Shake, Rattle and Roll


 Linda Cummings
 2 years ago
 Views:
Transcription
1 00 College Board. All rights reserved. 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Visualization, Interactive Word Wall Roller coasters are scar and fun to ride. Wooden roller coasters shake and rattle as part of the thrill of the ride. Below is the graph of the heights reached b the cars of the wooden roller coaster, Thunderball, over its first 0 feet of track. The graph displas a function because each input value has one and onl one output value. You can see this visuall using the vertical line test. Stud this graph to determine the domain and range. Height Above Ground (feet) _SB_A__SE.indd 77 Shake, Rattle and Roll Thunderball Roller Coaster Graph Distance Along the Track (feet) The domain gives all values of the independent variable: distance along the track in feet. These values are graphed along the horizontal or ais. The domain can be written in set notation as: {all real values of : 0 0} Read this notation as: the set of all real values of, between 0 and 0, inclusive. The range gives the values of the dependent variable: height above the ground in feet. The values are graphed on the vertical or ais. The range can be written in set notation as: {all real values of : 0 0 Read this notation as: the set of all real values of, between 0 and 0, inclusive. The graph above shows data that are continuous. The points in the graph are connected, indicating that domain and range are sets if real numbers with no breaks in between. A graph of discrete data consists of individual points that are not connected b a line or curve. MINILESSON: Reading Set Notation M Notes ACADEMIC VOCABULARY The independent variable is the input of a function. The dependent variable is the output of a function. Its value depends on the value of the independent variable. ACTIVITY. MATH TERMS An The ordered vertical pair line shows test is the a visual relationship check to see between if a graph two appears values, to be a function. written in a For specific a function, order ever using vertical parentheses line drawn in n the otation coordinate and a comma plane will separating intersect the the graph two values. in at most one point. This is equivalent to having each domain element associated with one and onl one element of the range. Unit Linear Functions 77 /7/09 0:8: AM Give practice in reading set notation, where { } is read the set of all values and : or is read such that and then an restrictions on the set are given. a. {real : = } [ the set of all real values of, such that is less than or equal to ] b. {odd numbers z : < z < } [ the set of all odd numbers z, such that z is between and ] c. {integer < 7} [ the set of all integer values of, such that is less than 7 ] d. {even numbers < or > } [ the set of all even numbers, such that is less than and greater than ] ACTIVITY. Domain and Range of Continuous Functions Activit Focus Domain and range of continuous functions Set notation Vertical line test Additional Materials Graphing calculators (optional) Whiteboards or chart paper Chunking the Activit Roller Coaster Eample Minilesson on reading set notation #a b Minilesson on vertical line test Eample Tr These Technolog Time EXAMPLE Shared Reading, Marking the Tet, Visualization, Interactive Word Wall The roller coaster eample introduces the vertical line test. Demonstrating was to visualize a vertical line moving from left to right along the ais can assist students as the begin to appl this test for themselves. Some methods that ou can demonstrate are: pass a ruler along the ais perpendicular to it displa the graph on an overhead slide then pass a second acetate over the top of the first with a vertical line drawn on it Technolog such as a projected graph or an overhead graphing calculator provides opportunities to help students with their visualization of the vertical line test. Unit Linear Functions 77
2 ACTIVITY. Continued EXAMPLE () Since the roller coaster graph passes the vertical line test and is therefore a function, ou should solicit eamples of nonfunctions that can be drawn to illustrate how some relations (like a circle graph, vertical line, or an polgon) fail the vertical line test and are not functions. See minilesson below. Identifing the domain and range of this finite, continuous graph is the objective of this activit. Helping students practice reading and writing the set notation that describes the domain and range is another skill that students will need to practice. See minilesson on the previous page. Group Discussion, Debriefing In part a, ask students to come up with a realworld scenario that might be represented b this graph. If the have difficult, scaffold b offering four scenarios and asking them which one is represented. ACTIVITY. M Notes Domain: {all real values of :  < } Range: {all real values of :  } This graph represents a function because it passes the vertical line test. The data are continuous because the function includes all real values of between  and, and all real values of between  and, inclusive. The independent variable is t, the minutes since the bath began, and the dependent variable is d, the depth of the bath water, since the depth of the water depends on how man minutes the water has been running. Domain {all real values of t: 0 t } Range {all real values of d: 0 d 8} The data is continuous because the function includes all real values of t between 0 and, inclusive, and all real values of d between 0 and 8, inclusive. SUGGESTED LEARNING STRATEGIES: Group Discussion a. Use set notation to write the domain and range for the graph below. Does this graph appear to represent a function? Justif our answer. Are the data discrete or continuous? Wh? b. The graph below shows the relationship between t, the length of time of the bath (from the time water starts running through the time the tub is drained) and d, the depth of the water in the bath tub. The graph represents function d (bath water depth). What are the dependent and independent variables? Eplain. Use set notation to write the domain and range of function d. Are the data discrete or continuous and wh? Depth of bath water (in.) Bath Water Depth d t Minutes since bath began 00 College Board. All rights reserved. 