1 st Quarter Notes Algebra I Name:

Transcription

1 1 st Quarter Notes Algebra I Name: 1

2 Table of Contents Unit 1 Pre Algebra Review Topics Integer Rules Review page 6 Translating and Evaluating page 7 Properties of Real Numbers page 10 Unit 2 Solving Equations and Inequalities One and Two Step Equations page 14 Proof for Equations page 16 Solving Multi-Step Equations page 19 Solving Linear Inequalities page 23 Solving & Graphing Compound Inequalities page 26 Unit 3 Literal Equations Solving Literal Equations page 29 Unit 4 Word Problems Word Problems with Proportions page 33 Word Problems with Equations page 35 Word Problems with Inequalities page 40 Unit 5 Functions and Linear Graphs Introduction to Functions page 44 Functional Notation page 46 Graphing with Table of Values page 49 Graphing using Intercepts page 53 What is Slope? page 57 Graphing in Slope Intercept Form page 60 2

3 Warm Ups 3

4 Warm Ups 4

5 Extra Graph Paper 5

6 Integer Rules Review Operation Rule Examples Addition Same signs: Add and keep the sign -7 + (-9) = = 16 Different signs: Subtract and keep the sign of the number with the larger absolute value = (-10) = -2 Subtraction Multiplication Division Use Keep, Change, Change Keep the first integer the same Change the operation to addition Change the sign of the second integer to its opposite Then follow the rules of addition. Positive Positive = Positive Negative Negative = Positive Positive Negative = Negative Negative Positive = Negative The rules for division are the same as the rules for multiplication = -9 + (-4) = (-6) = = 16-2 (-11) = = 9-13 (-5) = = = = = = = = = = -2 Evaluate (-14) (-12)

7 Translating and Evaluating Learning Target: SOL A.1: The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Operation Verbal Phrase Expression Addition: Sum, Plus, Total, More than, Increased by The sum of two and a number x A number n plus seven Subtraction: Difference, Less, Less than, Minus, Decreased by The difference of a number n and six. Six less a number y Six less than a number y Multiplication: Times, Product, Multiplied by, Of Division: Quotient, Divided by, Divided into The product of twelve and a number y One third of a number x The quotient of a number k and 2 Translating Expressions with Multiple Operations: (When order DOES NOT matter, always write the term with the VARIABLE 1 st and write the CONSTANT LAST) 1) Eight less than four times a number z. 2) Four times the quantity six less a number n. 3) Three times the sum of a number x and nine. 4) The quotient of twice a number x and ten is three. 5) Two more than the product of nine and a number is less than forty-seven. 7

8 Evaluating Expressions 6) ) a + 7 when a = 6 Order of Operations P E 8) when x = -2 M 9) ( ) ( ) when x = 5 and y = 2 D A S You Try It! Directions: Translate each mathematical statement below. Then, evaluate each expression if x = -4, y = 8, and z = -2 10) Twelve less a number y. 11) Three more than twice a number x. 12) Seven less than the product of x and z. 13) Five times the quotient of y and x. 14) Two more than the product of one half and the square of a number x. 15) Negative seven times the sum of a number z and ten. 8

9 Evaluate the following if a = 4, b = 1, and c = -5 16) 3 + ( ) 17) 18) ) 3c 2 2c ) 21) ( ) [( ) ( ) ] 22) ( ) ( )( ) 23) ( ) ( )( ) 9

10 Properties of Real Numbers Learning Target: SOL A.4b: The student will justify steps used in simplifying expressions and solving equations. Property Explanation Addition Multiplication Closure When two real numbers are added or multiplied, = 13 Real # + Real # = Real # 6 9 = 54 Real # Real # = Real # the result will always be a real number. Associative Numbers may be grouped differently without affecting the final value. Commutative Identity The numbers on each side of an operation sign may be commuted (switched) around without affecting the final value. Find a number that will produce an identical answer to the original number. Additive Identity Multiplicative Identity Inverse Find a number that will turn a number into the identity element. Additive Inverse Multiplicative Inverse Multiplicative Property of Zero Any number times zero is zero, and zero times any number is zero. Distributive This property is used to multiply a number by the sum or difference of two numbers in parentheses. The first number is distributed across the parentheses by multiplying it with both of the numbers inside the parentheses. 10

