Mathematics
Linear Functions
Linear functions are mathematical relationships between two variables that can be represented by a straight line on a graph. They have a constant rate of change, meaning that as one variable increases by a certain amount, the other variable changes by a consistent amount. The general form of a linear function is y = mx + b, where m is the slope and b is the y-intercept.
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eBook - PDFPractical Algebra
A Self-Teaching Guide
- Bobson Wong, Larisa Bukalov, Steve Slavin(Authors)
- 2022(Publication Date)
- Jossey-Bass(Publisher)
A formula enables us to substitute any value of x and immediately find the value of y . The verbal descriptions of the patterns above are usually written as the equations y = 2 x + 1, y = 2 x − 2, and y = − 3 x + 1, respectively. When we substitute many x -values into each equation and find their corresponding y -values, we can graph each equation. Let’s compare the tables, equations, and graphs in Figure 7.3: –10 –10 10 –5 5 –5 0 5 10 x y –10 –10 10 –5 5 –5 0 5 10 x y –10 –10 10 –5 5 –5 0 5 10 x y Table x y 0 1 1 3 2 5 3 7 x y 0 –2 1 0 2 2 3 4 x y 0 1 1 –2 2 –5 3 –8 Equation y = 2 x + 1 y = 2 x – 2 Graph y = –3 x + 1 Figure 7.3 Three views of a function. These examples show the following: • Points with a constant difference between y -values lie on a line. That’s why func-tions whose points have a constant difference between y -values are called Linear Functions . • Some lines are steeper than others. They can tilt upward or downward from left to right. Linear Functions AND THEIR GRAPHS 175 • Lines can have the same amount of steepness but pass through different points (parallel lines). • When we describe lines using equations such as y = 2 x + 1 or y = − 3 x + 1, the first number is related to the line’s steepness and the second is related to a point on the line. We describe the steepness of a line using the word slope . The slope of a line is the ratio of the amount of vertical change (or change in y ) to the amount of horizontal change (or change in x ). By convention, mathematicians use the letter m to represent slope. We express slope using the formula m = vertical change horizontal change . Slope expresses the rate of change of a linear function. Since a linear function has a constant rate of change, we interpret the slope of a line as follows: as x increases by units, y increases (or decreases) by units . Here are some examples: • A slope of − 2 means that as x increases by 1 unit, y decreases by 2 units.
eBook - PDFPrecalculus
A Prelude to Calculus
- Sheldon Axler(Author)
- 2016(Publication Date)
- Wiley(Publisher)
solution In this case the equation above becomes Line containing (2, 4) and (5, 1). y − 4 = 1 − 4 5 − 2 ( x − 2). Solving this equation for y, we get y = −x + 6. check If we take x = 2 in the equation above, we get y = 4, and if we take x = 5 in the equation above, we get y = 1. Thus the points (2, 4) and (5, 1) are indeed on this line. Section 2.1 Lines and Linear Functions 123 We have seen that a line in the xy-plane with slope m is characterized by the equation y = mx + b, where b is some number. To restate this conclusion in terms of functions, let f be the function defined by f ( x) = mx + b, where m and b are numbers. Then the graph of f is a line with slope m. Functions of this form are so important that they have a name—Linear Functions. Although we gave this definition in Section 1.4, it is sufficiently important to be worth repeating here. Linear function A linear function is a function f of the form f ( x) = mx + b, where m and b are numbers. Conversion between different units of measurement is usually done by a linear Differential calculus focuses on approximating an arbitrary function on a small part of its domain by a linear function. function, as shown in the next two examples. Example 4 A pound is officially defined to be exactly 0.45359237 kilograms. (a) Find a function f such that f ( p) is the weight in kilograms of an object weighing p pounds. (b) What is the interpretation of the inverse function f −1 ? (c) Find a formula for the inverse function f −1 . solution (a) Start with the equation 1 pound = 0.45359237 kilograms. Multiply both sides by p, getting p pounds = 0.45359237 p kilograms. Thus the desired function is given by the formula f ( p) = 0.45359237 p. (b) The function f converts pounds to kilograms. Thus the inverse function f −1 converts kilograms to pounds. Specifically, f −1 (k) is the weight in pounds of an object weighing k kilograms. (c) Solving the equation k = 0.45359237 p for p, we get p = k 0.45359237 .