78 SpringBoard Mathematics with Meaning Algebra 778_SB_A__SE.indd 78 MINILESSON: Vertical Line Test Provide additional eamples of graphs that will fail the vertical line test because the are not functions. Make coordinate graphs (without equations) of the items below on the board or overhead. Then ask students to determine, first for themselves and then b comparing answers with their neighbors, which of the following are functions. a. circle: b. vertical parabola: ( + ) = (  ) ( + ) + (  ) = c. hperbola:  = d. eponential function: = e e. horizontal parabola: f. right triangle: (, ), (, ), (, ) ( + ) = (  ) g. cubic: = ( + ) h. absolute value: = + /7/09 0:8: A7 00 College Board. All rights reserved. Items a, c, e, f, and h do not represent functions. 78 SpringBoard Mathematics with Meaning Algebra
3 ACTIVITY. ACTIVITY. Continued 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Marking the Tet, Questioning the Tet, Think Aloud EXAMPLE Give the domain and range of the function f () = (  ) graphed below. Step : Step : Stud the graph. The sketch of this graph is a portion of the function represented b the equation f () = (  ). Look for values for which the domain causes the function to be undefined. Look how the graph behaves near =. EXAMPLE Think Aloud, Question the Tet, Marking the Tet This eample uses the graph of a rational function. Students share what the notice or ask what the want to know about the graph. Reading graphs in mathematics to make meaning of the information represented is similar to reading tet in English class. Students ma not believe that can never take on the value of. Substituting into the equation f() = results in division (  ) b zero producing an undefined result for the function. This eample coupled with the visual clue of the dotted line can help students recognize undefined values in the future. TRY THESE Debriefing 00 College Board. All rights reserved. Solution: The domain and range for f () = can be written: (  ) Domain: {all real values of : } Range: {all real values of : > 0} TRY THESE a. Give the domain and range of the function f () = 8 + graphed below M Notes Notice the result when = is substituted into f (). f () = (  ) = 0 Division b zero is undefined in mathematics. Domain Range {all real numbers} {all real values of : 9} Unit Linear Functions 79 M78_SB_A__SE.indd 79 /7/09 0:8: AM Unit Linear Functions 79
4 ACTIVITY. Continued Suggested Assignment CHECK YOUR UNDERSTANDING p. 8, # UNIT PRACTICE p., # Technolog Time Create Representations, Group Discussion All of the equations can be viewed in the standard window where 0 = = 0 and 0 = = 0. However, for tracing purposes a friendl window or decimal window will serve students better. On a TI 8 this would be [9., 9.,, 0, 0, ]. Depending on how much practice students have had using the graphing calculator, ou ma need to assist students in entering the equations into the calculator. Functions ma need to be written out b kestrokes if students are not familiar with the calculator nomenclature. For eample, = + on some calculators is entered as = ( + ) and = is entered as = /. Allow students to work in partners or partners within foursomes so that the work can be collaborative and students can help each other with their questions. After students have had the opportunit to complete the chart and to compare answers with another pair, debrief the activit b asking students to share what the noticed about the results and the behavior of the graphs. As an etension, have students graph other linear, quadratic, and absolute value functions and ask how the restrictions on the domain and range of their new graphs compare with the restrictions on the previous set of functions. ACTIVITY. M Notes The domain is restricted to avoid situations where division b zero or taking the square root of a negative number would occur. TRY THESE () 80 SpringBoard Mathematics with Meaning Algebra 778_SB_A__SE.indd 80 SUGGESTED LEARNING STRATEGIES: Create Representations, Group Discussion b. Give the domain and range for the equation =. Eplain whether this equation represents a function and how ou determined this. Domain {all real numbers} Range {all real numbers} Answers ma var. Sample answer: Since the graph of this equation passes the vertical line test, this equation and its graph represent a function. Technolog Time Work with a partner to investigate the equations listed in the chart using graphing technolog. Ever equation given here is a function. Determine the domain and range for each function from the possibilities listed below the chart. Select the appropriate domain from choices and record our answer in the Domain column. Then select the appropriate range from choices a f and record the appropriate range in the Range column. When the chart is complete, compare our answers with those from another group. Function Domain Range. =  + a. =  +. = 9 . = +. = +. = Possible Domains: Possible Ranges: ) all real numbers a) all real numbers ) all real, such that  b) all real, such that 0 ) all real, such that 0 c) all real, such that  ) all real, such that d) all real, such that 0 ) all real, such that 0 e) all real, such that ) all real, such that 0 f) all real, such that Note on the Use of Graphing Calculators The purpose of this activit is to have students identif domain and range, not graph functions. If our students do not have graphing calculators or do not know how to graph the functions with a calculator, simpl displa the graphs on an overhead projector or on the board and have students identif the domain and range. c a d f b /7/09 0:8:7 A7 00 College Board. All rights reserved. 00 College Board. All rights reserved. 80 SpringBoard Mathematics with Meaning Algebra
