2.3 Solving Absolute Value Inequalities
|
|
- Lucy Baker
- 6 years ago
- Views:
Transcription
1 Name Class Date.3 Solving Absolute Value Inequalities Essential Question: What are two was to solve an absolute value inequalit? Resource Locker Eplore Visualizing the Solution Set of an Absolute Value Inequalit You know that when solving an absolute value equation, it s possible to get two solutions. Here, ou will eplore what happens when ou solve absolute value inequalities. A Determine whether each of the integers from -5 to 5 is a solution of the inequalit + < 5. Write es or no for each number in the table. If a number is a solution, plot it on the number line. Number = -5 = - Solution? = -3 = = -1 = = 1 = = 3 = = 5 B Determine whether each of the integers from -5 to 5 is a solution of the inequalit + > 5. Write es or no for each number in the table. If a number is a solution, plot it on the number line Number = -5 = - = -3 = - = -1 = = 1 = Solution? = 3 = = 5 Module 71 Lesson 3
2 State the solutions of the equation + = 5 and relate them to the solutions ou found for the inequalities in Steps A and B. If is an real number and not just an integer, graph the solutions of + < 5 and + > 5. Graph of all real solutions of + < 5: Graph of all real solutions of + > 5: Reflect 1. It s possible to describe the solutions of + < 5 and + > 5 using inequalities that don t involve absolute value. For instance, ou can write the solutions of + < 5 as > -3 and < 3. Notice that the word and is used because must be both greater than -3 and less than 3. How would ou write the solutions of + > 5? Eplain.. Describe the solutions of + 5 and + 5 using inequalities that don t involve absolute value. Eplain 1 Solving Absolute Value Inequalities Graphicall You can use a graph to solve an absolute value inequalit of the form ƒ () > g () or ƒ () < g (), where ƒ () is an absolute value function and g () is a constant function. Graph each function separatel on the same coordinate plane and determine the intervals on the -ais where one graph lies above or below the other. For ƒ () > g (), ou want to find the -values for which the graph ƒ () is above the graph of g (). For ƒ () < g (), ou want to find the -values for which the graph of ƒ () is below the graph of g (). Eample 1 A > Solve the inequalit graphicall. The inequalit is of the form ƒ () > g (), so determine the intervals on the -ais where the graph of ƒ () = lies above the graph of g () = The graph of ƒ () = lies above the graph of g () = to the left of = -6 and to the right of =, so the solution of > is < -6 or >. - Module 7 Lesson 3
3 B < 1 The inequalit is of the form f () < g (), so determine the intervals on the -ais where the graph of f () = lies the graph of g () = 1. The graph of f () = lies the graph of g () = 1 between = and =, so the solution of < 1 is > and < Reflect 3. Suppose the inequalit in Part A is instead of >. How does the solution change?. In Part B, what is another wa to write the solution > - and < 6? 5. Discussion Suppose the graph of an absolute value function ƒ () lies entirel above the graph of the constant function g (). What is the solution of the inequalit ƒ () > g ()? What is the solution of the inequalit ƒ () < g ()? Your Turn 6. Solve graphicall. Module 73 Lesson 3
4 Eplain Solving Absolute Value Inequalities Algebraicall To solve an absolute value inequalit algebraicall, start b isolating the absolute value epression. When the absolute value epression is b itself on one side of the inequalit, appl one of the following rules to finish solving the inequalit for the variable. Solving Absolute Value Inequalities Algebraicall 1. If > a where a is a positive number, then < -a or > a.. If < a where a is a positive number, then -a < < a. Eample A Solve the inequalit algebraicall. Graph the solution on a number line > 1 - > 6 B < -6 or - > 6 - < -1 or - > > 1 or < - The solution is > 1 or < -. + and + and The solution is and, or Reflect 7. In Part A, suppose the inequalit were > 1 instead of > 1. How would the solution change? Eplain. 8. In Part B, suppose the inequalit were instead of How would the solution change? Eplain. Module 7 Lesson 3
