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ADVANCED ALGEBRA II CHAPTER 2 -- LINEAR EQUATIONS AND FUNCTIONS ADVANCED ALGEBRA II CHAPTER 2 β LINEAR EQUATIONS AND FUNCTIONS 2.1 REPRESENT RELATIONS AND FUNCTIONS WARM-UP 1. Graph each ordered pair on the coordinate plane. A. (-4. -3) B. (3, 2) C. (0, 0) D. (1, -3) E. (-1, 5) 2. Evaluate each expression for x = -2, 0, and 5 A. π₯ + 2 B. β2π₯ + 3 C. 2π₯ 2 + 1 GRAPHING RELATIONS (V) Relation - Example: Graphing a Relation Graph the relation {(β2, 4), (3, β2), (β1, 0), (1, 5)} (V) Domain β (V) Range Example: Finding Domain and Range Write the ordered pairs for the relation shown to the right. Find the domain and range. D. |β2π₯ + 5 | (V) Mapping Diagram β Example: Make a Mapping Diagram Make a mapping diagram for the relation {(β2, 4), (3, β2), (3, 0), (1, 4), (β2, 7)} IDENTIFYING FUNCTIONS (V) Function - Example: Identifying Functions Determine whether each relation is a function. (V) β Vertical-Line Test - Example: Using the Vertical-Line Test Use the vertical-line test to determine whether each represents a function. (V) Function Notation β Example: Using function notation 1 3 Find π(β3), π(0), πππ π(5) for the function: π(π¦) = β 5 π¦ + 5 EQUATIONS IN TWO VARIABLES Example: Graph an equation in two variables A. π¦ = β3π₯ + 5 B. π₯ = π¦ 2 + 1 ADVANCED ALGEBRA II CHAPTER 2 β LINEAR EQUATIONS AND FUNCTIONS SECTIONS 2.2 β 2.4 LINEAR EQUATIONS WARM-UP Evaluate if π₯ = β2, 0, 1 πππ 4 A. 3 5 π₯β7 B. β2π₯ β 3 GRAPHING LINEAR EQUATIONS (V) Linear Function β (V) Linear Equation β (V) Independent Variable β (V) Dependent Variable β Example: Graphing a Linear Equation 2 A. π¦ = 3 π₯ + 3 B. 2π₯ β π¦ = 8 (V) y-intercept β (V) x-intercept β (V) Standard Form of a Linear Equation Example: Transportation The equation 3π₯ + 2π¦ = 120 models the number of passengers who can sit in a train car where x is the number of adults and y is the number of children. Graph the equation. Explain what the xand y-intercepts represent. Describe the domain and range. (V) Slope β Slope Formula Example: Finding Slope Find the slope of the line through the points A. (-2, 7) and (4, -1) B. (3, 7) and (3, -2) C. (4, -6) and (-1, -6) WRITING EQUATIONS OF LINES There are several ways to write the equation of a line. We will discuss them here. Point-Slope Form Example: Writing the Equation Given the Slope and a Point 1 Write in point-slope form an equation of the line with the slope β 2 through the point (8, -1). Example: Write an Equation Given Two Points Write in point-slope form the equation of the line through (1, 5) and (4, -1). The other forms of a line are Slope-Intercept and Standard Form. Slope-Intercept β Standard Form β Example: Write an Equation in All Three Forms Find the equation of the line through (-3, 6) and (4, 8). Please write in all three forms. SPECIAL LINES AND RELATIONSHIPS Horizontal Line Vertical Line Perpendicular Lines Parallel Lines Example: Writing the Equation of a Perpendicular Line 3 Write an equation of the line through each point and perpendicular to π¦ = π₯ + 2. 4 A. (0, 4) B. (6, 1) ADVANCED ALGEBRA II CHAPTER 2 β LINEAR EQUATIONS AND FUNCTIONS 2.5 MODEL DIRECT VARIATION WARM UP 1. Solve each equation for y. A. 12π¦ = 3π₯ B. 12π¦ = 5π₯ C. 3 4 π¦ = 15 WRITING AND INTERPRETING A DIRECT VARIATION (V) Direct Variation β (V) Constant of Variation β Example: Identifying Direct Variation From a Table For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. A. B. Example: Identifying Direct Variation From an Equation For each function, determine whether y varies directly with x. If so, find the constant of variation. A. 3y = 2x B. 7x + 4y = 10 Example: Water Conservation A dripping faucet wastes a cup of water if it drips for three minutes. The amount of water wasted varies directly with the amount of time the faucet drips. A. Find the constant of variation k and write an equation to model the direct variation. B. Find how long the faucet must drip to waste 4 ½ c of water. Example: Using a Proportion Suppose y varies directly with x, and x = 27 when y = -51. Find x, when y = -17. ADVANCED ALGEBRA II CHAPTER 2 β LINEAR EQUATIONS AND FUNCTIONS 2.6 DRAW SCATTER PLOTS AND BEST-FITTING LINES WARM UP 1. Find the change in x and the change in y between each pair of points. A. (10, 17) and (11.5, 13.5) B. (0, 3/10) and (-1, 2/5) 2. Evaluate each function for the given values. 