11 Using the Distributive Property The distributive property is often used to simplify expressions that cannot be simplified using the order of operations. Simplify each expression below. Combine like terms when necessary. 3(4x + 7) -5(3x 6) -7(2x + 8) 9(x 7) (4x + 5) Division and the Distributive Property The distributive property can also be used to simplify fractions with a monomial expression in the denominator. Instead of multiplying each coefficient, divide each coefficient by the number in the denominator. Try it! Simplify the following expressions using the distributive property. Combine like terms when necessary. 1. 5(6a + 7b c) 2. -3(12x 8y) 3. -5(2x + 1) 3(4x 9)

12 Properties of Equality Property Explanation Arithmetic Example Algebra Example Reflexive A quantity is equal to itself. x = x Symmetric Transitive Substitution Quantities on each side of an equal sign may be switched. If a first quantity equals a second, and the second equals a third, then the first and third quantities are also equal. Equal quantities may be substituted for each other. If x = y, then y = x. If a = b and b = c, Then a = c. If y = 3, then 2y = 2(3). Try it! Identify the property displayed in each example (2 + 6) = (-3 + 2) (x 4) 9 = (4 x) = x so x = y = y 5. x = z and z = 5, so x = = (8 3) = (-6 + 6) + 19 = If x = 2, 3x = 3(2) 11. 3(5x + 4) = 15x a = = 3(4x + 1) 14. A = A 15. (-6 4) 9 = -6 (4 9) so 3(4x + 1) = 9 12

13 Unit 1 Scratch Paper 13

14 One-Step & Two-Step Equations Learning Target: SOL A.4d: The student will solve multistep linear equations algebraically. What does it mean to solve something? Equations with Addition x + 12 = = n + 5 Equations with Subtraction x 12 = 23 n 5 = x = 19 y + (-12) = 33 z (-7) = = y - 12 Equations with Multiplication Equations with Division 5x = 35-3x = 27 x 5 6 x = 4x -8x = -44 x x 3 7 Solve by Combining Like Terms 7x 4x = = 5d 9d 14

15 The Basics of Solving Two Step Equations 8x + 3 = = 3x Backwards PEMDAS! m x You Try It! Solve the following equations. Show all of your work. Any non-integer answers should be expressed as fractions. 1) -12x = -60 2) 28 = 5x + 8 3) 4) 16 x = 12 5) 6) 60x x = 16 7) 15

16 Proofs for Equations Practice: Identify the property displayed in each example. Learning Target: SOL A.4b: The student will justify steps used in simplifying expressions and solving equations. 1. (2 + x) + 5 = 2 + (x + 5) 2. 2x 1 = 2x 3. 2(x - 7) = 2x If 2 = x, then x = = (x + 4) + 6 = (4 + x) a = 0 8. ( ) + 7 = Properties Explanations Algebraic Example Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Adding the same number to each side of an equation produces an equivalent equation. Subtracting the same number from each side of an equation produces an equivalent equation. Multiplying both sides of an equation by the same number produces an equivalent equation. Dividing both sides of an equation by the same number produces an equivalent equation. If a = b Then a + x = b + x If a = b Then a - x = b - x If a = b Then If a = b Then What property is being used when simplifying the expression or equation below? 1. 5x + 2 = a (7 b) 3. 5x = (a 7) b ( ) ( ) 16