5 Your Turn Solve the inequalit algebraicall. Graph the solution on a number line < Eplain 3 Solving a Real-World Problem with Absolute Value Inequalities Absolute value inequalities are often used to model real-world situations involving a margin of error or tolerance. Tolerance is the allowable amount of variation in a quantit. Eample 3 A machine at a lumber mill cuts boards that are 3.5 meters long. It is acceptable for the length to differ from this value b at most. meters. Write and solve an absolute value inequalit to find the range of acceptable lengths. Analze Information Identif the important information. The boards being cut are meters long. The length can differ b at most. meters. Formulate a Plan Let the length of a board be l. Since the sign of the difference between l and 3.5 doesn t matter, take the absolute value of the difference. Since the absolute value of the difference can be at most., the inequalit that models the situation is l -. Solve l l and l l and l So, the range of acceptable lengths is l. Module 75 Lesson 3
6 Justif and Evaluate The bounds of the range are positive and close to, so this is a reasonable answer. The answer is correct since +. = 3.5 and -. = 3.5. Your Turn 11. A bo of cereal is supposed to weigh 13.8 oz, but it s acceptable for the weight to var as much as.1 oz. Write and solve an absolute value inequalit to find the range of acceptable weights. Elaborate 1. Describe the values of that satisf the inequalities < a and > a where a is a positive constant. 13. How do ou algebraicall solve an absolute value inequalit? 1. Eplain wh the solution of > a is all real numbers if a is a negative number. 15. Essential Question Check-In How do ou solve an absolute value inequalit graphicall? Module 76 Lesson 3
7 Evaluate: Homework and Practice 1. Determine whether each of the integers from -5 to 5 is a solution of the inequalit If a number is a solution, plot it on the number line Online Homework Hints and Help Etra Practice Determine whether each of the integers from -5 to 5 is a solution of the inequalit If a number is a solution, plot it on the number line Solve each inequalit graphicall > _ + < Module 77 Lesson 3
8 Match each graph with the corresponding absolute value inequalit. Then give the solution of the inequalit. A. + 1 > 3 B. + 1 < 3 C. - 1 > 3 D. - 1 < Solve each absolute value inequalit algebraicall. Graph the solution on a number line _ + 3 > Module 78 Lesson 3
9 < < < Module 79 Lesson 3
10 Solve each problem using an absolute value inequalit. 17. The thermostat for a house is set to 68 F, but the actual temperature ma var b as much as F. What is the range of possible temperatures? 18. The balance of Jason s checking account is $3. The balance varies b as much as $8 each week. What are the possible balances of Jason s account? 19. On average, a squirrel lives to be 6.5 ears old. The lifespan of a squirrel ma var b as much as 1.5 ears. What is the range of ages that a squirrel lives? Image Credits: (t) Burwell and Burwell Photograph/iStockPhoto.com; (b) Flickr/Nanc Rose/Gett Images Module 8 Lesson 3
11 . You are plaing a histor quiz game where ou must give the ears of historical events. In order to score an points at all for a question about the ear in which a man first stepped on the moon, our answer must be no more than 3 ears awa from the correct answer, What is the range of answers that allow ou to score points? 1. The speed limit on a road is 3 miles per hour. Drivers on this road tpicall var their speed around the limit b as much as 5 miles per hour. What is the range of tpical speeds on this road? Image Credits: (t) NASA Headquarters - Great Images in NASA (NASA-HQ-GRIN); (b) Image Source/ Gett Images Module 81 Lesson 3
12 H.O.T. Focus on Higher Order Thinking. Represent Real-World Problems A poll of likel voters shows that the incumbent will get 51% of the vote in an upcoming election. Based on the number of voters polled, the results of the poll could be off b as much as 3 percentage points. What does this mean for the incumbent? 3. Eplain the Error A student solved the inequalit > 1 graphicall. Identif and correct the student s error. I graphed the functions () = and g() = 1. Because the graph of g() lies above the graph of () between = -3 and = 5, the solution of the inequalit is -3 < < Module 8 Lesson 3
13 . Multi-Step Recall that a literal equation or inequalit is one in which the constants have been replaced b letters. a. Solve a + b > c for. Write the solution in terms of a, b, and c. Assume that a > and c. b. Use the solution of the literal inequalit to find the solution of > 1. c. In Part a, eplain how the restrictions a > and c affect finding the solutions of the inequalit. Module 83 Lesson 3