5 00 College Board. All rights reserved. M78_SB_A__SE.indd 8 00 College Board. All rights reserved. CHECK YOUR UNDERSTANDING ACTIVITY. Write our answers on notebook paper. Show our work.. The graph below shows five points that our work. make up the function h. Give the domain. Give the domain and range for the and the range for the function h. function graphed below. Eplain wh this graph represents a function A student calculates how far awa a lightning strike is, based on when the thunder is heard. The student makes the table below using km/sec as the average speed of sound under rain conditions. If the thunder is onl heard when the lightning strike is within km of the listener, what are the domain and range for this model? Is this relation a function? How do ou know? Time until thunder is heard (sec) Distance from lightning strike (km). Give the domain and range of the function f () =. Yards Left to Walk. Jeff walks at an average rate of ards per minute. Mark s house is located 000 ards from Jeff s house. The graph below shows how far Jeff still needs to walk to reach Mark s house. Give the domain and range for this model. Is this model a function? Eplain Jeff Walks to Mark s House 8 0 Minutes Walking Unit Linear Functions 8 /7/09 0:8:0 AM ACTIVITY. Continued Suggested Assignment CHECK YOUR UNDERSTANDING p. 8, # 7 UNIT PRACTICE p., #,. domain: {all real values of : 0 9}; range: {all real values of : 0 }; This graph represents a function because it passes the vertical line test.. Since the time it takes to hear the thunder is dependent upon how far awa the lightning strikes, the domain will address the distances [d] and the range will address the time [t] in seconds since the lightning struck. Domain: {all real values of d: 0 d kilometers]. Range:{all real values of t : 0 t seconds} The pattern ehibits a constant change in time over distance, and if graphed, the pattern would look like a line. This relationship has onl one dependent value for ever independent value, so it is a function. If graphed, it would pass the vertical line test.. domain: {all real numbers} range: {all real numbers}. domain: {, ,,, } range: {, ,, }. domain {all real values of : 0 minutes}; range: {all real values of : ards}. This model is a function because no domain value is paired with more than one range value and its graph passes the vertical line test. Unit Linear Functions 8
6 ACTIVITY. Continued. Depending on how the letters are drawn, most letters will not pass the vertical line test. The letter V is one that could possibl pass the test. Another possibilit would be the letter W. Students should include a sketch of the letters chosen on grid paper to fit each categor along with a written justification that includes information about the vertical line test. 7. Answers ma var. Sample answer: Eamine a set of ordered pairs to see if an value is repeated; use the vertical line test on a graph; eamine a mapping to see if one input value is mapped to more than one output value. Students preferences will var. ACTIVITY. CHECK YOUR UNDERSTANDING () Write. Capital our answers letters sketched on notebook in the paper. coordinate Show our work. 7. MATHEMATICAL plane ma or ma not be functions. Pick REFLECTION one letter that represents a function and two that do not. Use the vertical line test as part of the eplanation for our selections. Describe at least three different methods for determining if a relation is a function. Which method do ou prefer and wh? 00 College Board. All rights reserved. 00 College Board. All rights reserved. 8 SpringBoard Mathematics with Meaning Algebra 778_SB_A__SE.indd 8 /7/09 0:8: A 8 SpringBoard Mathematics with Meaning Algebra
Translating Points. Subtract 2 from the ycoordinates
CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationy y y 5
Sstems of Linear Inequalities SUGGESTED LEARNING STRATEGIES: Marking the Tet, Quickwrite, Create Representations. Graph each inequalit on the number lines and grids provided. M Notes ACTIVITY.7 Inequalit
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a (  h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b (  h) 3 ) + k Resource Locker Eplore 1
More information2 Analysis of Graphs of
ch.pgs116 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
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 informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
More informationSection 105 Parametric Equations
88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationTransformations of Function Graphs
   0        Locker LESSON.3 Transformations of Function Graphs Teas Math Standards The student is epected to: A..C Analze the effect on the graphs of f () = when f () is replaced b af (), f (b),
More informationRational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza
Rational Functions 7 The ozone laer protects Earth from harmful ultraviolet radiation. Each ear, this laer thins dramaticall over the poles, creating ozone holes which have stretched as far as Australia
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationAlex and Morgan were asked to graph the equation y = 2x + 1
Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and intercept wa First, I made a table. I chose some values, then plugged
More informationEXPLORE EXPLAIN 1. Representing an Interval on a Number Line INTEGRATE TECHNOLOGY. INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling
Locker LESSON 1.1 Domain, Range, and End Behavior Teas Math Standards The student is epected to: A.7.1 Write the domain and range of a function in interval notation, inequalities, and set notation. Mathematical
More informationSTRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs
6 CHAPTER Analsis of Graphs of Functions. STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Ais Combining Transformations of Graphs In the previous
More informationLesson 6: Linear Functions and their Slope
Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationSymmetry. A graph is symmetric with respect to the yaxis if, for every point (x, y) on the graph, the point (x, y) is also on the graph.
Symmetry When we graphed y =, y = 2, y =, y = 3 3, y =, and y =, we mentioned some of the features of these members of the Library of Functions, the building blocks for much of the study of algebraic functions.
More information25. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math
Chapter 2 Lesson 25 The Graph of = k 2 BIG IDEA The graph of the set of points (, ) satisfing = k 2, with k constant, is a parabola with verte at the origin and containing the point (1, k). Vocabular
More informationContents. How You May Use This Resource Guide
Contents How You Ma Use This Resource Guide ii 9 Fractional and Quadratic Equations 1 Worksheet 9.1: Similar Figures.......................... 5 Worksheet 9.: Stretch of a Spring........................
More informationQuadratic Functions. Unit
Quadratic Functions Unit 5 Unit Overview In this unit ou will stud a variet of was to solve quadratic functions and appl our learning to analzing real world problems. Academic Vocabular Add these words
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be realvalued functions in a single variable. A function is realvalued if the input and output are real numbers
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More information4 NonLinear relationships
NUMBER AND ALGEBRA NonLinear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas
More informationPolynomial and Rational Functions
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
1. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationSection 0.2 Set notation and solving inequalities
Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.
More informationMULTIPLE REPRESENTATIONS through 4.1.7
MULTIPLE REPRESENTATIONS 4.1.1 through 4.1.7 The first part of Chapter 4 ties together several was to represent the same relationship. The basis for an relationship is a consistent pattern that connects
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationx 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac
Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square
More informationSECTION 25 Combining Functions
2 Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More information17.1 Connecting Intercepts and Zeros
Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
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 informationHIRES STILL TO BE SUPPLIED
1 MRE GRAPHS AND EQUATINS HIRES STILL T BE SUPPLIED Differentshaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object
More information6.3 Parametric Equations and Motion
SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationSome Tools for Teaching Mathematical Literacy
Some Tools for Teaching Mathematical Literac Julie Learned, Universit of Michigan Januar 200. Reading Mathematical Word Problems 2. Fraer Model of Concept Development 3. Building Mathematical Vocabular
More informationHigher. Functions and Graphs. Functions and Graphs 18
hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms
More informationExponential Functions
Eponential Functions In this chapter we will study the eponential function and its inverse the logarithmic function. These important functions are indispensable in working with problems that involve population
More informationIn this unit, students study arithmetic and geometric
Unit Planning the Unit In this unit, students stud arithmetic and geometric sequences and implicit and eplicit rules for defining them. Then the analze eponential and logarithmic patterns and graphs as
More informationSection C Non Linear Graphs
1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,
More informationChapter 8. Examining Rational Functions. You see rational functions written, in general, in the form of a fraction: , where f and g are polynomials
Chapter 8 Being Respectful of Rational Functions In This Chapter Investigating domains and related vertical asymptotes Looking at limits and horizontal asymptotes Removing discontinuities of rational functions
More informationCHAPTER 9. Polynomials
CHAPTER 9 In this chapter ou epand our knowledge of families of functions to include polnomial functions. As ou investigate the equation! graph connection for polnomials, ou will learn how to search for
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
More informationHow can you tell if a relation is a function? Time Worked (h) Amount Earned ($) B: Number of Weeks Worked and Amount Earned by 10 Different Students
. Functions, Domain, and Range When mathematicians and scientists recognize a relationship between items in the world around them, the tr to model the relationship with an equation. The concept of developing
More informationQ (x 1, y 1 ) m = y 1 y 0
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
More informationGraphing and transforming functions
Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6
More informationCourse Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics
Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)
More informationQuadratic Functions. MathsStart. Topic 3
MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE
More information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
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 informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationWarmUp y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.
