Chapters 1 and 2 Algebra 2

Real Numbers Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Imaginary Numbers

The opposite or additive inverse of any number a is –a The sum of opposites is 0 The reciprocal or multiplicative inverse of any number a is 1/a The product of reciprocals is 1

Properties of Real Numbers Commutative Associative Identity Inverse Distributive

Absolute Value The absolute value of a number is its distance from zero on the number line | -4| = 4 | 0 | = 0 | -1 ∙ (-2) | = |2| = 2

Evaluating Algebraic Expressions When you substitute numbers for the variables in an expression and follow the order of operations you evaluate the expression evaluate a – 2b + ab for a = 3 and b = -1 a – 2b + ab = 3 – 2(-1) + 3(-1) = 3 – (-2) + (-3) = 3 + 2 – 3 = 2

Combining Like Terms A term is a number, variable or the product of a number and one or more variables. The coefficient is the numerical factor in a term. Like terms have the same variables raised to the same powers. Combine like terms by adding coefficients

Try these – in your notebooks Evaluate 7x – 3xy for x = -2, y = 5 16 Evaluate (k-18)2 -4k for k = 6 120 Combine Like Terms 2x2 + 5x – 4x2 + x – x2 -3x2 + 6x -2(r + s) – (2r + 2s) -4r – 4s

Practice: p15 (1-45)odd Please check you answers in the back of your book when you are done

1.3 & 1.4 Solving Equations and Inequalities EQ: What are the steps to solving linear equations and inequalities? Warm Up: Solve these problems in your notebook. (Left hand side) Simplify each expression 5x – 9x – 3 2y + 7x + y – 1 10h + 12g – 8h – 4g ( x + y ) – ( x – y ) - (3 – c) – 4(c – 1)

1.3 & 1.4 Solving Equations and Inequalities EQ: What are the steps to solving linear equations and inequalities? 5x – 9x – 3 = -4x - 3 2y + 7x + y – 1 = 7x + 3y - 1 10h + 12g – 8h – 4g = 8g + 2h ( x + y ) – ( x – y ) = 2y - (3 – c) – 4(c – 1) = -3c + 1

Solving Equations – by steps Distribute Combine Like Terms Combine constants Solve for variable

Solving Equations A number that makes an equation true is the solution to the equation. Try these: 8z + 12 = 5z – 21 z = -11 6(t – 2)= 2 (9 – 2t) T = 3

Stations: Pair up - pick an A and a B. You will turn in ONE sheet of paper with all the problems solved. Begin at the station on your table. Student A does the A problem explaining each step to Student B Student B does the B problem explaining each step to Student A Add your answers together. They should add to the number on the equation paper. Once they do, you may move to the next station.

Solving For a Variable Solving for a variable means isolating that variable on one side of the equation. Solve d = rt for t Solve A = ½ h ( b1 + b2) for h Try these: Solve P = 2L + 2W for W Solve E = ½ mv2 for v

Solving Inequalities Solve just like equations. Reverse the direction of the inequality symbol if you multiply or divide by a negative. Graph the solution. Example: 6 + 5 (2 – x) ≤ 41

Solving Inequalities – Try these Solve and graph 3x – 6 < 27 X < 11 12 ≤ 2 ( 3n + 1) + 22 N ≥ -2

Compound Inequalities A pair of inequalities joined by and or or 3x – 1 > -28 and 2x + 7 < 19 Try this: X – 1 < 3 or x + 3 > 8 2x > x + 6 and x – 7 < 2

Exit Pass: Solve these equations and inequalities on a sheet of paper Exit Pass: Solve these equations and inequalities on a sheet of paper. Place in the Algebra 2 basket on your way out the door. 1. 16x – 15 = -5x + 48 2. 4w – 2(1 - w) = -38 3. -2x < 3 ( x – 5) graph the solution 4. 3x + 4 ≥ 1 and -2x + 7 ≥ 5 graph the solution Homework: p21 (1-27) odd p29 (1-33) odd

1-5 Absolute Value Equations and Inequalities EQ: How do you solve equations with absolute value? warm up Solve these equations 5(x-6) = 40 5b = 2(3b-8) 2y + 6y = 15 – 2y + 8 4x + 8 > 20 3a – 2 ≥ a + 6 4(t-1) < 3t + 5 .

1-5 Absolute Value Equations and Inequalities EQ: How do you solve equations with absolute value? The absolute value of a number is its distance from zero on the number line and distance is non-negative.