17 Proofs Solve each two-step equation below. Justify each step in the boxes provided. Solution 15y + 31 = 61 15y (-31) = 61 + (-31) Justification Given 15y + 0 = 30 15y = 30 ( ) ( ) 1y = 2 y = 2 Find the three mistakes in the following proof. Identify and correct these mistakes. Given Symmetric Property of Equality ( ) ( ) Inverse Property of Addition Addition Property of Equality Additive Identity Property Division Property of Equality 1x = x = Solve the given problem. Justify each step. Inverse Property of Multiplication Multiplicative Identity Property 17

18 You Try It! 1. Justify the steps for proof in the boxes provided. Solution Justification 35 = 7 ( x +3) Given 7( x + 3) = 35 7x + 21 = 35 7x = x + 0 =14 7x = 14 1x = 2 x = 2 2. Find the three mistakes in the following proof. Identify and correct these mistakes. 1x = x = Given Subtraction Property of Equality Additive Inverse Zero Product Property Division Property of Equality Multiplicative Identity Multiplicative Inverse 3. What property is being used when simplifying the expression or equation below? a. 7x = 21 b. 15 = 3x + 2 c. 18z + 0 = 2634 d. 1x = 5 3x + 2 = 15 18z = 2634 x = 5 18

19 Solving Multi-Step Equations Learning Target: SOL A.4d: The student will solve multistep linear equations in two variables including solving multistep linear equations algebraically and graphically. Solve each equation. Leave any non-integer answers as fractions unless decimals were used in the original problem. Round any decimals according to the decimals in the problem. 1. 6x + 22 = -3x x = x 3. (3x + 9) = 5x + 3x x 3(x 2) = (x 4) = =

20 Solve and discuss with your neighbor. 9. 3(x 3) = 3x (y 7) = 2y 14 Proofs 11. Using the given proof, justify each step in the boxes provided. Steps Justification 5x + 3 = y if y = 23 Given 5x + 3 = 23 5x = 20 x = Using the given proof, justify each step in the spaces provided. Solution 3x x = 43 3x + 7x + 5 = 43 Justification Given 10x + 5 = 43 10x (-5) = 43 + (-5) 10x + 0 = 38 10x = 38 ( ) ( ) 1x = x = 20

21 You try it! Solve each equation. Leave any non-integer answers as fractions unless decimals were used in the original problem. Round any decimals according to the decimals in the problem x = x 2. 4(1 y) + 3y = -2(y + 1) 3. -2(2x 6) + (12x + 8) = 5 3(2x + 1) 4. (10x + 15) = 18 4(x 3) Hermione, Harry, and Ron were solving three different math problems. The last step of their work is given below. Determine which person s problem has a solution that is all real numbers, whose problem has no solution and whose problem has the solution x = 0. Hermione Harry Ron 6x = 0 0x = 5 5 = 5 21

22 8. Jerri wrote these steps when solving an equation: Steps Justification 17(x + 3) = 6 4 Given 17x + 51 = x + 51 = 2 17x = -49 x = Select a property from the box to justify each step. Write your answer in each box. Substitution Property Subtraction Property of Additive Identity Equality Division Property of Equality Distributive Property Associative Property 9. Using the given proof, justify each step in the spaces provided. Solution 8x + (7x +3) = 78 (8x + 7x) + 3 = 78 Justification Given 15x + 3 = 78 15x (-3) = 78 + (-3) 15x + 0 = 75 15x = 75 ( ) ( ) 1x = 5 x = 5 22

23 Solving Linear Inequalities Learning Target: SOL A.5a: The student will solve multistep linear inequalities in two variables including solving multistep linear inequalities algebraically and graphically. Addition and Subtraction Properties of Inequality Adding or subtracting the same number to each side of an inequality produces an equivalent inequality. If a > b, then a + c > b + c. If a < b, then a c < b c. Multiplication and Division Properties of Inequality If you multiply or divide each side of an inequality by a.. Positive number you produce an equivalent inequality. Negative number you have to flip the sign to produce an equivalent inequality. If a < b and c is positive, then ac < bc. If a < b and c is negative, then ac > bc. Solving inequalities is very similar to solving equations with one notable difference! 1. 6x 2 > 4x x + 4 < 2(x 8) x > > Graph the solution for number #4. Then give two possible values of x that will make the inequality true. 23