4 A. π(π₯) = 3 π₯ β 2 for x = -3, 0, ½ B. π(π₯) = 3(2 β π₯) for x = 0, 1/6, a MODELING REAL-WORLD DATA Example: Transportation Jacksonville, Florida has an elevation of 12 ft above sea level. A hot-air balloon taking off from Jacksonville rises at a rate of 50 ft/min. Write an equation to model the balloonβs elevation as a function of time. Interpret the intercept at which the graph intersects the vertical axis. Example: Science A candle is 6 in. tall after burning for 1 hr. After 3 hr., it is 5 ½ in. tall. Write a linear equation to model the height y of the candle after burning x hours. PREDICTING WITH LINEAR MODELS (V) Scatter Plot β (V) Correlation β (V) Correlation Coefficient - (V) Trend Line (aka Line of Best Fit) Example: Winter Olympics The table shows the numbers of countries that participated in the Winter Olympics from 1980 to 2014. 1980 37 1984 49 1988 57 1992 64 1994 67 1998 72 A. Create a scatter plot of the data pairs. Use the graph to the right. B. Describe the correlation between the two variables. C. Write an equation that approximates the best fitting line and use it to predict the number of participating countries in 2018. ACTIVITY LAB β FINDING A LINE OF BEST FIT HANDOUT 2002 77 2006 80 2010 82 2014 88 ADVANCED ALGEBRA II CHAPTER 2 β LINEAR EQUATIONS AND FUNCTIONS 2.7 USE ABSOLUTE VALUE FUNCTIONS AND TRANSFORMATIONS WARM UP Graph each equation for the given domain and range. 1. π¦ = π₯ for real numbers x and π¦ β₯ 0 2. π¦ = 2π₯ β 4 for real numbers x and π¦ β₯ 0 3. π¦ = βπ₯ + 6 for π₯ β€ 5 GRAPHING ABSOLUTE VALUE FUNCTIONS (V) Absolute Value Function β Parent Function for Absolute Value Function: Translations: Stretches, Shrinks, and Reflections: Graph a Function of the Form y = | x β h | + k: ADVANCED ALGEBRA II CHAPTER 2 β LINEAR EQUATIONS AND FUNCTIONS EXTENSION β PIECEWISE FUNCTIONS Example: Writing a Piecewise Function Write a piecewise function to represent the graph on the right. Example: Graphing a Piecewise Function Graph the function π(π₯) = [π₯] (The Greatest Integer Function) Example: Graphing a Piecewise Function 2π₯ β 1, π₯ < 0 Graph π(π₯) = {β2, 0 β€ π₯ < 3 1 π₯ β 1, π₯ > 3 2 Example: Evaluating a Piecewise Function 1 β π₯ β 1, π₯ < 2 If π(π₯) = { 2 , find π(β1), π(2), πππ π(4) 3π₯ β 7, π₯ > 2 ADVANCED ALGEBRA II CHAPTER 2 β LINEAR EQUATIONS AND FUNCTIONS 2.8 GRAPH LINEAR INEQUALITIES IN TWO VARIABLES WARM UP 1. Solve each inequality. Graph the solution on a number line and write in interval notation. A. 12π β€ 15 B. 5 β 2π‘ β₯ 11 2. Solve and graph each absolute value equation or inequality. Then write in interval notation. A. |4π| = 18 B. |5 β 2π| = β2 GRAPHING LINEAR INEQUALITIES (V) Linear Inequality (in two variables) - C. 2|3β β 6| β₯ 10 Example: Graphing a Linear Inequality 1 Graph the inequality π¦ < π₯ β 3 2 Example: At least 35 performers of the Big Tent Circus are in the grand finale. Some pile into cars, while others balance on bicycles. Seven performers are in each car, and five performers are on each bicycle. Draw a graph showing all the combinations of cars and bicycles possible for the finale. GRAPHING TWO-VARIABLE ABSOLUTE VALUE INEQUALITIES Example: Graphing Absolute Value Inequalities A. π¦ β€ |π₯ β 4| + 5 B. β π¦ + 3 > |2π₯ β 1| Example: Writing Inequalities Write an inequality for each graph. A. B. ADVANCED ALGEBRA II CHAPTER 2 β LINEAR EQUATIONS AND FUNCTIONS CHAPTER 2 REVIEW In preparation for the Chapter 2 Examination, you may do the following things: 1. Study / Review Notes 2. Chapter 2 Review: p. 141 - 144 # 1 β 34 3. Chapter 2 Test: p. 145 # 1 β 27 4. Chapter 1 Extra Practice: p. 1011 # 1 β 42 Supplemental Notes / Problems: