Solution Manual for College Algebra: Concepts Through Functions, 4th Edition

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Solution Manual for College Algebra: Concepts Through Functions, 4th Edition - Page 1 preview image

MINILECTURENOTESPAMELATRIMSouthwest Tennessee Community CollegeCOLLEGEALGEBRA:CONCEPTSTHROUGHFUNCTIONSFOURTH EDITIONMichael SullivanChicago State UniversityMichael Sullivan, IIIJoliet Junior College

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Solution Manual for College Algebra: Concepts Through Functions, 4th Edition - Page 2 preview image

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Table of ContentsMini Lecture NotesF.1The Distance and Midpoint FormulasF.2Graphs of Equations in Two Variables; Intercepts; SymmetryF.3LinesF.4Circles1.1Functions1.2The Graph of a Function1.3Properties of Functions1.4Library of Functions: Piecewise-defined Functions1.5Graphing Techniques: Transformations1.6Mathematical Models: Building Functions1.7Building Mathematical Models Using Variation2.1Properties of Linear Functions and Linear Models2.2Building Linear Models from Data2.3Quadratic Functions and Their Zeros2.4Properties of Quadratic Functions2.5Inequalities Involving Quadratic Functions2.6Building Quadratic Models from Verbal Descriptions and from Data2.7Complex Zeros of a Quadratic Function2.8Equations and Inequalities Involving the Absolute Value Function3.1Polynomial Functions and Models3.2The Real Zeros of a Polynomial Function3.3Complex Zeros; Fundamental Theorem of Algebra3.4Properties of Rational Functions3.5The Graph of a Rational Function3.6Polynomial and Rational Inequalities4.1Composite Functions4.2One-to-One Functions; Inverse Functions4.3Exponential Functions4.4Logarithmic Functions4.5Properties of Logarithms4.6Logarithmic and Exponential Equations4.7Financial Models4.8Exponential Growth and Decay Models; Newton’s Law; Logistic Growth andDecay Models4.9Building Exponential, Logarithmic, and Logistic Models from Data

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5.1No mini lecture notes5.2The Parabola5.3The Ellipse5.4The Hyperbola6.1Systems of Linear Equations: Substitution and Elimination6.2Systems of Equations: Matrices6.3Systems of Equations: Determinants6.4Matrix Algebra6.5Partial Fraction Decomposition6.6Systems of Nonlinear Equations6.7Systems of Inequalities6.8Linear Programming7.1Sequences7.2Arithmetic Sequences7.3Geometric Sequences; Geometric Series7.4Mathematical Induction7.5The Binomial Theorem8.1Counting8.2Permutations and Combinations8.3Probability

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Mini-Lecture F.1The Distance and Midpoint FormulasLearning Objectives:1. Use the Distance Formula2. Use the Midpoint FormulaExamples:1. Find the distance between the points3, 7and4,10.2. Determine whether the triangle formed by points2, 2A,2,1B,and5, 4Cform a right triangle.3. Find the midpoint of the line segment joining the points16,3Pand24, 2P.Teaching Notes:Go over the terms used in introducing the rectangular coordinate system.Tell the students that the distance formula will be used in several applicationslater in the course.Most students don’t have too much trouble with the distance formula, but theywill make careless arithmetic mistakes such as using addition instead ofsubtraction.Go over simplifying radicals and adding the values under the radical prior totaking the square root.The midpoint formula is also fairly easy for students, but they will sometimeshave difficulty if the coordinates include fractions.Answers:1.582. No225AB,234BC,253AC3.15,2

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Mini-Lecture F.2Graphs of Equations in Two Variables; Intercepts; SymmetryLearning Objectives:1. Graph Equations by Plotting Points2. Find Intercepts from a Graph3. Find Intercepts from an Equation4. Test an Equation for Symmetry5. Know How to Graph Key EquationsExamples:1. Determine whether the points0,3,2, 0,2, 7are on the graph of theequation323yxx.2. Find the intercepts and graph21yx.3. List the intercepts and test for symmetry for each equation.a)240yxb)24xyxTeaching Notes:When graphing by plotting points, be sure to emphasize that they-coordinate isdetermined by the value ofx. This will help establish the function concept later.Emphasize how to find intercepts algebraically by setting0xto find they-intercept(s) and then0yto find thex-intercept(s).Symmetry can be seen on a graph and identified, but students will often havetrouble testing for symmetry algebraically. They may make a lot of sign errors sothat needs to be reinforced.Emphasize graphing of the key functions. It is important that they know the basicshapes of these graphs when this topic is revisited later in the course.Answers:1. Yes, No, Yes2.1 , 02,0,1