24 6. < (8x 1) > x 8. 3(x + 1) < 3x Using the axioms of inequality and the properties of real numbers, justify each step in the solutions given below. Solution Justification 4 2x > -6 Given -2x + 4 > -6-2x > -10 x < 5 Solution Let y = 2x and y 4 > -18 2x 4 > -18 2x > x + 0 > -14 2x > -14 Justification Given ( ) (2x) >( )(-14) 1x > -7 x > -7 24

25 Try it! 1. -3(2x + 1) > 1 8x 2. 5x < x (-3) > (x + 3) < 4x x < > Graph the solution for number #1. Then give two possible values of x that will make the inequality true. 8. Using the axioms of inequality and the properties of real numbers, justify each step in the solutions given below. Solution Justification 3(2x 1) < 2(4x + 3) Given 6x 3 < 8x + 6 6x < 8x x 8x < 8x 8x + 9-2x < 9 x > 25

26 Solving and Graphing Compound Inequalities sandwich inequalities Learning Target: SOL A.5a: The student will solve multistep linear inequalities in two variables including solving multistep linear inequalities algebraically and graphically. OR inequalities x is at least negative six and at x is either less than negative 2 most five or x is greater than seven Solve each compound inequality. Solve each compound inequality < 2x x 3 < 5 or 3x < -3x + 8 < x or 2x -6 26

27 Special Cases Solve and graph each compound inequality. 7. 3x + 1 < 4 or -2x 5 > x 6-11 or -3x 7 > -13 Solve each compound inequality. Try it! < 9x 1 < x 7 8 or -2x x + 7 < 3 or 5x < -2x Graph the answer to #3. 6. Graph the answer to #4. 7. Write an inequality that corresponds to each graph below:

28 Unit 2 Scratch Paper 28

29 Solving Literal Equations (Formulas) Goal: Rearrange a formula ( Solve the Formula ) so that a new variable is isolated. Learning Target: SOL A.4a: The student will solve literal equations (formulas) for a given variable. Solve for the indicated variable. 1. A = ½bh, solve for h 2. C = 2 r, solve for r 3. A = ½h(b 1 + b 2 ), solve for h 4. A = ½h(b 1 + b 2 ), solve for b 1 5. P = 2w + 2l, solve for l 6. F = C + 32, solve for C 29

30 Function Form: A two-variable equation (usually x and y) is written in function form if one of its variables is isolated on one side of the equation. Rewrite the following equations so y is a function of x. (Write y in terms of x in other words, isolate y!) Write all answers in simplest form! 7. -7x + y = y 3x = y = 8x 2x ⅓(y + 2) + 3x = 7x 30

31 Try it! Solve for the indicated variable. 1. I = Prt, solve for r 2. A = ½h(b 1 + b 2 ), solve for b 2 Rewrite each equation so that y is a function of x. Write your answer in simplest form = 12x 2y 4. y 7 = -2x 5 5. (25 5y) = 4x 9y x = 4y 4 31

32 Unit 3 Scratch Paper 32

33 Using Proportions to Solve Word Problems Learning Target: SOL A.4f The student will solve realworld problems involving multi-step linear equations in two variables. Steps: A. Determine the object/units being compared. B. Write a proportion that represents the situation. C. Solve your proportion. D. Circle your final answer. Include units. 1. The ship model kits sold at a hobby store have a scale of 1ft : 600ft. A completed model of the Queen Elizabeth II is 1.6 feet long. Estimate the actual length of the Queen Elizabeth II. 2. Mr. Land is trying to decide how my pizzas to buy for the 7 th grade picnic. If he usually needs 9 large pizzas to feed his class of 26 students, how many large pizzas should he buy if there are 204 students in the entire 7 th grade? 3. Based on the 2000 census, each member of the U.S. House of Representatives represents an average population of 646,952 people. If Virginia currently has 11 representatives, what was the approximate population of Virginia in 2000? 4. The ratio of male students to female students in the freshman class is 2:3. There are 216 girls in the freshman class. Find the number of males % of the books on Ms. Park s book shelf are mystery novels. If she has 40 books on her bookshelf, how many of them are mystery novels? 33