eBook - PDF- Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Represent each relationship with an equation. 106 CHAPTER 2 Rates of Change and Linear Functions Horizontal and Vertical Lines The slope, m, of any horizontal line is 0. So the general form for the equation of a horizontal line is y b x 0 = + or just y b = EXAMPLE 4 A Horizontal Line a. Construct a table (with both positive and negative values for x) and corresponding graph for the line y = 1. b. Is y a function of x? c. Generate the slope of the line using two points from the table. Solution a. Table 2.10 and Figure 2.33 show points that satisfy the equation y = 1. 6. Write a formula to describe each situation. a. y is directly proportional to x, and y is 4 when x is 12. b. d is directly proportional to t, and d is 300 when t is 50. 7. Write a formula to describe the following: a. The diameter, d, of a circle is directly proportional to the cir- cumference, C. b. The amount of income tax paid, T, is directly proportional to income, I. c. The tip amount t, is directly proportional to the cost of the meal, c. 8. Assume that a is directly proportional to b. When a = 10, b = 15. a. Find a if b is 6. b. Find b if a is 4. 3. In January 2010 the exchange rate was $1.00 U.S. to 0.70 euros, the common European currency. a. Find a linear function that converts U.S. dollars to euros. b. Find a linear function that converts U.S. dollars to euros with a service fee of $2.50. c. Which function represents a directly proportional relationship and why? 4. Suppose you go on a road trip, driving at a constant speed of 60 miles per hour. Create an equation relating distance d in miles and time traveled t in hours. Does it represent direct proportionality? What happens to d if the value for t doubles? If t triples? 5. The total cost C for football tickets is directly proportional to the number of tickets purchased, N. If two tickets cost $50, construct the formula relating C and N.
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Chapter 6 Functions and Graphs Harnessing this new power [of computer technology] within mathematics and school mathematics is the challenge for the 21st century. (RS/JMC, 1997, p. 6) STRAIGHT-LINE GRAPHS Straight-line graphs were discussed in Chapter 3 as one of a number of ways of introducing algebraic ideas and symbols. They are particularly attractive in this respect because they provide a ready link between numbers, symbols and pictures. An equation provides a way of encapsulating the patterns in the co-ordinates of a set of points that lie on a straight line by acting as a unique label which highlights key properties. Although a graph is an abstract representation it has a visual appeal and looks interesting, particularly when a family of related graphs is depicted. Students need to understand the links between the equation, the table of values or set of co-ordinates and the graph, and to be able to move fluently between these different representa-tions. In Chapter 3 it was suggested that introductory work on straight-line graphs should be confined to positive whole numbers and should begin by looking at a set of points on a straight line, using the pattern in the numbers to determine the equation of the line. This builds on the idea of representing the terms of a linear sequence algebraically and makes clear from the start where the equation comes from and what it means. Text books often start with equations and show students how to produce a table of values and then plot the corresponding lines. Whilst this may seem simpler as it is a more routine task, it starts from something that is unfamiliar, namely the equation, which can set up an immediate barrier because it looks strange and new and seems to have appeared for no apparent reason. Co-ordinates and their graphical representation should already be familiar and therefore provide a more reassuring start to a new idea.
eBook - PDFAlgebra
Form and Function
- William G. McCallum, Eric Connally, Deborah Hughes-Hallett(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
. . . . . . . . . . . . 81 Solving Using Substitution . . . . . . . . . . . . . . . . . 81 Solving Using Elimination. . . . . . . . . . . . . . . . . . 83 Visualizing Solutions Using Graphs . . . . . . . . . . 84 Modeling with Systems of Linear Equations. . . . 87 REVIEW PROBLEMS . . . . . . . . . . . . . . . . . . . . . 91 SOLVING DRILL . . . . . . . . . . . . . . . . . . . . . . . . . 97 42 Chapter 2 Linear Functions 2.1 INTRODUCTION TO Linear Functions Linear Functions describe quantities which change at a constant rate. For instance, suppose a pizza place charges $9 for a cheese pizza, and $2 for each topping. If = () is the total cost in dollars as a function of , the number of toppings, then (0) = Cost of pizza with 0 toppings = 9 = 9 + 0 ⋅ 2 = 9 (1) = Cost of pizza with 1 topping = 9 + 2 = 9 + 1 ⋅ 2 = 11 (2) = Cost of pizza with 2 toppings = 9 + 2 + 2 ⏟ ⏟ ⏟ 2 = 9 + 2 ⋅ 2 = 13 (3) = Cost of pizza with 3 toppings = 9 + 2 + 2 + 2 ⏟⏞ ⏟⏞ ⏟ 3 = 9 + 3 ⋅ 2 = 15 () = Cost of pizza with toppings = 9 + 2 + ⋯ + 2 ⏟⏞⏞ ⏟⏞⏞ ⏟ = 9 + ⋅ 2. So cost for a pizza with toppings is = () = 9 + 2. See Table 2.1 and Figure 2.1. Notice that the values of go up by 2 each time increases by 1. This means that the points in Figure 2.1 lie on a line. Table 2.1 0 9 1 11 2 13 3 15 4 17 1 2 3 3 6 9 12 15 18 (0, 9): No toppings costs $9 (1, 11): 1 topping costs $11 (2, 13): 2 toppings cost $13 (3, 15): 3 toppings cost $15 Figure 2.1: Cost, , of pizza as a function of , number of toppings The Family of Linear Functions Functions describing quantities that have a constant rate of change are Linear Functions. In the pizza example, since the cost increases by 2 dollars per topping, the cost function, = 9 + 2, has a constant rate of change; thus the cost function belongs to the family of Linear Functions. A linear function is a function that can be written () = + , for constants and .
