Presentation on theme: "Linear Functions and Relations"— Presentation transcript:
1Linear Functions and Relations Chapter 2Linear Functionsand Relations
2In Chapter 2, You Will…Move from simplifying variable expressions and solving one-step equations and inequalities to working with two variable equations and inequalities.Learn how to represent function relationships by writing and graphing linear equations and inequalities.By graphing data and trend lines, you will understand how the slope of a line can be interpreted in real-world situations.
32-1 Relations and Functions What You’ll Learn …To graph relations.To identify functions.
4Example 1 Graphing a Relation A relation is aset of pairs ofinput andoutput numbers.Example 1 Graphing a Relation[(-2,4), (3,-2), (-1,0), (1,5)][(0,4),(-2,3),(-1,3),(-2,2),(1,-3)]
5Finding Domain and Range (2,4),(3,4.5),(4,7.5),(5,7),(6,5),(6,7.5)D= _____________R= _____________The domain of a relation is the set of all inputs, or x-coordinates of the ordered pairs.The range of a relation is the set of all outputs, or y-coordinates of the ordered pairs.D= _____________R= _____________
6Using a Mapping (-2,-1) (-1,-1) (-2,1) (6,3) -2 -1 Another way to show arelation is to use amapping diagram, whichlinks elements of thedomain with correspondingelements of the range.-1163
7Example 3 Making a Mapping Diagram (0,2) (1,3) (2,4)(2,8) (-1,5) (0,8) (-1,3) (-2,3)
8Are the x's different?A function is a relation that assigns exactly one value in the range to each value in the domainXY1-36-29-13XY-4-13
10One way you can tell whether a relation is a function is to analyze the graph of the relation using the vertical line test. If any vertical line passes through more than one point of the graph, the relation is NOT a function.
12Function Notation Another way to write a function y = 3x + 4 is f(x)= 3x + 4.You read f(x) as “f of x” or“f is a function of x”.The notations g(x) and h(x) also indicatefunctions of x.
13Evaluating FunctionsFunction RuleA function rule is an equation that describes a function. You can think of a function rule as an input-output machineInputOutput
14Evaluating a Function Function Rule y = 3x + 4 3x + 4 X Y 1 2 3 x y InputOutputXY1233x + 4xy
15Evaluating a Function Rule f(n)= -3n – 10Find f(6).g(x) = -2x² + 7Find g(6).
16Example 6 Real World Connection The area of a square tile is a function of the length of a side of the square. Write a function rule for the area of a square. Evaluate the function for a square tile with side length 3.5 in.
177-6 Function Operations2.01 Use the composition of functions to model and solve problems; justify results.What you’ll learn …To add, subtract, multiply and divide functionsTo find the composite of two numbers
19Example 1 Adding and Subtracting Functions Let f(x) = 3x +8 and g(x) = 2x-12.Find f+g and f - g and their domain.Let f(x) = 5x2 - 4x and g(x) = 5x+1.Find f+g and f - g and their domain.
20Example 2 Multiplying and Dividing Functions Let f(x) = x2 - 1 and g(x) = x+1.Find fg and and their domain.fgLet f(x) = 6x2 +7x - 5 and g(x) = 2x-1.Find fg and and their domain.fg
21Composition of Functions The composition of function g with function f iswritten as g°f and is defined as (g°f)(x)= g(f(x)),where the domain of g°f consists of the values a inthe domain of f such that f(a) is in the domain of g.(g°f)(x) = g( f(x) )Evaluate the inner functionf(x) first.2. Then use your answer asthe input of the outer function g(x).
22Example 3 Composition of Functions Let f(x) = x-2 and g(x) = x2. Find (g°f)(-5).Let f(x) = x-2 and g(x) = x2. Find (f°g)(x) and evaluate (f°g)(-5).
23Example 4a Real World Connection Suppose you are shopping in the store in the photo. You have a coupon worth $5 off any item.Use functions to model discounting an item by 20% and to model applying the coupon.Use a composition of your two functions to model how much you would pay for an item if the clerk applies the discount first and then the coupon.Use a composition of your two functions to model how much you would pay for an item if the clerk applies the coupon first and then the discount.How much more is any item if the clerk applies the coupon first?
24Example 4b Real World Connection A store is offering a 10% discount on all items. In addition, employees get a 25% discount.Write a composite function to model taking the 10% discount first.Write a composite function to model taking the 25% discount first.Suppose you are an employee. Which discount would you prefer to take first?
257-7 Inverse Relations and Functions 2.01 Use the composition of functions to model and solve problems; justify results.What you’ll learn …To find the inverse of a relation or function.
26The Inverse of a Function If a relation maps element a of its domain to element b of its range., the inverse relation “undoes” the relation and maps b back to a.Relation r Inverse of r220.127.116.11.18.104.22.168.912
27Example 1 Finding the Inverse of a Relation Find the inverse of relation s.Graph s and its inverse.x-11y234
28Example 2 Interchanging x and y Find the inverse of y = x2 + 3.Does y = x2 + 3 define a function? Is its inverse a function? Explain.Find the inverse of y = 3x - 10.Is its inverse a function? Explain.