14 Lesson Performance Task The distance between the Sun and each planet in our solar sstem varies because the planets travel in elliptical orbits around the Sun. Here is a table of the average distance and the variation in the distance for the five innermost planets in our solar sstem. Average Distance Variation Mercur 36. million miles 7.39 million miles Venus 67. million miles.3 million miles Earth 93. million miles 1.55 million miles Mars 1 million miles 13. million miles Jupiter 8 million miles 3. million miles a. Write and solve an inequalit to represent the range of distances that can occur between the Sun and each planet. b. Calculate the percentage variation (variation divided b average distance) in the orbit of each of the planets. Based on these percentages, which planet has the most elliptical orbit? Image Credits: JPL/NASA Module 8 Lesson 3
2.3 Solving Absolute Value Inequalities
.3 Solving Absolute Value Inequalities Essential Question: What are two was to solve an absolute value inequalit? Resource Locker Eplore Visualizing the Solution Set of an Absolute Value Inequalit You
More information13.1 Understanding Piecewise-Defined Functions
Name Class Date 13.1 Understanding Piecewise-Defined Functions Essential Question: How are piecewise-defined functions different from other functions? Resource Locker Eplore Eploring Piecewise-Defined
More information10.2 Graphing Exponential Functions
Name Class Date 10. Graphing Eponential Functions Essential Question: How do ou graph an eponential function of the form f () = ab? Resource Locker Eplore Eploring Graphs of Eponential Functions Eponential
More information2.3. Solving Absolute Value Inequalities. Inequalities ENGAGE. 2.3 Solving Absolute Value
Resource Locker LESSO N 2.3 Solving Absolute Value Inequalities Name Class Date 2.3 Solving Absolute Value Inequalities Texas Math Standards The student is expected to: A2.6.F Solve absolute value linear
More information15.4 Equation of a Circle
Name Class Date 1.4 Equation of a Circle Essential Question: How can ou write the equation of a circle if ou know its radius and the coordinates of its center? Eplore G.1.E Show the equation of a circle
More information11.1 Solving Linear Systems by Graphing
Name Class Date 11.1 Solving Linear Sstems b Graphing Essential Question: How can ou find the solution of a sstem of linear equations b graphing? Resource Locker Eplore Tpes of Sstems of Linear Equations
More information5.2 Solving Linear-Quadratic Systems
Name Class Date 5. Solving Linear-Quadratic Sstems Essential Question: How can ou solve a sstem composed of a linear equation in two variables and a quadratic equation in two variables? Resource Locker
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 information13.1 Exponential Growth Functions
Name Class Date 1.1 Eponential Growth Functions Essential Question: How is the graph of g () = a b - h + k where b > 1 related to the graph of f () = b? Resource Locker Eplore 1 Graphing and Analzing f
More information7.2 Connecting Intercepts and Linear Factors
Name Class Date 7.2 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 informationName Class Date. Inverse of Function. Understanding Inverses of Functions
Name Class Date. Inverses of Functions Essential Question: What is an inverse function, and how do ou know it s an inverse function? A..B Graph and write the inverse of a function using notation such as
More information13.2 Exponential Growth Functions
Name Class Date. Eponential Growth Functions Essential Question: How is the graph of g () = a b - h + k where b > related to the graph of f () = b? A.5.A Determine the effects on the ke attributes on the
More information9.5 Solving Nonlinear Systems
Name Class Date 9.5 Solving Nonlinear Sstems Essential Question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? Eplore Determining the Possible Number of
More information4.1 Identifying and Graphing Sequences
Name Class Date 4.1 Identifing and Graphing Sequences Essential Question: What is a sequence and how are sequences and functions related? Resource Locker Eplore Understanding Sequences A go-kart racing
More informationDomain, Range, and End Behavior
Locker LESSON 1.1 Domain, Range, and End Behavior Common Core Math Standards The student is epected to: F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship
More information10.2 Graphing Square Root Functions
Name Class Date. Graphing Square Root Functions Essential Question: How can ou use transformations of a parent square root function to graph functions of the form g () = a (-h) + k or g () = b (-h) + k?
