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**Linear Functions and Relations**

Chapter 2 Linear Functions and Relations

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In 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.

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**2-1 Relations and Functions**

What You’ll Learn … To graph relations. To identify functions.

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**Example 1 Graphing a Relation**

A relation is a set of pairs of input and output numbers. Example 1 Graphing a Relation [(-2,4), (3,-2), (-1,0), (1,5)] [(0,4),(-2,3),(-1,3),(-2,2),(1,-3)]

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**Finding 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= _____________

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**Using a Mapping (-2,-1) (-1,-1) (-2,1) (6,3) -2 -1**

Another way to show a relation is to use a mapping diagram, which links elements of the domain with corresponding elements of the range. -1 1 6 3

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**Example 3 Making a Mapping Diagram**

(0,2) (1,3) (2,4) (2,8) (-1,5) (0,8) (-1,3) (-2,3)

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Are the x's different? A function is a relation that assigns exactly one value in the range to each value in the domain X Y 1 -3 6 -2 9 -1 3 X Y -4 -1 3

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**Example 4 Identifying Functions**

-2 5 -1 3 4 -1 2 3 -1 3 5

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One 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.

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Which are Functions? • • • • • •

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**Function 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 indicate functions of x.

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Evaluating Functions Function Rule A function rule is an equation that describes a function. You can think of a function rule as an input-output machine Input Output

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**Evaluating a Function Function Rule y = 3x + 4 3x + 4 X Y 1 2 3 x y**

Input Output X Y 1 2 3 3x + 4 x y

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**Evaluating a Function Rule**

f(n)= -3n – 10 Find f(6). g(x) = -2x² + 7 Find g(6).

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**Example 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.

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7-6 Function Operations 2.01 Use the composition of functions to model and solve problems; justify results. What you’ll learn … To add, subtract, multiply and divide functions To find the composite of two numbers

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**Function Operations Addition (f+g)(x) = f(x)+g(x)**

Multiplication (fg)(x) = f(x) g(x) Subtraction (f-g)(x) = f(x) – g(x) Division (x)= , g(x)≠0 f g f(x) g(x)

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**Example 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.

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**Example 2 Multiplying and Dividing Functions**

Let f(x) = x2 - 1 and g(x) = x+1. Find fg and and their domain. f g Let f(x) = 6x2 +7x - 5 and g(x) = 2x-1. Find fg and and their domain. f g

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**Composition of Functions**

The composition of function g with function f is written as g°f and is defined as (g°f)(x)= g(f(x)), where the domain of g°f consists of the values a in the domain of f such that f(a) is in the domain of g. (g°f)(x) = g( f(x) ) Evaluate the inner function f(x) first. 2. Then use your answer as the input of the outer function g(x).

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**Example 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).

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**Example 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?

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**Example 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?

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**7-7 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.

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**The 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 r 1 2 1.2 1.4 1.6 1.9 1.2 1.4 1.6 1.9 1 2

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**Example 1 Finding the Inverse of a Relation**

Find the inverse of relation s. Graph s and its inverse. x -1 1 y 2 3 4

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**Example 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.

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**Example 3 Graphing a Relation and Its Inverse**

Graph y= x2 + 3 and its inverse, y = +√x -3 . Graph y= 3x-10 and its inverse.

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**The 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.

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**Example 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.

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**Example 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)).

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**Composite of Inverse Functions**

If f and f-1 are inverse functions then, (f-1°f)(x) and (f°f-1)(x) = x.

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**Example 6 Composite of Inverse Functions**

For f(x) = 5x + 11, find (f-1°f)(777). For f(x) = 5x + 11, find (f°f-1)(-5802).

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**2-2 Linear Equations What you’ll learn … To graph linear equations.**

To write equations of lines.

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**Graphing 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.

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**Example 1 Graphing a Linear Equation**

Graph the equations using a table. y=-3x y=½x+3 x y x y

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**The 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.

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**Example 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.

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Slope The 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.

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**Slope Formula = Vertical change (rise) Horizontal change (run) y2 – y1**

x2 – x1 =

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**Example 3 Finding Slope Find the slope of the line through the points**

(3,2) and (-9,6). Find the slope of the line through the points (5,2) and (-6,2).

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Point-Slope Form When 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)

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**Example 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).

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**Example 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).

