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

Struggling with textbook problems? Solution Manual for College Algebra: Concepts Through Functions, 4th Edition offers a clear breakdown of every exercise for easy understanding.

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    M INI L ECTURE N OTES P AMELA T RIM Southwest Tennessee Community College C OLLEGE A LGEBRA : C ONCEPTS T HROUGH F UNCTIONS F OURTH EDITION Michael Sullivan Chicago State University Michael Sullivan, III Joliet Junior College

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    Table of Contents Mini Lecture Notes F.1 The Distance and Midpoint Formulas F.2 Graphs of Equations in Two Variables; Intercepts; Symmetry F.3 Lines F.4 Circles 1.1 Functions 1.2 The Graph of a Function 1.3 Properties of Functions 1.4 Library of Functions: Piecewise-defined Functions 1.5 Graphing Techniques: Transformations 1.6 Mathematical Models: Building Functions 1.7 Building Mathematical Models Using Variation 2.1 Properties of Linear Functions and Linear Models 2.2 Building Linear Models from Data 2.3 Quadratic Functions and Their Zeros 2.4 Properties of Quadratic Functions 2.5 Inequalities Involving Quadratic Functions 2.6 Building Quadratic Models from Verbal Descriptions and from Data 2.7 Complex Zeros of a Quadratic Function 2.8 Equations and Inequalities Involving the Absolute Value Function 3.1 Polynomial Functions and Models 3.2 The Real Zeros of a Polynomial Function 3.3 Complex Zeros; Fundamental Theorem of Algebra 3.4 Properties of Rational Functions 3.5 The Graph of a Rational Function 3.6 Polynomial and Rational Inequalities 4.1 Composite Functions 4.2 One-to-One Functions; Inverse Functions 4.3 Exponential Functions 4.4 Logarithmic Functions 4.5 Properties of Logarithms 4.6 Logarithmic and Exponential Equations 4.7 Financial Models 4.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 4.9 Building Exponential, Logarithmic, and Logistic Models from Data

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    5.1 No mini lecture notes 5.2 The Parabola 5.3 The Ellipse 5.4 The Hyperbola 6.1 Systems of Linear Equations: Substitution and Elimination 6.2 Systems of Equations: Matrices 6.3 Systems of Equations: Determinants 6.4 Matrix Algebra 6.5 Partial Fraction Decomposition 6.6 Systems of Nonlinear Equations 6.7 Systems of Inequalities 6.8 Linear Programming 7.1 Sequences 7.2 Arithmetic Sequences 7.3 Geometric Sequences; Geometric Series 7.4 Mathematical Induction 7.5 The Binomial Theorem 8.1 Counting 8.2 Permutations and Combinations 8.3 Probability

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    Mini-Lecture F.1 The Distance and Midpoint Formulas Learning Objectives: 1. Use the Distance Formula 2. Use the Midpoint Formula Examples: 1. Find the distance between the points   3, 7  and   4,10 . 2. Determine whether the triangle formed by points   2, 2 A   ,   2, 1 B   , and   5, 4 C  form a right triangle. 3. Find the midpoint of the line segment joining the points   1 6, 3 P   and   2 4, 2 P  . 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 applications later in the course.  Most students don’t have too much trouble with the distance formula, but they will make careless arithmetic mistakes such as using addition instead of subtraction.  Go over simplifying radicals and adding the values under the radical prior to taking the square root.  The midpoint formula is also fairly easy for students, but they will sometimes have difficulty if the coordinates include fractions. Answers: 1. 58 2. No 2 25 AB  , 2 34 BC  , 2 53 AC  3. 1 5, 2       