34 Try it! Directions: Write a proportion to represent each word problem below. Then, find the solution. SHOW ALL OF YOUR WORK! 1. Tara babysits every Saturday afternoon. She typically gets paid $42 for four hours worth of work. If she is asked to stay late, how much should she be paid for six hours worth of work? 2. Triangle ABC and Triangle DEF are similar. The height of ABC is 4 cm and the base is 7 cm. If the height of DEF is 14 cm, how long is the base? 3. Elizabeth is standing next to a flagpole that is 24 feet high. If the flagpole s shadow is 13 feet and Elizabeth s shadow is 3 feet, how tall is Elizabeth? Round to the nearest 100 th if necessary. 4. Biologists wanted to know how many fish were in Lake Neterer. Last week, they tagged 220 fish. This week, the biologists counted 15 tagged fish out of a sample of 300 fish from the same lake. Estimate the total number of fish in Lake Neterer. 5. Tyler is preparing for a college entrance exam. On a practice test, he answered 8 problems in 15 minutes. At this rate, will he be able to finish a 90 question exam in 150 minutes? 34

35 Using Equations to Solve Word Problems Learning Target: SOL A.4f The student will solve realworld problems involving multi-step linear equations in two variables. Steps: A. Define your variables. B. Use your variables to write an equation that represents the situation. C. Solve your equation D. Circle your final answer. Include units. 1. As a lifeguard, you earn $6 per day plus $2.50 per hour. How many hours must you work to earn $16 in one day? 2. The sum of the ages of three brothers is 59. Jason is twice as old as Brian. Alex is five more than three times Brian s age. How old is each brother? 3. The perimeter of a rectangle is 168 feet. Its length is 5 times the width. Find the dimensions of the rectangle. 35

36 4. The sum of three consecutive integers is 270. Find the numbers. 5. Find three consecutive odd integers such that the sum of the third and three times the first is the same as thirty more than twice the second. 6. In triangle ABC the measure of angle B is fifteen degrees less than the measure of angle A. The degree of angle C is ten degrees more than the sum of the measures of angles A and B. How much does each angle measure? 36

37 Try It! 1. A lifeguard at the community pool makes $9.50 per hour. A lifeguard at the country club makes $8.25 per hour, but has a weekly bonus of $40. How many hours do the lifeguards need to work in one week to earn the same amount of money? 2. Tess is hiking the entire length of the Superior Trail in Minnesota. She has already hiked 80 miles. If she continues to hike at a constant rate of 18 miles per day, how many days will it take her to reach the end of this 275 mile trail along the edge of Lake Superior? 3. Find three consecutive even integers such that the sum of the first and twice the third is the same as twenty-eight less than four times the second. 37

38 4. Jared is training for a marathon. His goal is to run a total of 27 miles over the course of this threeday holiday weekend (Saturday, Sunday, and Monday). If he plans to decrease the length of his run by 2 miles each day, how many miles will he need to run on Saturday if he is going to meet his goal? 5. When Sarah went to college she had $15,000 in a savings account to pay the rent on her apartment. Every month she makes a $1,300 withdrawal to pay rent. If she only has $3,300 left, how many months has she been paying rent? 6. John is five years older than Henry and Henry is 3 years older than Fred. The sum of their ages is 32. Find their ages. 38