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3.a)0, 2,0,2,4, 0, Symmetric with respect to the x-axisb)0, 0,Symmetric with respect to the origin

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Mini-Lecture F.3LinesLearning Objectives:1. Calculate and Interpret the Slope of a Line2. Graph Lines Given a Point and a Slope3. Find the Equation of a Vertical Line4. Use the Point-Slope Form of a Line; Identify Horizontal Lines5. Write the Equation of a Line in Slope-Intercept Form6. Identify the Slope and y-Intercept of a Line from its Equation7. Find the Equation of a Line Given Two Points8. Graph Lines Written in General Form Using Intercepts9. Find Equations of Parallel Lines10. Find Equations of Perpendicular Lines.Examples:1. Determine the slope of the line containing the points5, 4,0, 7.2. Graph the line containing the point2, 4with slope23m .3. Write an equation of the line satisfying the given conditions:a) Slope34, containing the point2, 4b) Containing the points4, 2and3,4c)x-intercept3,y-intercept2 d) Vertical line containing5,1e) Parallel to the line345xyand containing the point3,6f) Perpendicular to the line237xyand containing the point1,14. Find the slope andy-intercept of the line463xy .5. Find the intercepts and graph the line24xy.Teaching Notes:Emphasizing that the definition of the slope is “the change inyover the changeinx” will help students remember the formula for the slope.Showing the students how to derive the point-slope formula from the slopeformula will help students remember the point-slope form.Emphasize care with signs when using both the slope formula and the point-slopeformula.

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When using the slope andy-intercept to graph a linear function, encouragestudents to count down and to the right when the slope is negative. Consistencywill help students successfully graph all lines.Emphasize the slopes and equations of horizontal and vertical lines.Use the graphing utility and graph several lines on the same square screen. Thiswill help students see the role that the slope plays in graphing lines.Emphasize the relationship between the slopes of parallel lines and perpendicularlines.Answers:1.7430( 5)5m 2.3.a)31142yxb)622yxc)223yxd)5xe)33344yxf)3122yx4.23m,12b5.x-intercept2 ,y-intercept4

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Mini-Lecture F.4CirclesLearning Objectives:1. Write the Standard Form of the Equation of a Circle2. Graph a Circle3. Work with the General Form of the Equation of a CircleExamples:1. Write the standard form and general form of the equation of each circle with radiusrand center,h k. Graph each circle.a)3;,2, 3rh kb)2 ;,0, 03rh k2. Find the center,h kand radiusrof each circle.a)2222238xyb)226240xyxy3. Find the general form of the equation of each circle.a) Center2,3and containing the point0, 4b) Endpoints of a diameter at6,10and4,4Teaching Notes:Emphasize taking the opposite signs of those in the parentheses when finding thecenter of a circle.Some students will not recognize that a circle of the form222xyrhas theorigin as its center. Show them this form written as22200xyr.Many students will forget to add the same numbers to both sides of the equationwhen completing the squares in the equation of a circle.Emphasize the difference between the standard form of the equation of a circleand the general form.Have students review the method of completing the square.

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Answers:1. a)2222239;4640xyxyxyb)222244;099xyxya)b)2. a)2,3 ;2crb)3,1 ;6cr3. a)222353xyb)221374xy

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Mini-Lecture 1.1FunctionsLearning Objectives:1. Determine Whether a Relation Represents a Function2. Find the Value of a Function3. Find the Difference Quotient of a Function4. Find the Domain of a Function Defined by an Equation5. Form the Sum, Difference, Product, and Quotient of Two FunctionsExamples:1. Determine whether the equation definesyas function ofx.a)22yxxb)234yxc)5710xyd)23yx2. For223fxxx finda)(0)fb)( 1)fc)(3)fd)(1)fae)()fxh.3. Find the domain of each function.a)23fxxb)22fxxc)54fxxd)221xfxx4. For23fxxand22gxxfinda)fgxb)fgxc)2fgd) 3fg5. Find and simplify the difference quotient()( ) ,fxhfxh0hfor2( )23fxxx .