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Slope Intercept Form Another 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 +b Slope y intercept

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**Example 6 Finding Slope Using Slope-Intercept Form**

Find the slope of 4x + 3y = 7. Find the slope of ½x + ¾y = 1

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**Summary: Equations of Lines**

Point Slope Form y-y1 = m(x-x1) y- 2= -3(x+4) Standard Form Ax + By = C 3x + y = -10 Slope Intercept Form y = mx + b y = -3x - 10

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**Special Slopes Vertical Line Horizontal Line Zero Slope**

Undefined Slope

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**Special Slopes Perpendicular Lines Parallel Lines Have same slopes**

Have reciprocal slopes

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**Example 7 Writing an Equation of a Perpendicular or Parallel Line**

Write an equation perpendicular to y=5x-3 And through the point (-1,3). Write an equation parallel to y=2/3x+5/8 and through the point (2,1).

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**2-4 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.

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**Example 1a Real World Connection**

Jacksonville, Florida has an elevation of 12 ft 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. Graph the equation.

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**Example 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.

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**Example 2a Real World Connection**

A candle is 6 inches tall after burning for 1hour. After 3 hours, it is 5½ inches tall. Write a linear equation to model the height y of the candle after burning x hours. What does the slope represent ? The y intercept? Graph the equation.

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**Example 2b Real World Connection**

Another candle is 7 inches tall after burning for 1hour. After 2 hours, it is 5 inches tall. Write a linear equation to model the height y of the candle after burning x hours. How tall will the candle be after burning 11 hours? What was the original height of the candle? When will the candle burn out?

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Scatter Plots 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 of a scatter plot. You can use a trend line to make predictions.

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**Correlation Strong, Positive Correlation Weak, Positive Correlation**

No Correlation Weak, Negative Correlation Strong, Negative Correlation

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**Calculator Steps Enter data into lists. Turn on Stat Plot. Zoom 9**

Stat Edit Turn on Stat Plot. 2nd y = Zoom 9 Turn on Diagnostic 2nd Zero Find the trend line. Stat Calc 4

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**Trend Line Make a scatter plot of the data. Draw a trend line.**

Fat Calories 6 267 7 260 10 220 19 388 20 430 27 550 36 633 Make 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.

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**Trend 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. Hawk Length Wingspan Cooper’s 21 36 Crane 41 Gray 18 38 Harris’s 24 46 Roadside 16 31 Broad-winged 19 39 Short-tailed 17 35 Swanson’s’

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**2-5 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.

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**Graphing 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.

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**Example 1 Graphing an Absolute Value Function**

Graph y= 3x+12 Graph y= - x

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**Example 2 Using a Graphing Calculator**

Graph y= 3 - ½x Vertex _________ Graph y= - 3x+4 +6 Vertex __________

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**Example 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 House Minutes before arrival Minutes after departure

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**Example 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 House Minutes before arrival Minutes after departure

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**2-6 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.

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**Translating 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 Translation Horizontal Translation

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**Example 1 Comparing Graphs**

Compare the graphs y = x and y = x -3 y = x and y = x +5

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Rules of Translations Given 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.

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A 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.

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**Example 2 Graphing a Vertical Translation**

For each function, identify the parent function and the value of k. Then graph the function by translating the parent function. y = x – 1 y = 3x + 5 y = x – 3 y = - x +2 y = x y = - x

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**Example 3a Writing Equations for Vertical Translations**

Write an equation for each translation. y = 2x, 4 units down. y = 3x , 2 units down.

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**Example 3b Writing Equations for Vertical Translations**

Write an equation for each translation given y = x .

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**Example 4 Graphing a Horizontal Translation**

For each function, identify the parent function and the value of k. Then graph the function by translating the parent function. y = x + 3 y = - x - 2 y = x - 1 y = - x + ¾

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**Example 5 Writing Equations for Horizontal Translations**

Write an equation for each translation given y = x .

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**Example 6 Real World Connection**

Describe a possible translation of Figures A and B in the Nigerian textile design below. A B

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**2-7 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.

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**Graphing 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.

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**Example 1a Graphing an Inequality**

y > 2x + 3 m= ____ b = ____

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**Example 1b Graphing an Inequality**

3x - 5y ≥ 10 m= ____ b = ____

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**Example 2 Real World Connection**

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 possible combination of cars and bicycles that could be use in the finale.

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**Example 3 Graphing Absolute Value Inequalities**

y ≤ x – -y + 3 > x + 1

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**Example 4 Writing Inequalities**

Write an inequality for each graph. The boundary line is given.

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**In 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.

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