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    Mini-Lecture F.2 Graphs of Equations in Two Variables; Intercepts; Symmetry Learning Objectives: 1. Graph Equations by Plotting Points 2. Find Intercepts from a Graph 3. Find Intercepts from an Equation 4. Test an Equation for Symmetry 5. Know How to Graph Key Equations Examples: 1. Determine whether the points   0,3 ,   2, 0  ,   2, 7 are on the graph of the equation 3 2 3 y x x    . 2. Find the intercepts and graph 2 1 y x   . 3. List the intercepts and test for symmetry for each equation. a) 2 4 0 y x    b) 2 4 x y x   Teaching Notes:  When graphing by plotting points, be sure to emphasize that the y - coordinate is determined by the value of x . This will help establish the function concept later.  Emphasize how to find intercepts algebraically by setting 0 x  to find the y - intercept(s) and then 0 y  to find the x - intercept(s).  Symmetry can be seen on a graph and identified, but students will often have trouble testing for symmetry algebraically. They may make a lot of sign errors so that needs to be reinforced.  Emphasize graphing of the key functions. It is important that they know the basic shapes of these graphs when this topic is revisited later in the course. Answers: 1. Yes, No, Yes 2. 1 , 0 2       ,   0, 1 

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

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    Mini-Lecture F.3 Lines Learning Objectives: 1. Calculate and Interpret the Slope of a Line 2. Graph Lines Given a Point and a Slope 3. Find the Equation of a Vertical Line 4. Use the Point-Slope Form of a Line; Identify Horizontal Lines 5. Write the Equation of a Line in Slope-Intercept Form 6. Identify the Slope and y-Intercept of a Line from its Equation 7. Find the Equation of a Line Given Two Points 8. Graph Lines Written in General Form Using Intercepts 9. Find Equations of Parallel Lines 10. 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, 4 with slope 2 3 m   . 3. Write an equation of the line satisfying the given conditions: a) Slope 3 4  , containing the point   2, 4  b) Containing the points   4, 2 and   3, 4  c) x -intercept 3  , y -intercept 2   d) Vertical line containing   5, 1  e) Parallel to the line 3 4 5 x y   and containing the point   3, 6  f) Perpendicular to the line 2 3 7 x y   and containing the point   1, 1   4. Find the slope and y -intercept of the line 4 6 3 x y    . 5. Find the intercepts and graph the line 2 4 x y    . Teaching Notes:  Emphasizing that the definition of the slope is “the change in y over the change in x ” will help students remember the formula for the slope.  Showing the students how to derive the point-slope formula from the slope formula will help students remember the point-slope form.  Emphasize care with signs when using both the slope formula and the point-slope formula.

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     When using the slope and y -intercept to graph a linear function, encourage students to count down and to the right when the slope is negative. Consistency will 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. This will help students see the role that the slope plays in graphing lines.  Emphasize the relationship between the slopes of parallel lines and perpendicular lines. Answers : 1. 7 4 3 0 ( 5) 5 m      2. 3. a) 3 11 4 2 y x   b) 6 22 y x   c) 2 2 3 y x   d) 5 x  e) 3 33 4 4 y x   f) 3 1 2 2 y x   4. 2 3 m  , 1 2 b  5. x -intercept 2   , y -intercept 4 

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    Mini-Lecture F.4 Circles Learning Objectives: 1. Write the Standard Form of the Equation of a Circle 2. Graph a Circle 3. Work with the General Form of the Equation of a Circle Examples: 1. Write the standard form and general form of the equation of each circle with radius r and center   , h k . Graph each circle. a)     3; , 2, 3 r h k    b)     2 ; , 0, 0 3 r h k   2. Find the center   , h k and radius r of each circle. a)     2 2 2 2 2 3 8 x y     b) 2 2 6 2 4 0 x y x y      3. Find the general form of the equation of each circle. a) Center   2, 3  and containing the point   0, 4 b) Endpoints of a diameter at   6,10 and   4, 4   Teaching Notes:  Emphasize taking the opposite signs of those in the parentheses when finding the center of a circle.  Some students will not recognize that a circle of the form 2 2 2 x y r   has the origin as its center. Show them this form written as     2 2 2 0 0 x y r     .  Many students will forget to add the same numbers to both sides of the equation when completing the squares in the equation of a circle.  Emphasize the difference between the standard form of the equation of a circle and the general form.  Have students review the method of completing the square.