39 Word Problems Warm-Up! 1. Plumber Joe charges a flat fee of $45 plus $15 per hour for every house call he makes. If he charges Mr. and Mrs. Centennial $112.50, how many hours did he work at their house? 2. A person that weighs 135 pounds on earth would weigh pounds on the moon. What would a rock that weighs 7.06 pounds on earth weigh on the moon? 3. You purchase 4 tickets to a baseball game from an internet agency. In addition to the cost per ticket, the agency charges a convenience charge of $2.75 per ticket. You also choose to pay for rush delivery, which costs $18. The total cost of your order is $157. What is the price per ticket before the convenience charge? 4. John is eight years older than Henry and Henry is seven years older than Fred. The sum of their ages is 49. Find their ages. 39

40 Using Inequalities to Solve Word Problems Learning Target: SOL A.5c: The student will solve realworld problems involving multi-step linear inequalities in two variables. Recognizing Inequalities x is at most 4 x is at least 4 x is no more than 4 x is no less than 4 Steps: A. Define your variables. B. Use your variables to write an inequality that represents the situation. C. Solve your inequality. D. Circle your final answer. Include units. 1. Sally wants to rent tables for her outdoor wedding. The rental shop in town will charge her $11.25 to rent the long rectangular table for her bridal party that day. Each of the round tables her guests will sit at cost $8.75. If she can spend no more than $160 on tables, what are the possible numbers of round tables she can rent? Represent your answer algebraically and graphically. 2. A blank CD can hold at most 70 minutes of music. So far you have burned 25 minutes of music onto the CD. You estimate that each song lasts 4 minutes. What are the possible numbers of additional songs that you can burn onto the CD? 3. A gym is offering a trial membership for 3 months by discounting the regular monthly rate by $50. You will consider joining the gym if the total cost of the trial membership is less than $100. What must the price of a regular monthly membership be in order for you to take advantage of this deal? 40

41 4. Your cell phone plan costs $49.99 per month for a given number of minutes. Each additional minute or part of a minute costs $0.40. You budgeted $55 per month for phone costs. What are the possible additional minutes (x) that you can afford each month? 5. Pretty Mountain State Park rents cabins for guests to stay in for $110 per night. If you have purchased a year-long state parks pass, they will discount your nightly rate by $15. At the time of rental, guests can also opt to pay $55 for an unlimited supply of firewood. You have a state parks pass and you will choose to pay for the unlimited supply of firewood. For how many nights can you rent the cabin if you are determined to spend less than $1,100? Try It! Directions: Write an inequality to represent each word problem below. Then, find the solution. SHOW ALL OF YOUR WORK! 1. A gas station charges $0.10 less per gallon of gasoline if a customer also gets a car wash. The price of gas is regularly $2.09 a gallon, and a car wash is $8.00. If you get a car wash, what are the possible amounts (in gallons) of gasoline that you can buy if you can spend at most $20? 2. Tony is a new waiter at the family restaurant in town. He is hoping to earn at least $100 during his 8 hour shift. If he makes $70 in tips, what would his hourly rate need to be to reach this goal? 41

42 3. To become a member of an ice skating rink, you have to pay a $30 membership fee. The cost of admission to the rink is $5 for members and $7 for nonmembers. After how many visits to the rink is it less expensive to be a member than a nonmember? In other words, at what point is it worth it to get the membership? 4. Bryan s dog is three years more than twice his cat s age. Find all possible ages of the cat if the sum of their ages is at most 18. Represent your answer algebraically and graphically. 5. Jacob is training for a marathon and is using a pyramid training pattern for the next five days. He plans to increase the number of miles he runs by a single mile each day from the first to the third day, peaking on the third day, and then decreasing the number of miles run by a single mile per day for the last two days. a) If x represents the number of miles Jacob runs on Day 1, write expressions for how many miles, in terms of x, he runs from Days 2 through 5. Day 1 = Day 2 = Day 3 = Day 4 = Day 5 = b) Find all possible values that Jacob can run on Day 1 such that his total number of miles run over the five days is at least 64. Represent your answer algebraically and graphically. 42

43 Unit 4 Scratch Paper 43

44 An Introduction to Functions Learning Target: SOL A.7a and b: The student will investigate and analyze linear families and their characteristics both algebraically and graphically, including determining whether a relation is a function and finding the domain and range of a function. Key Definitions Relation: Function: x y One Way Street! Only check the x s! Decide if the given values represent a function. Then state the domain and range. 1. x y 2. x y 3. {(0, 5);(1,5);(2,5)}

45 The Vertical Line Test: Use the vertical line test to determine if the relation is a function State the domain and range of each function

46 Functional Notation Learning Target: SOL A.7e: The student will investigate and analyze linear families and their characteristics both algebraically and graphically, including finding the values of a function for elements in its domain. What is functional notation? The symbol is another name for y and is read as. It does not mean f times x. You can use letters other than f, such as g or h to name functions. Let f(x) = 4x 3 and g(x) = x Evaluate each function for the given x values. 1. f(-5) 2. g(4) 3. g(-3) 4. f(g(6)) Let f(x) = 6x + 9 and g(x) = -x + 5. Find the value of x so that the function has the given value. 5. f(x) = 3 6. g(x) = 2 46

47 Try it! 1. Give an example of a graph, a list of ordered pairs, and a table of values that each represent a function. Your examples must be different than the examples cited in the notes. State the domain and range of each relation below. Then, determine if it represents a function. 2. {(-2, -8),(-1, -1), (0, 0), (1, 1), (2, 8)} 3. x y Draw the graph of a function with a domain that is all real numbers and a range where y

48 7. True or False: A relation is always a function. Support your reasoning with an example or a counter-example. (Note: A counter-example is an example proving the statement is false.) Let f(x) = 2x and g(x) = 4x 6. Evaluate each function for the given domain values. 8. f(-4) 9. g(7) 10. g(f(3)) Let f(x) = -7x + 12 and g(x) = 8x 32. Find the value of x so that the function has the given value. 11. f(x) = g(x) = 4 48

49 Graphing Linear Equations Using a Table of Values Learning Target: SOL A.6, A.7e, f: The student will graph linear equations in two variables; find the values of a function for elements in its domain; and make connections between and among multiple representations of a function. The graph of an equation in two variables is the set of points in a coordinate plane that represents all solutions of the equation. Make a table of values for the following equations. Then, graph. 1. y = 2x 4 x y 2. f(x) = -3x + 6 x f(x) 3. y + 10 = 3x + 2x x y 49

50 4. x y Horizontal and Vertical Lines! Make a table of values for the following equations. Then, graph each equation. x = 5 y = -2 x y x y x = -3 y = 5 x y x y 50

51 It s Your Turn! Make a table of values for the following equations. Then graph. Use the domain {0, 1, 2, 3, 4}. Use the domain {-2, -1, 0, 1, 2}. f(x) = 3x 5 x f(x) x y x = -2. Choose your own domain. x y x y 51

52 1. Which of the graphs on the previous page does not represent a function? Explain your reasoning. 2. Three of the graphs on the previous page represent functions. Identify them and state their domain and range. 3. Write an equation for a function whose domain is all real numbers and whose range is 3. What kind of line is this? Graph the line. 52

53 Graphing Equations in Standard Form Using Intercepts Learning Target: SOL A.6, A.7f: The student will graph linear equations in two variables and make connections between and among multiple representations of a function. Standard Form: You can easily graph an equation written in standard form by finding two convenient points. X - Intercept Y - Intercept Draw the line that has the given intercepts. 1. x-intercept: 3 2. x-intercept:-5 y-intercept: 5 y-intercept: -3 Graph the following equations by finding the x- and y-intercepts. 3. 3x + 2y = x + 5y =

54 Finding the Roots of a Function Method 1: Graphically Find the root(s) of each graph. (In other words, find the x-intercepts.) 1. y = -3x x 4y = Method 2: Algebraically Find the root(s) of each equation. (In other words, find solution when f(x)=0 or y=0.) 5. y = 3x x + 3y = f(x) = - x