More information10.1 Inverses of Simple Quadratic and Cubic Functions
Name Class Date 10.1 Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of quadratic functions and cubic functions, and how can ou find them? Resource
More information15.2 Graphing Logarithmic
Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > and b 1 related to the graph of f () = log b? Resource Locker Eplore 1 Graphing
More information13.2 Exponential Decay Functions
Name Class Date 13. Eponential Deca Functions Essential Question: How is the graph of g () = a b h + k where < b < 1 related to the graph of f () = b? Eplore 1 Graphing and Analzing f () = ( 1 and f ()
More information4.1 Circles. Deriving the Standard-Form Equation of a Circle. Explore
Name Class Date 4.1 Circles ssential Question: What is the standard form for the equation of a circle, and what does the standard form tell ou about the circle? plore Deriving the Standard-Form quation
More information15.2 Graphing Logarithmic
Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > 0 and b 1 related to the graph of f () = log b? Resource Locker A.5.A Determine
More information11.1 Inverses of Simple Quadratic and Cubic Functions
Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A,
More informationMaintaining Mathematical Proficiency
Name Date Chapter 5 Maintaining Mathematical Proficienc Graph the equation. 1. + =. = 3 3. 5 + = 10. 3 = 5. 3 = 6. 3 + = 1 Solve the inequalit. Graph the solution. 7. a 3 > 8. c 9. d 5 < 3 10. 8 3r 5 r
More information5.1 Understanding Linear Functions
Name Class Date 5.1 Understanding Linear Functions Essential Question: What is a linear function? Resource Locker Eplore 1 Recognizing Linear Functions A race car can travel up to 210 mph. If the car could
More informationEssential Question: How can you compare linear functions that are represented in different ways? Explore Comparing Properties of Linear Functions
Locker LESSON 6.5 Comparing Properties of Linear Functions Common Core Math Standards The student is epected to: F-IF.9 Compare properties of two functions each represented in a different wa (algebraicall,
More information6.5 Comparing Properties of Linear Functions
Name Class Date 6.5 Comparing Properties of Linear Functions Essential Question: How can ou compare linear functions that are represented in different was? Resource Locker Eplore Comparing Properties of
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 information7.1 Connecting Intercepts and Zeros
Locker LESSON 7. Connecting Intercepts and Zeros Common Core Math Standards The student is epected to: F-IF.7a Graph linear and quadratic functions and show intercepts, maima, and minima. Also A-REI.,
More informationx. 4. 2x 10 4x. 10 x
CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Reasoning with Equations and Quantities Name: Date: Understand solving equations as a process of reasoning and eplain the reasoning MCC9-1.A.REI.1 Eplain
More informationGraphs of Nonlinear Inequalities
3-3 BJECTIVES Graph polnomial, absolute value, and radical inequalities in two variables. Solve absolute value inequalities. Graphs of Nonlinear Inequalities PHARMACLGY Pharmacists label medication as
More information6.3 Interpreting Vertex Form and Standard Form
Name Class Date 6.3 Interpreting Verte Form and Standard Form Essential Question: How can ou change the verte form of a quadratic function to standard form? Resource Locker Eplore Identifing Quadratic
More information14.3 Constructing Exponential Functions
Name Class Date 1.3 Constructing Eponential Functions Essential Question: What are discrete eponential functions and how do ou represent them? Resource Locker Eplore Understanding Discrete Eponential Functions
More informationLESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II
LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS
More information10.1 Inverses of Simple Quadratic and Cubic Functions
COMMON CORE Locker LESSON 0. Inverses of Simple Quadratic and Cubic Functions Name Class Date 0. Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of
More informationSystems of Linear Inequalities
. Sstems of Linear Inequalities sstem of linear inequalities? How can ou sketch the graph of a ACTIVITY: Graphing Linear Inequalities Work with a partner. Match the linear inequalit with its graph. + Inequalit
More informationExplore 1 Graphing and Analyzing f(x) = e x. The following table represents the function ƒ (x) = (1 + 1 x) x for several values of x.
1_ 8 6 8 Locker LESSON 13. The Base e Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes of the graphs of ƒ () = b and ƒ () = log b () where b is, 1, and e
More informationGraph and Write Equations of Ellipses. You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses.
TEKS 9.4 a.5, A.5.B, A.5.C Before Now Graph and Write Equations of Ellipses You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses. Wh? So ou can model
More informationLESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II
LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS
More informationSummer MA Lesson 14 Section 1.7 (part 2) and Sections 1.1 & 2.8
Summer MA 1500 Lesson 14 Section 1.7 (part ) and Sections 1.1 &.8 I Solving Absolute Value Inequalities Absolute Value Inequalities: u < c or u c, if c 0 The inequalit u < cindicates all values less than
More information9.1 Adding and Subtracting Rational Expressions
Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions? Resource Locker Eplore Identifying Ecluded Values Given a rational epression,
More informationLaurie s Notes. Overview of Section 3.5
Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson.