CST/CAHSEE: WarmUp Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationFamilies of Quadratics
Families of Quadratics Objectives To understand the effects of a, b, and c on the graphs of parabolas of the form a 2 b c To use quadratic equations and graphs to analze the motion of projectiles To distinguish
More informationFUNCTIONS. Introduction to Functions. Overview of Objectives, students should be able to:
FUNCTIONS Introduction to Functions Overview of Objectives, students should be able to: 1. Find the domain and range of a relation 2. Determine whether a relation is a function 3. Evaluate a function 4.
More informationReasoning with Equations and Inequalities
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations.
More informationChapter 3A  Rectangular Coordinate System
 Chapter A Chapter A  Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been
More information5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
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 informationChapter 3. Curve Sketching. By the end of this chapter, you will
Chapter 3 Curve Sketching How much metal would be required to make a ml soup can? What is the least amount of cardboard needed to build a bo that holds 3 cm 3 of cereal? The answers to questions like
More information1 Quadratic Functions
C h a p t e r 1 Quadratic Functions Quadratic Functions Eplain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic
More informationEssential Question How can you describe the graph of the equation Ax + By = C? Number of adult tickets. adult
3. Graphing Linear Equations in Standard Form Essential Question How can ou describe the graph of the equation A + B = C? Using a Table to Plot Points Work with a partner. You sold a total of $16 worth
More informationWhy should we learn this? One realworld connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the intercept. One realworld connection is to find the rate
More informationRepresenting Quadratic Functions Graphically
CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing Quadratic Functions Graphicall Mathematics Assessment Resource Service Universit of Nottingham
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More information8.7 Systems of NonLinear Equations and Inequalities
8.7 Sstems of NonLinear Equations and Inequalities 67 8.7 Sstems of NonLinear Equations and Inequalities In this section, we stud sstems of nonlinear equations and inequalities. Unlike the sstems of
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 informationGraph each function. Compare to the parent graph. State the domain and range. 1. SOLUTION:
 Root Functions Graph each function. Compare to the parent graph. State the domain and range...5.. 5. 6 is multiplied b a value greater than, so the graph is a vertical stretch of. Another wa to identif
More information13 Graphs, Equations and Inequalities
13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,
More informationCalculus Card Matching
Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
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 informationI think that starting
. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries
More informationCoordinate Geometry. Positive gradients: Negative gradients:
8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight
More informationCalculus and Vectors. Day Lesson Title Math Learning Goals Expectations 1, 2, 3,
Unit Applying Properties of Derivatives Calculus and Vectors Lesson Outline Day Lesson Title Math Learning Goals Epectations 1,,, The Second Derivative (Sample Lessons Included) 4 Curve Sketching from
More informationTesting Center Student Success Center x x 18 12x I. Factoring and expanding polynomials
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 informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More informationGraph Ordered Pairs on a Coordinate Plane
Graph Ordered Pairs on a Coordinate Plane Student Probe Plot the ordered pair (2, 5) on a coordinate grid. Plot the point the ordered pair (2, 5) on a coordinate grid. Note: If the student correctly plots
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationGraphing Nonlinear Systems
10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities
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 informationLet (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. SlopeIntercept Form
8 () Chapter Linear Equations in Two Variables and Their Graphs In this section SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications
More informationImagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x
OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations
More informationLINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,
LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are
More informationGraphing Linear Equations in SlopeIntercept Form
4.4. Graphing Linear Equations in SlopeIntercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope
More information