29Example 3 Graphing a Relation and Its Inverse Graph y= x2 + 3 and its inverse, y = +√x -3 .Graph y= 3x-10 and its inverse.
30The inverse of a function is denoted by f-1 The inverse of a function is denoted by f-1. Read f-1 as “the inverse of f” or as “f inverse”. The notation f(x) is used for functions, but f-1(x) may be a relation that is not a function.
31Example 4a Finding an Inverse Function Consider the function f(x) = √x+1.Find the domain and range of f.Find f-1.Find the domain and range of f-1.Is f-1 a function? Explain.
32Example 4b Finding an Inverse Function Consider the function f(x) = 10 – 3x.Find the domain and range of f.Find f-1.Find the domain and range of f-1.Find f-1(f(3)).Find f-1(f(2)).
33Composite of Inverse Functions If f and f-1 are inverse functions then,(f-1°f)(x) and (f°f-1)(x) = x.
352-2 Linear Equations What you’ll learn … To graph linear equations. To write equations of lines.
36Graphing Linear Equations A function whose graph is a line is a linear function. You can represent a linear function with a linear equation, such as y=3x+2. A solution is any ordered pair (x,y) that makes the equation true.Because the y depends on the value of x, y is called the dependent variable and the x is called the independent variable.
37Example 1 Graphing a Linear Equation Graph the equations using a table.y=-3xy=½x+3xyxy
38The y intercept of a line is the point in which the line crosses the y-axis. The x intercept of a line is the point in which the line crosses the x-axis.The standard form of a linear equation is Ax +By = C, where A,B and C are real numbers and A and B are not both zero.
39Example 2 Real World Connection The equation 3x +2y =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. Describe the domain and range. Explain what the x and y intercepts represent.
40SlopeThe slope of a non-vertical line is the ratio of the vertical change to a corresponding horizontal change.You can calculate the slope by subtracting the corresponding coordinates of two points on the line.
42Example 3 Finding Slope Find the slope of the line through the points (3,2) and (-9,6).Find the slope of the linethrough the points(5,2) and (-6,2).
43Point-Slope FormWhen you know the slope and a point on a line, you can use the point-slope form to write an equation of the line.y – y1 = m (x - x1)
44Example 4 Writing an Equation Given the Slope and a Point Write in standard form an equation of the line with slope -½ through the point (8,-1).Write in standard form an equation of the line with slope 2 through the point (4, -2).
45Example 5 Writing an Equation Given Two Points Write in point slope form an equation of the line through (1,5) and (4,-1).Write in point slope form an equation of the line through (-2,-1) and (-10,17).
46Slope Intercept FormAnother form of the equation of a line is slope intercept form, which you can use to find the slope by examining the equation.y= mx +bSlopey intercept
47Example 6 Finding Slope Using Slope-Intercept Form Find the slope of 4x + 3y = 7.Find the slope of ½x + ¾y = 1
48Summary: Equations of Lines Point SlopeFormy-y1 = m(x-x1)y- 2= -3(x+4)Standard FormAx + By = C3x + y = -10Slope Intercept Formy = mx + by = -3x - 10
49Special Slopes Vertical Line Horizontal Line Zero Slope Undefined Slope
50Special Slopes Perpendicular Lines Parallel Lines Have same slopes Have reciprocal slopes
51Example 7 Writing an Equation of a Perpendicular or Parallel Line Write an equationperpendicular to y=5x-3And through the point (-1,3).Write an equationparallel to y=2/3x+5/8and through the point (2,1).
522-4 Using Linear Models What you’ll learn … 2.04 Create and use best-fit mathematical models of linear, exponential, and quadratic functions to solve problems involving sets of data.a) Interpret the constants, coefficients, and bases in the context of the data.b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.What you’ll learn …To write linear equations that model real-world data.To make predictions from linear models.
53Example 1a Real World Connection Jacksonville, Florida has anelevation of 12 ft above sealevel. A hot air balloontaking off from Jacksonvillerises 50 ft/min. Write anequation to model theballoon’s elevation as afunction of time.Graph the equation.
54Example 1b Real World Connection Suppose a balloon begins descending at a rate of 20 ft/min from an elevation of 1350 ft. Write an equation to model the balloon’s elevation as a function of time.What is true about the slope of this line?Graph the equation.Interpret the h-intercept.
55Example 2a Real World Connection A candle is 6 inches tallafter burning for 1hour.After 3 hours, it is 5½inches tall. Write alinear equation to model theheight y of the candle afterburning x hours.What does the sloperepresent ? The y intercept?Graph the equation.
56Example 2b Real World Connection Another candle is 7inches tall after burning for1hour. After 2 hours, it is 5inches tall. Write a linearequation to model theheight y of the candle afterburning x hours.How tall will the candlebe after burning 11 hours?What was the originalheight of the candle?When will the candleburn out?