Absolute Value Equations Usually have two solutions | 2y – 4 | = 12 means 2y – 4 = 12 or 2y – 4 = -12 Isolate the absolute value Rewrite as two equations Solve both equations Be sure to check your answers – they may not always work.

Try these | 3x + 2 | = 7 X = 5/3, -3 3|4w – 1| - 5 = 10 W = -13/5, 5 X = 1, -9/5 is an extraneous solution

1-5 Absolute Value Inequalities | 3x + 6 | ≥ 12 - rewrite the equation as: 3x + 6 ≥ 12 or 3x + 6 ≤ -12 Note: The inequality symbol changes direction for the negative solution Solve |2x – 3| ˃ 7, graph the solution

1-5 Absolute Value Inequalities First isolate the absolute value expression 3|2x + 6| -9 ˂ 15

1-5 Absolute Value Inequalities Exit Pass: | x + 3 | = 9 |3x – 6| - 7 = 14 |6 – 5x| = -18 2 | x + 3 | ≥ 10 | 2x + 4 | - 6 < 0

homework p 36 1-53 every other odd, except 29 (1,5,9,13,17,… etc)

Warm up: Complete a 2 minute quick write in your notebook about how to solve absolute value equations and inequalities.

There will be a test next Tuesday/Wednesday on solving linear equations and inequalities, including absolute value problems. There will be basic probability questions.

1-6 Probability EQ: How do you calculate experimental and theoretical probability? Probability measures how likely it is for an event to occur. Expressed as a percent- 0% to 100% or as a number 0 to 1 The probability of an impossible event is 0% The probability of a certain event is 100%

When you gather data from observations you can calculate an experimental probability.

The set of all possible outcomes is called the sample space You can calculate theoretical probability as a ratio of outcomes.

Carnival Fish! Homework: page 42 (7-21, 25-33)odd page 45 (51-61) odd

Warm Up Glue the warm up slip into your notebook and complete (page 56)

Stations Review Fold a sheet of binder paper in half lengthwise and width wise so there are four sections on each side. You will move from station to station completing each set of review problems in a section. You answers should add together to get the number on the station poster. Show your work!

2-1 The Coordinate Plane In an ordered pair ( x,y) the first number is the x coordinate and the second number is the y coordinate The x-y coordinate plane is divided into four quadrants by the x and y axes

2-1 Relations and Functions A relation is a set of pairs of input and output values The domain is the set of all inputs, or x values of the ordered pairs The range is the set of all outputs, or y values of the ordered pairs

2-1 Relations and Functions

2-1 Relations and Functions What is the domain and range of this relation? Domain {-3, -1, 1} Range {-4, -2, 1, 3}

2-1 Relations and Functions What is the domain and range of this relation? D {-2, -1, 1, 3} R { -2, 0, 4, 5}

2-1 Relations and Functions A function is like a machine. Put an input (x) in and get an output (y) out. A function is a relation in which each element of the domain is matched with exactly one element in the range.

2-1 Relations and Functions

2-1 Relations and Functions Vertical line test – If a vertical line passes through at least two points on a graph, then the relation is NOT a function

2-1 Relations and Functions Function notation Y = 2x can be rewritten as f(x) = 2x, and read “f of x” It does not mean f times x To evaluate the function at x = 3 write f(3), read “f of 3”

Use the function f(a) = 2a + 3 Evaluate the function at: f(-5) f(-3) f(1/2) f(4)

2-1 Relations and Functions Homework p 50 (3-35) odd: Chapter 1 Test

2-2 Linear Equations EQ: How do you graph a line in standard form? A function whose graph is a line is a linear equation Because the value of y depends on the value of x, y is called the dependent variable and x is the independent variable The y intercept is the point where the line crosses the y axis (x = 0) The x intercept is the point where the line crosses the x axis (y = 0)

2-2 Linear Equations The standard form of a linear equation is Ax + By = C and is graphed by finding the x and y intercepts Example: 3x + 2y = 120 Graph 2x + y = 20

2-2 Linear Equations Slope is the ratio of the vertical change to the horizontal change Slope = vertical change (rise) horizontal change (run) Given two points (x1, y1) and (x2, y2) Slope = y2 – y1 x2 – x1

2-2 Linear Equations Find the slope between (3,2) and (-9,6) Change in y = 6 - 2 = 4 Change in x = -9 – 3 = -12 Slope = 4/-12 = -1/3 Try these: ( -2, -2) and (4,2) 2/3 (0, -3) and (7, -9) -6/7

Practice P69 (1-7) odd, (11,13,15)