More information10.1 Adding and Subtracting Rational Expressions
Name Class Date 10.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions? A2.7.F Determine the sum, difference of rational epressions with integral
More informationMHF 4U Unit 1 Polynomial Functions Outline
MHF 4U Unit 1 Polnomial Functions Outline Da Lesson Title Specific Epectations 1 Average Rate of Change and Secants D1., 1.6, both D1.1A s - Instantaneous Rate of Change and Tangents D1.6, 1.4, 1.7, 1.5,
More informationSystems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.
NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations
More informationName Class Date. Deriving the Standard-Form Equation of a Parabola
Name Class Date 1. Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Eplore Deriving the Standard-Form Equation of
More information15.2 Graphing Logarithmic
_ - - - - - - Locker LESSON 5. Graphing Logarithmic Functions Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes on the graphs of f () = b and f () = log b
More informationQuick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.)
Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,,
More informationSolving Systems Using Tables and Graphs
3-1 Solving Sstems Using Tables and Graphs Vocabular Review 1. Cross out the equation that is NOT in slope-intercept form. 1 5 7 r 5 s a 5!3b 1 5 3 1 7 5 13 Vocabular Builder linear sstem (noun) LIN ee
More informationMath 115: Review for Chapter 2
Math 5: Review for Chapter Can ou determine algebraicall whether an equation is smmetric with respect to the - ais, the -ais, or the origin?. Algebraicall determine whether each equation below is smmetric
More informationFunctions. Introduction
Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of
More informationMATH 115: Review for Chapter 6
MATH 115: Review for Chapter 6 In order to prepare for our test on Chapter 6, ou need to understand and be able to work problems involving the following topics: I SYSTEMS OF LINEAR EQUATIONS CONTAINING
More information4.1 Circles. Explore Deriving the Standard-Form Equation
COMMON CORE r Locker LESSON Circles.1 Name Class Date.1 Circles Common Core Math Standards The student is epected to: COMMON CORE A-CED.A.3 Represent constraints b equations or inequalities,... and interpret
More information13.2 Exponential Decay Functions
6 6 - - Locker LESSON. Eponential Deca Functions Common Core Math Standards The student is epected to: F.BF. Identif the effect on the graph of replacing f() b f() + k, kf(), f(k), and f( + k) for specific
More informationFair Game Review. Chapter = How many calculators are sold when the profit is $425? Solve the equation. Check your solution.
Name Date Chapter 4 Fair Game Review Solve the equation. Check our solution.. 8 3 = 3 2. 4a + a = 2 3. 9 = 4( 3k 4) 7k 4. ( m) 2 5 6 2 = 8 5. 5 t + 8t = 3 6. 3 5h 2 h + 4 = 0 2 7. The profit P (in dollars)
More informationLearning Objective: We will construct and interpret scatterplots (G8M6L4)
Learning Objective: We will construct and interpret scatterplots (G8ML) Concept Development: A Scatter Plot is a graph of numerical data on two variables. Eamples: -- The number of hours ou stud for a
More information13.3 Exponential Decay Functions
6 6 - - Locker LESSON. Eponential Deca Functions Teas Math Standards The student is epected to: A.5.B Formulate eponential and logarithmic equations that model real-world situations, including eponential
More informationLESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II
LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will
More informationFinding Complex Solutions of Quadratic Equations
COMMON CORE y - 0 y - - 0 - Locker LESSON 3.3 Finding Comple Solutions of Quadratic Equations Name Class Date 3.3 Finding Comple Solutions of Quadratic Equations Essential Question: How can you find the
More information3.1 Solving Quadratic Equations by Taking Square Roots
COMMON CORE -8-16 1 1 10 8 6 0 y Locker LESSON.1 Solving Quadratic Equations by Taking Square Roots Name Class Date.1 Solving Quadratic Equations by Taking Square Roots Essential Question: What is an imaginary
More informationName Class Date. Solving Special Systems by Graphing. Does this linear system have a solution? Use the graph to explain.
Name Class Date 5 Solving Special Sstems Going Deeper Essential question: How do ou solve sstems with no or infinitel man solutions? 1 A-REI.3.6 EXAMPLE Solving Special Sstems b Graphing Use the graph
More informationEssential Question How can you solve a nonlinear system of equations?