eBook - PDFFunctions Modeling Change
A Preparation for Calculus
- Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
. . . . . . . . . . 40 Interpretation of the Parameters of a Line � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 41 Intersection of Two Lines � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 42 Linear Inequalities � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 43 Summary for Section 1�5 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 45 1.6 Fitting Linear Functions to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Laboratory Data: The Viscosity of Motor Oil � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 49 Interpolation and Extrapolation � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 50 How Regression Works � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 51 Correlation � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 51 Summary for Section 1�6 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 52 STRENGTHEN YOUR UNDERSTANDING � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 55 Chapter 1 Linear Functions AND CHANGE 2 Chapter 1 Linear Functions AND CHANGE 1.1 FUNCTIONS AND FUNCTION NOTATION In everyday language, the word function expresses the idea of dependence. For example, we might say that election results are a function of the economy, meaning that the winner is determined by how the economy is doing. Or we might claim that car sales are a function of the weather, meaning that the number of cars sold is affected by the weather. In mathematics, the meaning of the word function is similar, but more precise. A function is a relationship between two quantities.
eBook - PDFIntermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Since the values of y can be any real number, the range is the interval 1 2` , ` 2 shown on the y -axis. Graph the function f 1 x 2 5 3 x 2 1 and find its domain and range. In Section 2.1, we graphed equations whose graphs were lines. These equations define basic functions, called Linear Functions . EXAMPLE 8 Solution a SELF CHECK 8 6 EXAMPLE 9 Solution a SELF CHECK 9 Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2.4 Introduction to Functions 137 LINEAR FUNCTION A linear function is a function defined by an equation that can be written in the form f 1 x 2 5 mx 1 b or y 5 mx 1 b where m is the slope of the line graph and 1 0, b 2 is the y -intercept. Solve the equation 3 x 1 2 y 5 10 for y to show that it defines a linear function. Then graph the function and find its domain and range. We solve the equation for y as follows: 3 x 1 2 y 5 10 2 y 5 2 3 x 1 10 Subtract 3 x from both sides. y 5 2 3 2 x 1 5 Divide both sides by 2. Because the given equation can be written in the form f 1 x 2 5 mx 1 b , it defines a linear function. The slope of its line graph is 2 3 2 , and the y -intercept is 1 0, 5 2 . The graph appears in Figure 2-40. From the graph, we can see that both the domain and the range are the interval 1 2` , ` 2 . Solve the equation 5 x 2 2 y 5 20 for y to show that it defines a linear function. A special case of a linear function is the constant function , defined by the equation f 1 x 2 5 b , where b is a constant. Its graph, domain, and range are shown in Figure 2-41.