2-2 Writing Equations Point-Slope form of an equation y – y1 = m ( x – x1) Write equation when given a point and slope Ex: Write in standard form an equation of the line with slope -1/2 through the point (8, -1)

2-2 Try these Write in slope intercept the equation of the line with slope 2, through the point (4, -2) Write in slope intercept form the equation of the line with slope 3, through the point (-1, 5)

2-2 Writing an equation given two points. (1,5) and (4, -1)

2-2 Slope Intercept form Y = mx + b M is the slope B is the y intercept To find the slope of a line in standard form, solve the equation for y

2-2 Find the slope of 4x + 3y = 7 3x + 2y = 1 3x – 12y = 6

2-2 Parallel lines have the same slope Perpendicular lines have slopes that are opposite reciprocals of each other The line perpendicular to y = 3x +7 will have a slope of – 1/3

2-3 Direct Variation EQ: How do you determine if a function is a direct variation? Warm Up: find the slope between (3,-5) and (1,2) write in slope intercept form the equation of the line through (-3,-2) and (1,6) write in standard form the equation of the line with slope 2/5, through (-1,3)

2-3 Direct Variation EQ: How do you determine if a function is a direct variation? A linear function y = kx represents direct variation. The slope k is constant. You can write k = y/x, and y/x is the constant of variation The rate of change of the function k is constant. A direct variation function always contains the point (0,0)

2-3 Direct Variation EQ: How do you determine if a function is a direct variation? What does the graph of a direct variation look like?

2-3 Direct Variation EQ: How do you determine if a function is a direct variation? Direct Variation from a table. k = y/x For each table, find y/x for each pair of points.

2-3 Direct Variation Identify direct variation from an equation Must be able to put equation in the form y = kx 3y = 2x Y = 2x + 3 Y = x/2 7x + 4y = 10

2-3 Direct Variation EQ: How do you determine if a function is a direct variation? Direct Variation Activity – Rotate for each task Group chooses direct variation function. Writes an ordered pair that represents the function on their poster. Next group determines the constant of variation k for the given point. (k = y/x) Next group writes the equation for the direct variation in the form y=kx. Next group constructs a table containing 5 other points that would be on the line. Next group plots those points and constructs the line through them. Final group checks all the work and verifies that all parts have been done correctly.

2-3 Direct Variation EQ: How do you determine if a function is a direct variation? Homework assignment: page 76 (1-45) odd Chapter 1 make up test on Wednesday during enrichment.

2-3 Direct Variation Can use direct variation to solve some problems – set up as a proportion Suppose y varies directly with x, and x = 27 when y = -51. Find x when y = -17.

Homework P 70 (21 -33) odd, (39 – 57) odd P 76 (1 – 21) odd

2-4 Using Linear Models Both equations represent direct variations If y = 4 when x = 3, find y when x = 6 If y = 7 when x = 2, find y when x = 8

2-4 Using Linear Models EQ: How do you use linear equations to model real-world situations? y=mx + b m = slope which is a rate of change speed, rate of increase or decrease etc b = a starting value beginning height, distance, weight etc result = (rate of change) ∙ x + (start value)

2-4 Using Linear Models Jacksonville, FL has an elevation of 12 feet above sea level. A hot air balloon taking off from Jacksonville rises 50 ft/min. Write an equation to model the balloon’s elevation as a function of time result = (rate of change) ∙ x + (start value) Graph the equation Interpret the intercept at which the graph intersects the vertical axis.

Using two points to make a model A candle is 6 in. tall after burning for 1 hour. After 3 hours it is 5 ½ inches tall. What is the rate of change? (Slope) Write an equation in slope intercept form to model the height y of the candle after it has been burning x hours. What does the y intercept 6 ¼ represent?

Using models to make predictions Using the equation for the candle. In how many hours will the candle be 4 inches tall? How tall will the candle be after burning for 11 hours? When will the candle burn out?

whiteboard problems

Scatter plot A scatter plot is a graph that relates two different sets of data by plotting the data as ordered pairs. You can use a scatter plot to determine a relationship between the data sets. A trend line is a line that approximates the relationship between the data sets in a scatter plot.

Correlation in a scatter plot

Draw a trend line that has about the same number of points above and below it Use the slope and y intercepts to estimate the equation of the line

Group work whiteboard problems

page 83 (1-13) all

2-5: Absolute Value Functions and Graphs

2-5: Absolute Value Functions and Graphs

2-5: Absolute Value Functions and Graphs

2-5: Absolute Value Functions and Graphs

2-5: Absolute Value Functions and Graphs

2-5: Absolute Value Functions and Graphs

2-5: Absolute Value Functions and Graphs Homework: page 92 (1-9, 19-27) odds Please make your graphs large enough to read!

