2 y = x 4 4x 2 We see that h( x) h(x). Thus, h is not odd. Slide
39 Vertical Translation Vertical Translation For b > 0, the graph of y = f(x) + b is the graph of y = f(x) shifted up b units; y = 3x 2 +2 y = 3x 2 the graph of y = f(x) b is the graph of y = f(x) shifted down b units. y = 3x 2 3 Slide
40 Horizontal Translation Horizontal Translation For d > 0, the graph of y = f(x d) is the graph of y = f(x) shifted right d units; y = (3x 2) 2 the graph of y = f(x + d) is the graph of y = f(x) shifted left d units. y = (3x + 2) 2 y = 3x 2 Slide
41 Reflections The graph of y = f(x) is the reflection of the graph of y = f(x) across the xaxis. The graph of y = f( x) is the reflection of the graph of y = f(x) across the yaxis. If a point (x, y) is on the graph of y = f(x), then (x, y) is on the graph of y = f(x), and ( x, y) is on the graph of y = f( x). Slide
42 Example Reflection of the graph y = 3x 3 4x 2 across the xaxis. y = 3x 3 4x 2 y = 3x 3 + 4x 2 Slide
43 Example Reflection of the graph y = x 3 2x 2 across the yaxis. y = x3 + 2x 2 y = x 3 2x 2 Slide
44 Vertical Stretching and Shrinking The graph of y = af(x) can be obtained from the graph of y = f(x) by stretching vertically for a > 1, or shrinking vertically for 0 < a < 1. For a < 0, the graph is also reflected across the x axis. (The ycoordinates of the graph of y = af(x) can be obtained by multiplying the ycoordinates of y = f(x) by a.) Slide
45 Examples Stretch y = x 3 x vertically. Slide
46 Examples Shrink y = x 3 x vertically. Slide
47 Examples Stretch and reflect y = x 3 x across the x axis Slide
48 Horizontal Stretching or Shrinking The graph of y = f(cx) can be obtained from the graph of y = f(x) by shrinking horizontally for c > 1, or stretching horizontally for 0 < c < 1. For c < 0, the graph is also reflected across the y axis. (The xcoordinates of the graph of y = f(cx) can be obtained by dividing the xcoordinates of the graph of y = f(x) by c.) Slide
49 Examples Shrink y = x 3 x horizontally. Slide
50 Examples Stretch y = x 3 x horizontally. Slide
51 Examples Stretch horizontally and reflect y = x 3 x. Slide
52 2.5 Variation and Applications Find equations of direct, inverse, and combined variation given values of the variables. Solve applied problems involving variation.
53 Direct Variation If a situation gives rise to a linear function f(x) = kx, or y = kx, where k is a positive constant, we say that we have direct variation, or that y varies directly as x, or that y is directly proportional to x. The number k is called the variation constant, or constant of proportionality. Slide
54 Direct Variation The graph of y = kx, k > 0, always goes through the origin and rises from left to right. As x increases, y increases; that is, the function is increasing on the interval (0, ). The constant k is also the slope of the line. y kx, k 0 Slide
55 Direct Variation Example: Find the variation constant and an equation of variation in which y varies directly as x, and y = 42 when x = 3. Solution: We know that (3, 42) is a solution of y = kx. y = kx 42 = k k 14 = k The variation constant 14, is the rate of change of y with respect to x. The equation of variation is y = 14x. Slide
56 Application Example: Wages. A cashier earns an hourly wage. If the cashier worked 18 hours and earned $168.30, how much will the cashier earn if she works 33 hours? Solution: We can express the amount of money earned as a function of the amount of hours worked. I(h) = kh I(18) = k 18 $ = k 18 $9.35 = k The hourly wage is the variation constant. Next, we use the equation to find how much the cashier will earn if she works 33 hours. I(33) = $9.35(33) = $ Slide
57 Inverse Variation If a situation gives rise to a function f(x) = k/x, or y = k/x, where k is a positive constant, we say that we have inverse variation, or that y varies inversely as x, or that y is inversely proportional to x. The number k is called the variation constant, or constant of proportionality. For the graph y = k/x, k 0, as x increases, y decreases; that is, the function is decreasing on the interval (0, ). Slide
58 Inverse Variation For the graph y = k/x, k 0, as x increases, y decreases; that is, the function is decreasing on the interval (0, ). y k x, k 0 Slide
59 Inverse Variation Example: Find the variation constant and an equation of variation in which y varies inversely as x, and y = 22 when x = 0.4. Solution: k y x k (0.4)22 k 8.8 k The variation constant is 8.8. The equation of variation is y = 8.8/x. Slide
60 Application Example: Road Construction. The time t required to do a job varies inversely as the number of people P who work on the job (assuming that they all work at the same rate). If it takes 180 days for 12 workers to complete a job, how long will it take 15 workers to complete the same job? Solution: We can express the amount of time required, in days, as a function of the number of people working. k t varies inversely as P tp ( ) P k t(12) 12 k k This is the variation constant. Slide
61 Application continued The equation of variation is t(p) = 2160/P. Next we compute t(15). tp ( ) t(15) t 2160 P It would take 144 days for 15 people to complete the same job. Slide
62 Combined Variation Other kinds of variation: y varies directly as the nth power of x if there is some n positive constant k such that. y varies inversely as the nth power of x if k there is some positive constant k such that y. y varies jointly as x and z if there is some positive constant k such that y = kxz. y kx x n Slide
63 Example The luminance of a light (E) varies directly with the intensity (I) of the light and inversely with the square distance (D) from the light. At a distance of 10 feet, a light meter reads 3 units for a 50cd lamp. Find the luminance of a 27cd lamp at a distance of 9 feet. I E k D k 50 Solve for k k Substitute the second set of data into the equation. The lamp gives an luminance reading of 2 units. E E Slide
64 2.6 Solving Linear Inequalities Solve linear inequalities. Solve compound inequalities. Solve inequalities with absolute value. Solve applied problems using inequalities. Copyright 2008 Pearson Education, Inc.
