Lecture Guide Math 105 - College Algebra to accompan College Algebra, 2nd edition John Coburn Prepared b Stephen Toner Victor Valle College Last updated: 10/7/10 1
ALEKS Review Topics Simplif: 1.1 Linear Equations *Solve each of the following equations: 1.1 #12 Factor completel: Simplif: Evaluate: 1.1 #22 Rationalize the denominator: 2
*Identif the following equations as an identit, a contradiction, or a conditional equation, then state the solution. 1.1 #32 1.1 #64 The area near the joining of the Tigris and Euphrates Rivers (in modern Iraq) has often been called the Cradle of Civilization, since the area has evidence of man ancient cultures. The length of the Euphrates River eceeds that of the Tigris b 620 mi. If the have a combined length of 2880 mi., how long is each river? 1.1 #34 1.1 #66 Hi-Tech Home Improvements bus a fleet of identical trucks that cost $32,750 each. The compan is allowed to depreciate the value of their trucks for ta purposes b $5250 per ear. If compan policies dictate that older trucks must be sold once thier value declines to $6500, approimatel how man ears will the keep these trucks? 1.1 #47 Solve for P. 3
1.2 Linear Inequalities A. Interval Notation Inequalit Graph Interval Notation B. Solving Inequalities Ke idea: Whenever ou multipl or divide each side of an inequalit b a negative, switch the direction of the inequalit. Solve: Solve: 4
C. Compound Inequalities 4. Given:, Find each of the following: 5. 6. Find each union or intersection. Write our solution using interval notation. 1. 7. 2. 8. 3. 5
D. Finding Domains of Functions The domain of an epression (or function) is the set of values which satisf the epression. To find the domain, start with a "default" domain of and then take awa... ** E. Word Problems 1.2 #79 Jacques is going to college on an academic sholarship that requires him to maintain at least a 75% average in all of his classes. So far he has scored 82%, 76%, 65%, and 71% on four eams. What scores are possible on his last eam that will enable him to keep his scholarship? ** ** *Find the domain of each function: 1.2 #88 Davis Truck Rentals will rent a moving van for $15.75/da, plus $0.35 per mile. Bertz Van Rentals will rent the same van for $25/da, plus $0.30 per mile. How man miles m must the van be driven to make the cost at Bertz a better deal? 6
1.3 Absolute Value Equations & Inequalities 3. Properties: 1. 2. 3. 4. *Solve each of the following: 1. 2. 5. 7
ALEKS REVIEW PAGE: 5. Solve: 1. Factor: 2. Factor completel: 6. The sets C and D are defined as follows: 3. Simplif: Write and using interval notation. 4. Solve: 7. Rationalize the denominator and simplif: 8
1.4 Comple Numbers Evaluate: Definitions: Evaluate: Evaluate: Multipl: Add: Multipl: Comple numbers are of the form. Powers of i: Divide: 9
1.5 Solving Quadratic Equations Summar: a. If, use factoring. The epression found in the quadratic formula is called the. 4 Cases for the Discriminant: 1. If is negative, the quadratic equation has. 2. If is zero, the quadratic equation has. b. If, square root each side. 3. If is a positive perfect square, the quadratic equation has. 4. If is positive, but is not a perfect square, the quadratic equation has. c. If, it's not quadratic! *Find the value of the discriminant and state what that sas about the nature/tpe of solutions: Solve in two was. ALEKS PROBLEM: Find all values of k such that the quadratic equation has two real solutions. Write our solution as an equalit or inequalit: 10
*Solve each of the following: 4. 1. 2. Completing the Square Process: 1. Write our equation in the form: 3. 2. If there's a leading coefficient, divide both sides b it. 3. Cut the number in front of in half; write this new value on the line below. 4. Square this new value and write the product in the blank on the line above; add this product to the right side of the equation also. 5. Insert an, parentheses and eponent on the left to complete the square. 6. Add together the values on the right side. 7. Square root each side of the equation. Don't forget the plus-or-minus sign! 8. Solve for. ALEKS PROBLEM: Find the value of k needed in order to complete the square: 5. 6. 11
*Solve each b completing the square: 7. 10. The square of a number is subtracted from 60, resulting in Find all such numbers. 8. 11. The product of consecutive integers is 44 more than 14 times their sum. Find all such integers. 9. 12
12. The length of a rectangle is 1 ft. longer than twice its width. If the area is 78 ft 2, find the rectangle's dimensions. ALEKS PROBLEM: Find all real or imaginar roots of. 13. A right triangle has one leg that is 2 ft. longer than the other leg. The hpotenuse is 2 ft. less than twice the shorter leg. Find the lengths of all three sides of the triangle. ALEKS PROBLEM: When a ball is thrown, its height in feet after seconds is given b the equation, where is the initial upward velocit in feet per second. If ft/sec, find all values of for which feet. Do not round an intermediate results. Round our answer to 2 decimal places. (There ma be more than one answer.) h 13