57Scatter PlotsA 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 of a scatter plot. You can use a trend line to make predictions.
59Calculator Steps Enter data into lists. Turn on Stat Plot. Zoom 9 Stat EditTurn on Stat Plot.2nd y =Zoom 9Turn on Diagnostic2nd ZeroFind the trend line.Stat Calc 4
60Trend Line Make a scatter plot of the data. Draw a trend line. FatCalories626772601022019388204302755036633Make a scatter plot of the data.Draw a trend line.Estimate the number of calories in a fast-food item that has 14g of fat.
61Trend Line Make a scatter plot of the data on the calculator. Draw a trend line.Predict the wingspan of a hawk that is 28 inches long.HawkLengthWingspanCooper’s2136Crane41Gray1838Harris’s2446Roadside1631Broad-winged1939Short-tailed1735Swanson’s’
622-5 Absolute Value Functions and Graphs 2.08 Use equations and inequalities with absolute value to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.What you’ll learn …To graph absolute value functions.
63Graphing Absolute Value Functions A function of the form f(x) = mx+b +c, where m≠0, is an absolute value function.The vertex of a function is a point where the function reaches a maximum or minimum.
64Example 1 Graphing an Absolute Value Function Graph y= 3x+12Graph y= - x
65Example 2 Using a Graphing Calculator Graph y= 3 - ½xVertex _________Graph y= - 3x+4 +6Vertex __________
66Example 4 Real World Connection Suppose you pass the Betsy Ross House halfway along your trip to school each morning. You walk at a rate of one city block per minute. Sketch a graph of your trip to school based on your distance and time from the Betsy Ross House.Blocks from Ross HouseMinutes before arrivalMinutes after departure
67Example 4 Real World Connection Suppose you ride your bicycle to school at a rate of three city blocks per minute. How would the graph of your trip to school change? Sketch a new graph.Blocks from Ross HouseMinutes before arrivalMinutes after departure
682-6 Vertical and Horizontal Translations 2.08 Use equations and inequalities with absolute value to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.What you’ll learn …To analyze vertical translations.To analyze horizontal translations.
69Translating Graphs Vertically A translation is an operation that shifts a graph horizontally, vertically or both. It results in a graph of the same slope and size, in a different position.Vertical TranslationHorizontal Translation
70Example 1 Comparing Graphs Compare the graphsy = x and y = x -3y = x and y = x +5
71Rules of TranslationsGiven f(x), then –f(x) is a reflection about the x axis.Given f(x), then f(x) + k moves up k units.Given f(x), then f(x) - k moves down k units.Given f(x), then f(x+k) moves left k units.Given f(x), then f(x-k) moves right k units.
72A family of functions is a group of functions with common characteristics. A parent function is the simplest function with these characteristics. A parent function and one or more translations make up a family of functions.
73Example 2 Graphing a Vertical Translation For each function, identify the parent function andthe value of k. Then graph the function by translatingthe parent function.y = x – 1y = 3x + 5y = x – 3y = - x +2y = xy = - x
74Example 3a Writing Equations for Vertical Translations Write an equation for each translation.y = 2x, 4 units down.y = 3x , 2 units down.
75Example 3b Writing Equations for Vertical Translations Write an equation for each translation given y = x .
76Example 4 Graphing a Horizontal Translation For each function, identify the parent function andthe value of k. Then graph the function by translatingthe parent function.y = x + 3y = - x - 2y = x - 1y = - x + ¾
77Example 5 Writing Equations for Horizontal Translations Write an equation for each translation given y = x .
78Example 6 Real World Connection Describe a possible translation of Figures A and B in the Nigerian textile design below.AB
792-7 Two- Variable Inequalities 2.08 Use equations and inequalities with absolute value to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.What you’ll learn …To graph linear inequalities.To graph absolute value inequlaities.
80Graphing Linear Inequalities A linear inequality is an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line.To graph a linear inequality, first graph the boundary line.Then decide which side of the line contains solutions to the inequality and whether the boundary line is included.
81Example 1a Graphing an Inequality y > 2x + 3m= ____b = ____
83Example 2 Real World Connection At least 35 performers of the Big Tent Circus are in thegrand finale. Some pile into cars, while others balance onbicycles. Seven performers are in each car, and fiveperformers are on each bicycle. Draw a graph showing allthe possible combination of cars and bicycles that could beuse in the finale.
84Example 3 Graphing Absolute Value Inequalities y ≤ x –-y + 3 > x + 1
85Example 4 Writing Inequalities Write an inequality for each graph. The boundary line is given.
86In Chapter 2, You Should Have Moved from simplifying variable expressions and solving one-step equations and inequalities to working with two variable equations and inequalities.Learned how to represent function relationships by writing and graphing linear equations and inequalities.By graphing data and trend lines, you should understand how the slope of a line can be interpreted in real-world situations.