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    Answers: 1. a)     2 2 2 2 2 3 9; 4 6 4 0 x y x y x y          b) 2 2 2 2 4 4 ; 0 9 9 x y x y      a) b) 2. a)   2, 3 ; 2 c r    b)   3, 1 ; 6 c r    3. a)     2 2 2 3 53 x y     b)     2 2 1 3 74 x y    

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    Mini-Lecture 1.1 Functions Learning Objectives: 1. Determine Whether a Relation Represents a Function 2. Find the Value of a Function 3. Find the Difference Quotient of a Function 4. Find the Domain of a Function Defined by an Equation 5. Form the Sum, Difference, Product, and Quotient of Two Functions Examples: 1. Determine whether the equation defines y as function of x . a) 2 2 y x x   b) 2 3 4 y x   c) 5 7 10 x y   d) 2 3 y x   2. For   2 2 3 f x x x     find a) (0) f b) ( 1) f  c) (3) f d) ( 1) f a  e) ( ) f x h  . 3. Find the domain of each function. a)   2 3 f x x   b)   2 2 f x x  c)   5 4 f x x   d)   2 2 1 x f x x   4. For   2 3 f x x   and   2 2 g x x  find a)    f g x  b)    f g x  c)    2 f g  d)   3 f g       5. Find and simplify the difference quotient ( ) ( ) , f x h f x h   0 h  for 2 ( ) 2 3 f x x x     .

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    Teaching Notes:  Many students do not relate to f in ( ) f x as the name of the function. To make the point, give two functions names like Fred(x) and Ginger(x) .  When determining if an equation is a function, first substitute several values for x . This reinforces the concepts of “one input” leads to “one output”. Be sure to work at least one that is not a function. Then, work the examples in the book that solve for y .  Emphasize “Finding the Domain of a Function Defined by an Equation” in the book.  Remind students to review interval notation before discussing the domain.  Many students have trouble simplifying the difference quotient. Remind them that   2 2 2 x h x h    . Also, finding ( ) f x h  first and then substituting it and ( ) f x into the difference quotient seems to keep students from getting so overwhelmed by the algebra.  Emphasize that y is “what” (the value of the function) and x is “where” (the location of the value). This will be helpful later in discussing relative maxima and minima. Answers: 1. a) yes b) no c) yes d) yes 2. a) 3  b) 6  c) 6  d) 2 2 a   e) 2 2 2 2 2 3 x xh h x h       3. a)   ,   b)     ,0 0,    c)   4,   d)   ,   4. a) 2 2 2 3 x x   b) 2 2 2 3 x x    c) 8 d) 1 6 5. 4 2 1 x h   

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

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    Teaching Notes:  Be sure to draw a relation that is not a function. Have the points along a vertical line through the function labeled so that students can see the shared first coordinates.  Some students fail to see the connection between ( ) f x and y . When they are asked to find the value of x on the graph given ( ) f x , remind them that ( ) f x is the y -coordinate.  Remind students when they are finding the domain of a function from its graph that 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 it properly on the graph of the function. Students have the same trouble with the range.  Remind students that ( ) 0 f x  means the y -coordinates are positive and are located above the x -axis. Answers: 1) a) No b) Function, Domain=   2,  , Range =   ,   , x -intercept 3  , No symmetry 2) a) yes b)   1; 2,1  c) 6 x  (d)     , 2 2,    e) zero 0  ;   0, 0 f) y -intercept 0 

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    Mini-Lecture 1.3 Properties of Functions Learning Objectives: 1. Determine Even and Odd Functions from a Graph 2. Identify Even and Odd Functions from the Equation 3. Use a Graph to Determine Where a Function is Increasing, Decreasing, or Constant 4. Use a Graph to Locate Local Maxima and Local Minima 5. Use a Graph to Locate the Absolute Maximum and the Absolute Minimum 6. Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function is Increasing or Decreasing 7. Find the Average Rate of Change of a Function Examples: 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 function 3 ( ) 2 1 f x x x    is odd, even, or neither. 3. Find the average rate of change of 3 2 ( ) 3 f x x x    from 1 x   to 4 x  . Teaching Notes:  Reinforce that the values of the relative maxima and minima are y -values.  When finding even and odd functions, remind students that a negative number raised to an even power is positive and a negative number raised to an odd power is negative.  When determining where a function is increasing or decreasing by looking at its graph, emphasize that you should observe what is happening to the y -values as you move from the left to the right.  Show how to use MAXIMUM and MINIMUM on a graphing utility to approximate local maxima and minima. Using TRACE can also help students visualize the increasing or decreasing nature of the function.

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