1.6 Solving Other Tpes of Equations *Solve each of the following equations" 1.6 #40 1.6 #16 1.6 #28 1.6 #53b 14
1.6 #57a 1.6 #64 1.6 #57b 1.6 #66 15
1.6 #69 1.6 #84 Find 3 consecutive integers such that the sum of twice the largest and the fourth power of the smallest is equal to the square of the remaining integer increased b 75. 1.6 #73 16
2.1 Rectangular Coordinates A. Domain and Range 8 7 6 5 4 3 2 Algebraicall, we've alread discussed how to find the domain. (See section 1.2D of these notes.) Graphicall, we can find the domain of a graph b looking at where the graph etends, from 1-3 -2-1 1 2 3-1 -2-3 -4-5 -6-7 -8. *Find the domain and range of the relation: To determine the range, look at where the graph etends verticall. *Find the domain and range of each: 3 2 1 Graph the following relations: -2-1 1 2 1. 2. -1 4 3 2 1-3 -2-1 1 2 3-1 -2 3. 1-5 -4-3 -2-1 1 2 3 4 5-1 17
B. Distance and Midpoint Midpoint Formula: ALEKS PROBLEM: A line segment has one endpoint at and a midpoint at. Find the coordinates of the other endpoint. Distance Formula: C. Graphing Circles General Formula: has center with radius r. *Graph: *Find the distance and midpoint between and. *Find the equation of the circle having center at and radius 4. 18
*Find the equation of the circle having and as endpoints of its diameter. 2.1 #80 Find the center and radius: 2.1 #72 Find the center and radius: ALEKS PROBLEM: The point circle whose center is at equation of this circle. lies on a. Find the 19
2.2 Graphing Lines D. Horizontal and Vertical Lines A. Slope-Intercept Form *Graph E. Slope *Find the slope of each: B. Graphing b Intercepts 1. From graph: *Graph 2 1-2 -1 1 2-1 -2 2. Between points and. 3. Given the line's equation,. C. The - Chart 20
Parallel lines have the. The slopes of perpendicular lines are of each other. 2.3 Finding Equations of Lines, Given Clues Three Forms of a Line 1. Slope-Intercept 2. Point-Slope 2.2 #72 After purchasing an autographed baseball card for $85, its value increases b $1.50 per ear. a. Write this as a function. 3. Standard *Find the equation of the line having the following attributes: a. and has -intercept b. What is the card's value 7 ears after the purchase? c. How man ears will it take for the card's value to reach $100? b. and the line passes through 21
c. passes through and 2.3 #109 The number of households that are hooked up to the Internet has been increasing steadil in recent ears. In 1995, approimatel 9 million homes were online. b 2001, this figure had climbed to about 51 million. a. Use the relation (ear, homes online) with t=0 corresponding to 1995 to find an equation model for the number of homes online. d. passes through and is parallel to the line b. Discuss what the slope indicates in this contet. c. According to this model, in what ear did the first homes being to come online? d. If the rate of change stas constant, how man households will be on the internet in 2006? e. passes through and is perpendicular to the line e. How man ears after 1995 will there be over 100 million households connected? 22
ALEKS PROBLEM: Find the equation of the line below: ALEKS PROBLEM: Owners of a recreation area are filling a small pond with water. The are adding water at a rate of 35 liters per minute. There are 700 liters in the pond to start. Let W represent the amount of water in the pond (in liters), and let T represent the number of minutes that water has been added. Write an equation relating W to T, and then graph our equation using the aes below. 23
ALEKS PROBLEM: Choose the best graph that gives the best representation. Raina is hiking toward her campsite at a constant pace. A few kilometers from the campsite, she sees a snake and turns and runs the other wa. Minutes later, she sits to rest for awhile. Juan is driving at a constant speed. He then slows down to pass an accident. After passing it, he goes back to his original speed and continues driving at that speed. A sight-seeing ship is stopped in the water for an hour, miles from shore. Then the captain heads the ship back to shore at a constant rate. The ship docks at shore for awhile, then returns to the open sea. Charmaine begins her jogging workout b running faster and faster for about a minute. Once she hits a comfortable pace, she runs at that pace for several minutes. Then she graduall slows down to a stop over the net few minutes. 24