55 You Try It! 1. Graph the following equations. Label the points where the line crosses the axes. 4x + y = 4-3x + 9y = -18 4x + 3y = Is it possible for a line not to have an x-intercept? Is it possible for a line not to have a y-intercept? Explain. 3. The x-intercept of the graph of Ax + 5y = 20 is 2. What is the value of A? 4. Consider the equation 3x + 5y = k. What values could k have so that the x-intercept and the y- intercept of the equation s graph would both be integers? Explain. 55

56 5. You are helping to plan an awards banquet for your school, and you need to rent tables to seat 180 people. Tables come in two sizes. Small tables seat 4 people, and large tables seat 6 people. This situation can be modeled by the equation 4x + 6y = 180 Where x is the number of small tables and y is the number of large tables. Find the intercepts of the graph of the equation. Graph the equation. Give four possibilities for the number of each size table you could rent. 6. Find the roots of the following functions: a. b. c. f(x) = x d. 7. If a 0, find the intercepts of the graph of y = ax + b in terms of a and b. 56

57 What is slope? Learning Target: SOL A.6a The student will graph linear equations in two variables and determine the slope of a line when given the graph of the line or two points on the line. The slope of a non-vertical line is the ratio of the change to the change. We refer to this as the rate of change and use m to represent slope. Finding the slope of A Line Given a Graph: m = Read a graph like you read a book, from left to right. If the line is going, the slope should be. If the line is going, the slope should be. Find the slope of each line graphed

58 Finding slope of horizontal and vertical lines: HOY VUX H: O: Y: V: U: X: Finding the slope Given Two Points: m = Find the slope of each line passing through the given points. 5. (4, 3) and (-2, -3) 6. (-3, 5) and (8, 2) 7. (8, 0) and (8, 4) 8. (6, -2) and (5, -2) 58

59 Try it! Find the slope of each line What type of line has a slope of zero? 4. Is a line with undefined slope a function? Why or why not? Find the slope of the line passing through the given points. 5. (-1, 2) and (3, -4) 6. (0, 5) and (-3, 5) 7. (8, 5) and (2, -7) 8. (-1, 3) and (-1, 9) 9. The graph below shows the elevation of a hiker walking on a mountain trail. a. What does the slope represent? Elevation (ft) b. Describe the hike in terms of slope. Time (min) 59

60 Graph Using Slope-Intercept Form Learning Target: SOL A.6a The student will graph linear equations in two variables including determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Graph the line y = 2x + 1. What is the slope of the line? x y What is the y-intercept? Slope-Intercept Form Identify the slope and the y-intercept for each equation. Be sure to re-write it in slope-intercept form first! 1. y = 2x y = -3x 3. 3x + y = y = 2 6. x + 4y = 4 60

61 Graph each line using the slope-intercept method. Steps 1. Re-write equation in slope-intercept form. 2. Graph the y-intercept (this goes directly on the y-axis!) 3. Use the slope to find the next point. 4. Draw the line. 1. y = x y = 4 5x 3. 3x y = 0 4. y = x + 3 Decide which method of graphing would be the easiest to use based upon the given equation. (TOV, x/y Intercepts, or slope-intercept) 1. 3x 9y = y = 3x x y = 8 4. y = x + 7y 9 = 2 6. y = -2x

62 Try it! Determine the slope (m) and y-intercept (b) of each line Graph each line using slope intercept form Write the equation of a line with a slope of -4 and a y-intercept of 7. 62

63 Standard Form vs. Slope-Intercept Form Graph each equation. Use the most efficient method possible. Do not rearrange the equation. 1. 6x + y = 6 2. y = 3. y = 4 63

64 Graphing Linear Equations: Intercept Method vs. Slope-Intercept Form Graph each equation. Use the most efficient method possible. Do not rearrange the equation. 1. y = 3x 2. y = 4x 3. 3x 5y = y = -x y = 3x y x = 5 64

65 7. x = x + y = 3 9. y = -2x x + 4y = x y = y = 3x

66 Unit 5 Scratch Paper 66

67 67