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Teaching Notes:Many students do not relate tofin( )fxas the name of the function. To makethe point, give two functions names likeFred(x)andGinger(x).When determining if an equation is a function, first substitute several values forx.This reinforces the concepts of “one input” leads to “one output”. Be sure towork at least one that is not a function. Then, work the examples in the bookthat solve fory.Emphasize “Finding the Domain of a Function Defined by an Equation” in thebook.Remind students to review interval notation before discussing the domain.Many students have trouble simplifying the difference quotient. Remind themthat222xhxh. Also, finding()fxhfirst and then substituting it and( )fxinto the difference quotient seems to keep students from getting sooverwhelmed by the algebra.Emphasize thatyis “what” (the value of the function) andxis “where” (thelocation of the value). This will be helpful later in discussing relative maximaand minima.Answers:1. a) yesb) noc) yesd) yes2. a)3b)6c)6d)22ae)222223xxhhxh3.a), b),00,c)4,d), 4. a)2223xxb)2223xxc)8d)165.421xh

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Mini-Lecture 1.2The Graph of a FunctionLearning Objectives:1. Identify the Graph of a Function2. Obtain Information from or about the Graph of a Function (p. 58)Examples:a)b)2. For22xfxxanswer the following questions.a) Is the point3,6on the graph off?b) For2x what isfx? What point is on the graph off?c) If3fx, what isx?d) What is the domain off?e) List any zeros off. What points are on the graph off?f) List the y-intercept, if there is one, of the graph off.

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Teaching Notes:Be sure to draw a relation that is not a function. Have the points along a verticalline through the function labeled so that students can see the shared firstcoordinates.Some students fail to see the connection between( )fxandy.When they areasked to find the value ofxon the graph given( )fx, remind them that( )fxisthey-coordinate.Remind students when they are finding the domain of a function from its graphthat they should be looking to the left and right (or horizontally) on the graph.Even if they know the definition of the domain, sometimes they fail to visualize itproperly on the graph of the function. Students have the same trouble with therange.Remind students that( )0fxmeans they-coordinates are positive and arelocated above thex-axis.Answers:1) a) Nob) Function, Domain=2,, Range =, ,x-intercept3,No symmetry2) a) yesb)1;2,1c)6x(d), 22,e) zero0;0, 0f)y-intercept0

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Mini-Lecture 1.3Properties of FunctionsLearning Objectives:1. Determine Even and Odd Functions from a Graph2. Identify Even and Odd Functions from the Equation3. Use a Graph to Determine Where a Function is Increasing, Decreasing, or Constant4. Use a Graph to Locate Local Maxima and Local Minima5. Use a Graph to Locate the Absolute Maximum and the Absolute Minimum6. Use a Graphing Utility to Approximate Local Maxima and Local Minima and toDetermine Where a Function is Increasing or Decreasing7. Find the Average Rate of Change of a FunctionExamples:1.For the graph below(a) State the intervals where the function is increasing, decreasing, or constant.(b) State the domain and range.(c) State whether the graph is odd, even, or neither.(d) Locate the maxima and minima.2. Determine algebraically whether the function3( )21fxxxis odd, even, orneither.3. Find the average rate of change of32( )3fxxx from1x to4x.Teaching Notes:Reinforce that the values of the relative maxima and minima arey-values.When finding even and odd functions, remind students that a negative numberraised to an even power is positive and a negative number raised to an odd poweris negative.When determining where a function is increasing or decreasing by looking atits graph, emphasize that you should observe what is happening to they-valuesas you move from the left to the right.Show how to use MAXIMUM and MINIMUM on a graphing utility toapproximate local maxima and minima. Using TRACE can also help studentsvisualize the increasing or decreasing nature of the function.

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