2.4 Functions A function is a rule which assigns a -value to each -value. 2.4 #65 Find the domain of Is the following a function? 3 2 1-3 -2-1 1 2 3-1 -2 2.4 #94 John's old '87 LeBaron has a 15-gallon gas tank and gets 23 mpg. The number of miles he can drive is a function of how much gas is in the tank. If a graph passes BOTH the Vertical Line Test and the Horizontal Line Test, the is called a function. a. Write this relationship in equation form. b. Determine the domain and range of the function in contet. *Do the following represent functions? domain red white blue domain red white blue range stars stripes polka dots range stars stripes polka dots 2.4 #99 Paul's Plumbing charges a flat fee of $50 per service call plus an hourl rate of $42.50. This means tha cost is a function of the hours the job takes to complete plus the flat fee. a. Write this relationship in equation form. d. Determine the domain and range of the function in this contet, if ou have a maimum budget of $150 to spend on the rental. 25
ALEKS PROBLEM: g( 5) The functions f, g and h are defined as follows. f ( ) 4 2 2 1 g ( ) 1 h( ) 12 9 h 4 Find the following. Simplif our answers as much as possible. f (7) g( 3) 2 h 3 The functions f, g and h are defined as follows. f ( ) 3 8 3 g( ) 10 2 h ( ) 2 10 Find the following. Simplif our answers as much as possible. f (2) g (6) The functions f, g and h are defined as follows. f ( ) 16 17 g ( ) 5 h( ) 2 2 2 h 2 Find the following. Simplif our answers as much as possible. f 3 4 26
2.5 Analzing Graphs 4. A function is even if. Even functions are smmetric about (the - ais). A function is positive where. Graphicall, the graph is above the -ais. A function is odd if. Odd functions are smmetric about the origin. A function is negative where. Graphicall, the graph is below the -ais. A function is when it "goes up" from left to right, and it when it "goes down" from left to right. A function is where its graph is horizontal. *Determine whether the following functions are even, odd or neither. 1. Formal definition of an increasing function: A function is said to be increasing on an open interval if for all where,. 2. Local Etrema: 3. 27
End behavior: As As As As For each of the following functions, find... a. intervals on which is increasing, decreasing, or constant; b. intervals on which c. an local or global etrema; d. an -values for which e. the domain and range. 28
2.6 Transforming Graphs 29
"Code List" For Transformations 2. Recall that is a circle of radius 2 centered at the origin... Graph: 3. *For the following functions, describe the changes to the base toolbo function and then graph. 1. 4. 30
5. 6. *For the given graph below, graph. and 31
ALEKS PROBLEMS: f ( h) f ( ) Find the difference quotient h where h 0 for the function below. f ( h) f ( ) Find the difference quotient h where h 0 for the function below. 2 f ( ) 5 2 2 f ( ) 4 6 f ( h) f ( ) Find the difference quotient h where h 0 for the function below. f 2 ( ) 3 6 For each function, determine whether the equation defines as a function of. 2 9 6 5 2 2 25 25 36 9 2 36 32
2.7 Piecewise Defined Functions 2.7 #45 At Wet Will's Water World, infants under 2 are free, then admission is charged according to age. Children 2 and older but less than 13 pa $2, teenagers 13 and older, but less than 20 pa $5, adults 20 and older but less than 65 pa $7, and senior citizens 65 and older get in at the teenage rate. Write this information in the form of a piecewise-defined function and state the domain for each piece. Then sketch the graph and find the cost of admission for a famil of nine which includes: one grandparent (70), two adults (44/45), 3 teenagers, 2 children and one infant. Given a. b. c. Graph:, find each: Graph: 33
Graph: 2.8 The Algebra of Functions Given and, find the following: a. b. State the piecewise function which matches each graph: a. c. d. b. e. 34
Given, and 2.8 #73 find the following: a. b. c. in a different wa. 2.8 #74 d. e. f. 35
The domains of and will all be the same (the intersection of their separate domains). The domain of will be further restricted so that ALEKS PROBLEM: Two functions and are defined in the figures below: To find the domains of composite functions, compose them and then analze the function that results. *Given and, find... a. the domain of 1 g 0 0 f 2 2 2 1 3 3 3 2 5 7 4 7 7 8 7 8 8 9 b. the domain of Find the domain and range of the composite function Write our answers in set notation. c. the domain of and d. the domain of 36
Review of Chapters 1 and 2 2. Graph: 1. a. Find the domain of b. Find the domain of c. Find the domain of and 3. For the graph of above, state the intervals on which... a. is increasing. d. Find the domain of b. is decreasing. c.. e. Find the domain of d. 4. Find the center and radius: 37
5. Is the following function even, odd, or neither? 7. Find the distance and midpoint between the points and. a. b. 6. Find the equation of the line which passes through the points and. 8. Find three consecutive odd integers such that the square of the third is fift less than the product of the first two integers. 38
9. Find the value of the discriminant and use it to describe the nature/tpe of the solutions: 12. Solve: 10. Find the given union or intersection. Write our answer using interval notation: a. 13. Solve: b. 11. Solve: 39
14. Solve: 16. Solve: 15. Solve: 40