eBook - PDFCollege Algebra
Building Skills and Modeling Situations
- Charles P. McKeague, Katherine Yoshiwara, Denny Burzynski(Authors)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
282 Chapter 4 Slope, Rates of Change, and Linear Functions This form of the equation of a line is used to find the equation of a line, either given one point on the line and the slope, or given two points on the line. Find the equation of the line with slope −2 that contains the point (−4, 3). Write the answer in slope-intercept form. SOLUTION Using (x 1 , y 1 ) = (−4, 3) and m = −2 in y − y 1 = m(x − x 1 ) Point-slope form gives us y − 3 = −2(x + 4) Note: x − (−4) = x + 4 y − 3 = −2x − 8 Multiply out right side y = −2x − 5 Add 3 to each side Figure 4 is the graph of the line that contains (−4, 3) and has a slope of −2. Notice that the y-intercept on the graph matches that of the equation we found. Find the equation of the line that passes through the points (−3, 3) and (3, −1). SOLUTION We begin by finding the slope of the line: m = 3 − (−1) ________ −3 − 3 = 4 ___ −6 = − 2 __ 3 PROPERTY Point-Slope Form of the Equation of a Line The equation of the line through (x 1 , y 1 ) with slope m is given by y − y 1 = m(x − x 1 ) EXAMPLE 4 FIGURE 4 x y 2 1 –1 –3 –4 –5 3 4 5 1 –3 –5 2 3 4 5 –2 –1 1 –1 –3 –4 –5 3 4 5 1 –3 –5 2 3 4 5 –2 –1 1 1 Slope = − 2 __ 1 = −2 (–4, 3) (0, –5) –2 y-intercept = −5 EXAMPLE 5 4.2 Linear Functions and Equations of Lines 283 Using (x 1 , y 1 ) = (3, −1) and m = − 2 _ 3 in y − y 1 = m(x − x 1 ) yields y + 1 = − 2 __ 3 (x − 3) y + 1 = − 2 __ 3 x + 2 Multiply out right side y = − 2 __ 3 x + 1 Add −1 to each side Figure 5 shows the graph of the line that passes through the points (−3, 3) and (3, −1). As you can see, the slope and y-intercept are − 2 _ 3 and 1, respectively. The last form of the equation of a line that we will consider in this section is called the standard form. It is used mainly to write equations in a form that is free of fractions and is easy to compare with other equations.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
1 9780357452080_0200 08/28/20 Finals left, © iStockPhoto.com/Dsafanda; right, © Shaunl/Getty Images Functions and Their Graphs 2 2.3 Temperature (Exercise 87, p. 196) 2.7 Diesel Mechanics (Exercise 70, p. 229) 2.1 Linear Equations in Two Variables 2.2 Functions 2.3 Analyzing Graphs of Functions 2.4 A Library of Parent Functions 2.5 Transformations of Functions 2.6 Combinations of Functions: Composite Functions 2.7 Inverse Functions GO DIGITAL Chapter 2 Section 3 Exercise 27 NEXT PREV. 17 19 21 23 25 27 29 31 33 35 37 159 Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 160 Chapter 2 Functions and Their Graphs GO DIGITAL 2.1 Linear Equations in Two Variables Use slope to graph linear equations in two variables. Find the slope of a line given two points on the line. Write linear equations in two variables. Use slope to identify parallel and perpendicular lines. Use slope and linear equations in two variables to model and solve real-life problems. Using Slope The simplest mathematical model relating two variables x and y is the linear equation y = mx + b Linear equation in two variables x and y where m and b are constants. The equation is called linear because its graph is a line. (In mathematics, the term line means straight line.) By letting x = 0, you obtain y = m(0) + b = b. So, the line crosses the y-axis at y = b, as shown in Figure 2.1. In other words, the y-intercept is (0, b). The steepness, or slope, of the line is m.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Likewise, the fact that S (3) = 36 π tells us that the ordered pair (3, 36 π ) is a member of function S . EXAMPLE 9 r EXAMPLE 10 3.6 Evaluating Functions 229 If we look at Example 10, we see that when the radius of a sphere is 3, the numerical values of the volume and surface area are equal. How unusual is this? Are there other values of r for which V ( r ) and S ( r ) are equal? We can answer this question by looking at the graphs of both V and S . To graph the function V ( r ) = 4 _ 3 πr 3 , set Y 1 = 4 π X 3 /3. To graph S ( r ) = 4 πr 2 , set Y 2 = 4 π X 2 . Graph the two functions in each of the following windows: Window 1: X from − 4 to 4, Y from − 2 to 10 Window 2: X from 0 to 4, Y from 0 to 50 Window 3: X from 0 to 4, Y from 0 to 150 Then use the Trace and Zoom features of your calculator to locate the point in the first quadrant where the two graphs intersect. How do the coordinates of this point compare with the results in Example 10? More About Example 10 USING TECHNOLOGY We conclude this section by taking another look at the equation of a line from a function perspective. Linear Functions In Section 3.3, we introduced the slope-intercept form of a line, y = mx + b . As long as the slope is defined (the line is not vertical), the graph of the line will pass the vertical line test. Therefore, every non-vertical line represents a function. This leads us to the following definition. A linear function is any function that can be expressed in the form f ( x ) = mx + b The graph of f is a line with slope m passing through the y -axis at (0, b ). linear function DEFINITION 230 CHAPTER 3 Linear Equations in Two Variables and Functions The average price per gallon for gasoline between the years 2002 and 2008 can be modeled by the linear function f ( x ) = 0.32 x + 0.76 where x is the number of years from 2000 and f ( x ) is the average price in dollars per gallon. ( Source : U.S. Department of Energy) a. Graph the function.
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