.5 Solving Nonlinear Sstems Essential Question Essential Question How can ou solve a nonlinear sstem of equations? Solving Nonlinear Sstems of Equations Work with a partner. Match each sstem with its graph.
More informationEssential Question: How can you solve equations involving variable exponents? Explore 1 Solving Exponential Equations Graphically
6 7 6 y 7 8 0 y 7 8 0 Locker LESSON 1 1 Using Graphs and Properties to Solve Equations with Eponents Common Core Math Standards The student is epected to: A-CED1 Create equations and inequalities in one
More information1Write and graph. 2Solve problems. Now. Then. Why? New Vocabulary
Direct Variation Then You found rates of change of linear functions. (Lesson -) Now Write and graph direct variation equations. Solve problems involving direct variation. Wh? Bianca is saving her mone
More informationLESSON 4.3 GRAPHING INEQUALITIES
LESSON.3 GRAPHING INEQUALITIES LESSON.3 GRAPHING INEQUALITIES 9 OVERVIEW Here s what ou ll learn in this lesson: Linear Inequalities a. Ordered pairs as solutions of linear inequalities b. Graphing linear
More information5.3 Interpreting Rate of Change and Slope
Name Class Date 5.3 Interpreting Rate of Change and Slope Essential question: How can ou relate rate of change and slope in linear relationships? Resource Locker Eplore Determining Rates of Change For
More informationComparing Linear, Exponential, and Quadratic Functions
. Comparing Linear, Eponential, and Quadratic Functions How can ou compare the growth rates of linear, eponential, and quadratic functions? ACTIVITY: Comparing Speeds Work with a partner. Three cars start
More informationSECTION 8-7 De Moivre s Theorem. De Moivre s Theorem, n a Natural Number nth-roots of z
8-7 De Moivre s Theorem 635 B eactl; compute the modulus and argument for part C to two decimal places. 9. (A) 3 i (B) 1 i (C) 5 6i 10. (A) 1 i 3 (B) 3i (C) 7 4i 11. (A) i 3 (B) 3 i (C) 8 5i 12. (A) 3
More informationExploring Operations Involving Complex Numbers. (3 + 4x) (2 x) = 6 + ( 3x) + +
Name Class Date 11.2 Complex Numbers Essential Question: What is a complex number, and how can you add, subtract, and multiply complex numbers? Explore Exploring Operations Involving Complex Numbers In
More informationFair Game Review. Chapter 9. Find the square root(s) ± Find the side length of the square. 7. Simplify Simplify 63.
Name Date Chapter 9 Find the square root(s). Fair Game Review... 9. ±. Find the side length of the square.. s. s s Area = 9 ft s Area = 0. m 7. Simplif 0. 8. Simplif. 9. Simplif 08. 0. Simplif 88. Copright
More informationGraph and Write Equations of Circles
TEKS 9.3 a.5, A.5.B Graph and Write Equations of Circles Before You graphed and wrote equations of parabolas. Now You will graph and write equations of circles. Wh? So ou can model transmission ranges,
More information2.1 Rates of Change and Limits AP Calculus
.1 Rates of Change and Limits AP Calculus.1 RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
More information5.7 Start Thinking. 5.7 Warm Up. 5.7 Cumulative Review Warm Up
.7 Start Thinking Graph the linear inequalities < + and > 9 on the same coordinate plane. What does the area shaded for both inequalities represent? What does the area shaded for just one of the inequalities
More informationGetting ready for Exam 1 - review
Getting read for Eam - review For Eam, stud ALL the homework, including supplements and in class activities from sections..5 and.,.. Good Review Problems from our book: Pages 6-9: 0 all, 7 7 all (don t
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 informationAdding and Subtracting Rational Expressions
COMMON CORE Locker LESSON 9.1 Adding and Subtracting Rational Epressions Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions?