65 Inequalities An inequality is a sentence with <, >,, or as its verb. Examples: 5x 7 < 3 + 4x 3(x + 6) 4(x 3) Copyright 2008 Pearson Education, Inc. Slide
66 Principles for Solving Inequalities For any real numbers a, b, and c: The Addition Principle for Inequalities: If a < b is true, then a + c < b + c is true. The Multiplication Principle for Inequalities: If a < b and c > 0 are true, then ac < bc is true. If a < b and c < 0, then ac > bc is true. Similar statements hold for a b. When both sides of an inequality are multiplied or divided by a negative number, we must reverse the inequality sign. Copyright 2008 Pearson Education, Inc. Slide
67 Examples Solve: 4x 6 2x 10 4x 2x 4 2x 4 x 2 Solve: 6( x 3) 7( x 2) 6x 18 7x 14 x 4 x 4 {x x < 2} or (, 2) ) {x x 4} or [ 4, ) [ Copyright 2008 Pearson Education, Inc. Slide
68 Compound Inequalities When two inequalities are joined by the word and or the word or, a compound inequality is formed. Conjunction contains the word and. Example: 7 < 3x + 5 and 3x Disjunction contains the word or. Example: 3x or 3x + 6 > 12 Copyright 2008 Pearson Education, Inc. Slide
69 Examples Solve: 4 3x x x 3 4 x 1 Solve: 4x 5 3 or 4x 5 > 3 4x 5 3 or 4x 5 3 4x 2 4x 8 1 x 2 x 2 ( ] ] ( Copyright 2008 Pearson Education, Inc. Slide
70 Inequalities with Absolute Value Inequalities sometimes contain absolutevalue notation. The following properties are used to solve them. For a > 0 and an algebraic expression X: X < a is equivalent to a < X < a. X > a is equivalent to X < a or X > a. Similar statements hold for X a and X a. Copyright 2008 Pearson Education, Inc. Slide
71 Example Solve: 4x x x x 2 ( ) Copyright 2008 Pearson Education, Inc. Slide
72 Application Johnson Catering charges $100 plus $30 per hour to cater an event. Catherine s Catering charges $50 per hour. For what lengths of time does it cost less to hire Catherine s Catering? 1. Familiarize. Read the problem. 2. Translate. Catherine s is less than Johnson 50x < x Copyright 2008 Pearson Education, Inc. Slide
73 Application continued 3. Carry out. 50x x 20x 100 x 5 4. Check. 50(5)? (5) 250? State. For values of x < 5 hr, Catherine s Catering will cost less. Copyright 2008 Pearson Education, Inc. Slide
Algebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationList the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated
MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible
More information1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some
Section 3.1: First Derivative Test Definition. Let f be a function with domain D. 1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some open interval containing c. The number
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationSimplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 249x + 6 x  6 A) 1, x 6 B) 8x  1, x 6 x 
More informationAnalyzing Piecewise Functions
Connecting Geometry to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 04/9/09 Analyzing Piecewise Functions Objective: Students will analyze attributes of a piecewise function including
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More information1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each xcoordinate was matched with only one ycoordinate. We spent most
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: AAPR.3: Identify zeros of polynomials
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More information1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.
1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal
More informationTo define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions
Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationa. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F
FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationIntegers (pages 294 298)
A Integers (pages 294 298) An integer is any number from this set of the whole numbers and their opposites: { 3, 2,, 0,, 2, 3, }. Integers that are greater than zero are positive integers. You can write
More informationFunctions. MATH 160, Precalculus. J. Robert Buchanan. Fall 2011. Department of Mathematics. J. Robert Buchanan Functions
Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: determine whether relations between variables are functions, use function
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationAlgebra I Notes Relations and Functions Unit 03a
OBJECTIVES: F.IF.A.1 Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More informationCourse Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)
Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationEquations. #110 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0
College Algebra Review Problems for Final Exam Equations #110 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More information2.5 Transformations of Functions
2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More information7.1 Graphs of Quadratic Functions in Vertex Form
7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationWeek 1: Functions and Equations
Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.12.2, and Chapter
More information1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number
1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x  x) B. x 3 x C. 3x  x D. x  3x 2) Write the following as an algebraic expression
More informationElements of a graph. Click on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and yintercept in the equation of a line Comparing lines on
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina  Beaufort Lisa S. Yocco, Georgia Southern University
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationAlgebra II A Final Exam
Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationIV. ALGEBRAIC CONCEPTS
IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationNotes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions.
Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions. Basic Functions In several sections you will be applying shifts
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationIntroduction to Quadratic Functions
Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2
More informationMPE Review Section III: Logarithmic & Exponential Functions
MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure
More informationGraphing Quadratic Functions
Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x value and L be the yvalues for a graph. 1. How are the x and yvalues related? What pattern do you see? To enter the
More informationhttps://williamshartunionca.springboardonline.org/ebook/book/27e8f1b87a1c4555a1212b...
of 19 9/2/2014 12:09 PM Answers Teacher Copy Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach BellRinger Activity Students
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Prealgebra Algebra Precalculus Calculus Statistics
More informationUnit 1 Equations, Inequalities, Functions
Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1100 Overview: This unit models realworld situations by using one and twovariable linear equations. This unit will further expand upon pervious
More informationALGEBRA I (Common Core)
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Wednesday, June 17, 2015 1:15 to 4:15 p.m. MODEL RESPONSE SET Table of Contents Question 25..................
More informationMath 131 College Algebra Fall 2015
Math 131 College Algebra Fall 2015 Instructor's Name: Office Location: Office Hours: Office Phone: Email: Course Description This course has a minimal review of algebraic skills followed by a study of
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions  4, use the graph at the right.. Eplain wh the graph
More informationHow To Understand And Solve Algebraic Equations
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationContinuity. DEFINITION 1: A function f is continuous at a number a if. lim
Continuity DEFINITION : A function f is continuous at a number a if f(x) = f(a) REMARK: It follows from the definition that f is continuous at a if and only if. f(a) is defined. 2. f(x) and +f(x) exist.
More informationUnit 7: Radical Functions & Rational Exponents
Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More informationhttp://www.aleks.com Access Code: RVAE4EGKVN Financial Aid Code: 6A9DBDEE3B74F5157304
MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationChapter 2: Linear Equations and Inequalities Lecture notes Math 1010
Section 2.1: Linear Equations Definition of equation An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable
More informationVoyager Sopris Learning Vmath, Levels CI, correlated to the South Carolina College and CareerReady Standards for Mathematics, Grades 28
Page 1 of 35 VMath, Level C Grade 2 Mathematical Process Standards 1. Make sense of problems and persevere in solving them. Module 3: Lesson 4: 156159 Module 4: Lesson 7: 220223 2. Reason both contextually
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationObjectives. Materials
Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI84 Plus / TI83 Plus Graph paper Introduction One of the ways
More informationScope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B
Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced
More informationf(x) = g(x), if x A h(x), if x B.
1. Piecewise Functions By Bryan Carrillo, University of California, Riverside We can create more complicated functions by considering Piecewise functions. Definition: Piecewisefunction. A piecewisefunction
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationSome Lecture Notes and InClass Examples for PreCalculus:
Some Lecture Notes and InClass Examples for PreCalculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
More informationFunctions Modeling Change: A Precalculus Course. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Functions Modeling Change: A Precalculus Course Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 PREFACE This supplement consists of my lectures of a freshmenlevel mathematics class offered
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More information1) (3) + (6) = 2) (2) + (5) = 3) (7) + (1) = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = 7) (5)(4) = 8) (3)(6) = 9) (1)(2) =
Extra Practice for Lesson Add or subtract. ) (3) + (6) = 2) (2) + (5) = 3) (7) + () = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = Multiply. 7) (5)(4) = 8) (3)(6) = 9) ()(2) = Division is
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationWeek 2: Exponential Functions
Week 2: Exponential Functions Goals: Introduce exponential functions Study the compounded interest and introduce the number e Suggested Textbook Readings: Chapter 4: 4.1, and Chapter 5: 5.1. Practice Problems:
More informationCOMMON CORE STATE STANDARDS FOR MATHEMATICS 35 DOMAIN PROGRESSIONS
COMMON CORE STATE STANDARDS FOR MATHEMATICS 35 DOMAIN PROGRESSIONS Compiled by Dewey Gottlieb, Hawaii Department of Education June 2010 Operations and Algebraic Thinking Represent and solve problems involving
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationStudents will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.
Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable
More informationGraphing calculators Transparencies (optional)
What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity Teacher Version Lesson Objective: Length of Activity: Students will: Recognize piecewise functions and the notation used
More informationLogo Symmetry Learning Task. Unit 5
Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to
More informationDomain of a Composition
Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such
More informationAP Calculus AB 2004 FreeResponse Questions
AP Calculus AB 2004 FreeResponse Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More information