Practice Graph these absolute value functions Y = | 3x + 6 |

2 – 6 Families of functions EQ: How do translations affect the graph of a parent function? A family of functions is made up of functions with common characteristics A parent function is the simplest function with these characteristics A translation shifts a graph horizontally, vertically or both. It results in a graph of the same shape and size but possibly in a different position.

2 – 6 Families of functions EQ: How do translations affect the graph of a parent function? Absolute value functions y = |x| parent function y = |x| + k shifts vertex of function k units up (down if negative) y = | x – h | shifts vertex of function h units to the right (to the left if h is negative) y = a|x| stretches |x| by a factor of a (slope) y = -a|x| reflects the graph of |x| over the x axis

2 – 6 Families of functions EQ: How do translations affect the graph of a parent function? y = a|x – h| + k what does h do? what does k do? what does a do?

2 – 6 Families of functions EQ: How do translations affect the graph of a parent function? How is each graph different from the parent function y = |x|? y = |x+1| y = -|x| y = | x | - 3 y = | x - 2 | + 4

2 – 6 Families of functions EQ: How do translations affect the graph of a parent function? homework: page 99 (1-11, 17-19) all Chapter 2 test on Monday October 1

2 – 6 Families of functions Graph y = |x| On the same graph, graph y = |x| + 3 y = |x| - 2 Describe how adding a constant outside the absolute value affects the graph of the parent function

2 – 6 Families of functions Explain how a function of the form y= |x| + k is different from the parent function. A vertical translation moves the graph of the parent function up (or down) k units. Write the equation for the graph of y = |x| translated 5 units down. Y = |x| translated 7 units up.

2 – 6 Families of functions On a new graph, draw the parent function y = |x| On the same graph, draw y = |x + 2| On the same graph draw y = | x – 4| Describe how adding a number inside the absolute value affects the graph of the parent function

2 – 6 Families of functions For a positive number h, y = | x - h| is a horizontal translation of the parent function to the right h units Y = |x + h| is a horizontal translation h units to the left.

2 – 6 Families of functions

2 – 6 Families of functions Graph y = 2 |x| Graph y = - |x| Graph y = ½ |x| How does multiplying a graph by a number larger than one affect the graph? How does multiplying a graph by a number less than one affect the graph? How does multiplying by a negative affect the graph?

2 – 6 Families of functions A vertical stretch multiplies all y values by the same factor greater than one, stretching the graph vertically A vertical shrink multiplies all y values by a factor less than one, compressing the graph vertically Multiplying by a negative factor reflects the graph over the x axis

2 – 6 Families of functions A function is a vertical stretch of y = |x| by 5 – what is the equation? Reflect the function across the x axis. What is the equation?

2 – 6 Families of functions

2 – 6 Families of functions Write equations for the graphs obtained by translating y = |x| 10 units right 4 units down 7 units left, 6 units up Reflection across x axis Vertical shrink by a factor of 2/3

Homework Page 92 (33-43) odd Page 99 (1-13) odd Page 102 (1-10) Chapter 1 & 2 test next week Tuesday

Warm up Graph the following functions y = 2x + 3 y = -1/3x +1

2-7 Two Variable Inequalities A linear inequality is an inequality in two variables whose graph is a region in the coordinate plane that is bounded by a line. To graph a linear inequality: Graph the boundary line Determine which side of the line contains solutions Determine if the boundary line is included

2-7 Two Variable Inequalities A dashed boundary line indicates the line is not part of the solution A solid boundary line indicates the line is part of the solution Choose a test point to check if a region makes the inequality true – use (0,0), if it is not on the line Example: graph y > ½ x - 1

2-7 Two Variable Inequalities Try this on your whiteboard – graph: y ≤ 2x + 3 Graph the line y = 2x + 3 Check the test point (0,0) Is the line part of the solution?

2-7 Two Variable Inequalities Graph y ˃ -4x + 3

2-7 Two Variable Inequalities Graph the absolute value inequality y ≤ | x – 4 | + 5 -y + 3 > | x + 1 |

Homework: Page 106 (1,5,9,11,15,19, 25) Corrections to quiz – use quiz as study guide Chapter 2 test on Monday

Study Guide Answers a b c d d b a c a c d b