More information1.2 Characteristics of Function Graphs
Name Class Date 1.2 Characteristics of Function Graphs Essential Question: What are some of the attributes of a function, and how are the related to the function s graph? Resource Locker Eplore Identifing
More informationKeira Godwin. Time Allotment: 13 days. Unit Objectives: Upon completion of this unit, students will be able to:
Keira Godwin Time Allotment: 3 das Unit Objectives: Upon completion of this unit, students will be able to: o Simplif comple rational fractions. o Solve comple rational fractional equations. o Solve quadratic
More informationName Date. Work with a partner. Each graph shown is a transformation of the parent function
3. Transformations of Eponential and Logarithmic Functions For use with Eploration 3. Essential Question How can ou transform the graphs of eponential and logarithmic functions? 1 EXPLORATION: Identifing
More informationSolving Systems of Linear Equations by Graphing. ESSENTIAL QUESTION How can you solve a system of equations by graphing? 8.9 Slope-intercept form
? LESSN. Solving Sstems of Linear Equations b Graphing ESSENTIAL QUESTIN How can ou solve a sstem of equations b graphing? Epressions, equations, and relationships.9 Identif and verif the values of and
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More information2.4 Exercises. Interval Notation Exercises 1 8: Express the following in interval notation. Solving Linear Inequalities Graphically
_chpp7-8.qd //8 4: PM Page 4 4 CHAPTER Linear Functions and Equations.4 Eercises Interval Notation Eercises 8: Epress the following in interval notation... 7 -. Ú - 4. 7. { 8}. { - 4} 7. { } 8. { 7 } Solving
More informationThe second type of conic is called an ellipse, and is defined as follows. Definition of Ellipse
72 Chapter 10 Topics in Analtic Geometr 10.3 ELLIPSES What ou should learn Write equations of ellipses in standard form and graph ellipses. Use properties of ellipses to model and solve real-life problems.
More information10.3 Solving Nonlinear Systems of Equations
60 CHAPTER 0 Conic Sections Identif whether each equation, when graphed, will be a parabola, circle, ellipse, or hperbola. Then graph each equation.. - 7 + - =. = +. = + + 6. + 9 =. 9-9 = 6. 6 - = 7. 6
More information3.7 InveRSe FUnCTIOnS
CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
More information8.2 Graphing More Complicated Rational Functions
1 Locker LESSON 8. Graphing More Complicated Rational Functions PAGE 33 Name Class Date 8. Graphing More Complicated Rational Functions Essential Question: What features of the graph of a rational function
More informationApplications. 12 The Shapes of Algebra. 1. a. Write an equation that relates the coordinates x and y for points on the circle.
Applications 1. a. Write an equation that relates the coordinates and for points on the circle. 1 8 (, ) 1 8 O 8 1 8 1 (13, 0) b. Find the missing coordinates for each of these points on the circle. If
More information7.1 Guided Practice (p. 401) 1. to find an ordered pair that satisfies each of the equations in the system. solution of the system.
CHAPTER 7 Think and Discuss (p. 9). 6,00,000 units. 0,00,000 6,00,000 4,400,000 renters Skill Review (p. 96) 9r 4r 6r. 8.. 0.d.d d 4. w 4 w 4 w 4 w 4 w. 6. 7 g g 9 g 7 g 6 g 0 7 8 40 40 40 7. 6 8. 8 9....
More informationLESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8 - 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 information3.2 Understanding Relations and Functions-NOTES
Name Class Date. Understanding Relations and Functions-NOTES Essential Question: How do ou represent relations and functions? Eplore A1.1.A decide whether relations represented verball, tabularl, graphicall,
More informationStudy Guide and Intervention
6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a
More informationCh 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations
Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4
More information9.1 Adding and Subtracting Rational Expressions
9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions? Resource Locker Eplore Identifying Ecluded Values Given a rational epression, identify
More informationLinear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?
3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear
More informationPre-AP Algebra 2 Lesson 1-1 Basics of Functions
Lesson 1-1 Basics of Functions Objectives: The students will be able to represent functions verball, numericall, smbolicall, and graphicall. The students will be able to determine if a relation is a function
More informationNAME DATE PERIOD. Study Guide and Intervention
NAME DATE PERID Stud Guide and Intervention Graph To graph a quadratic inequalit in two variables, use the following steps: 1. Graph the related quadratic equation, = a 2 + b + c. Use a dashed line for
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 informationName Class Date. Solving by Graphing and Algebraically
Name Class Date 16-4 Nonlinear Sstems Going Deeper Essential question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? To estimate the solution to a sstem
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 informationFunctions. Introduction CHAPTER OUTLINE
Functions,00 P,000 00 0 970 97 980 98 990 99 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of
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 information