Mathematics
Graphs of Common Functions
Graphs of common functions represent the relationship between input and output values. Common functions include linear, quadratic, cubic, square root, absolute value, and exponential functions. Each type of function has a distinct shape when graphed, allowing for visual representation of their behavior and characteristics. These graphs are essential for understanding and analyzing mathematical relationships and real-world phenomena.
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7 Key excerpts on "Graphs of Common Functions"
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
chapter 3 Functions and Their Graphs Functions lie at the heart of modern mathematics. We begin this chapter Euclid explaining geometry (from The School of Athens, painted by Raphael around 1510). by introducing the notion of a function. Three key objects associated with each function are its graph, domain, and range. In the second section of this chapter, we will see how algebraic transformations of a function change these three key objects. The third section of this chapter deals with the composition of functions. As we will see, the operation of composition allows us to express complicated functions in terms of simpler functions. Inverse functions and their graphs become the center of attention in the last two sections of this chapter. Inverse functions will be key tools later in this book, for example in our treatment of roots and logarithms. 117 118 chapter 3 Functions and Their Graphs 3.1 Functions learning objectives By the end of this section you should be able to evaluate functions defined by formulas; work with graphical as well as algebraic representations of functions; apply the vertical line test to determine if a curve is the graph of some function; determine the domain and range of a function, either algebraically or from a graph; work with functions defined by tables. Definition and Examples Functions and their domains A function associates every number in some set of real numbers, called the domain of the function, with exactly one real number. We usually denote functions by letters such as f , g, and h. If f is a Although we do not need to do so in this book, functions can be defined more generally to deal with objects other than real numbers. function and x is a number in the domain of f , then the number that f associates with x is denoted by f (x) and is called the value of f at x. example 1 Suppose a function f is defined by the formula f (x) = x 2 for every real number x.
eBook - PDF- Cynthia Y. Young(Author)
- 2018(Publication Date)
- Wiley(Publisher)
We will discuss common functions, the domain and range of functions, and graphs of functions. We will determine whether a function is increasing, decreasing, or constant on an interval and calculate the average rate of change of a function. We will perform operations on functions and the composition of functions. Finally, we will discuss one-to-one functions and inverse functions. FUNCTIONS AND THEIR GRAPHS 1.1 FUNCTIONS 1.2 GRAPHS OF FUNCTIONS 1.3 GRAPHING TECHNIQUES: TRANSFORMATIONS 1.4 COMBINING FUNCTIONS 1.5 ONE-TO-ONE FUNCTIONS AND INVERSE FUNCTIONS • Definition of a Function • Functions Defined by Equations • Function Notation • Domain of a Function • Recognizing and Classifying Functions • Increasing and Decreasing Functions • Average Rate of Change • Piecewise- Defined Functions • Horizontal and Vertical Shifts • Reflection about the Axes • Stretching and Compressing • Adding, Subtracting, Multiplying, and Dividing Functions • Composition of Functions • One-to-One Functions • Inverse Functions • Graphical Interpretation of Inverse Functions • Finding the Inverse Function 102 CHAPTER 1 Functions and Their Graphs 1.1.1 Definition of a Function What do the following pairs have in common? ■ ■ Every person has a blood type. ■ ■ Temperature is some typical value at a particular time of day. ■ ■ Every working household phone in the United States has a 10-digit phone number. ■ ■ First-class postage rates correspond to the weight of a letter. ■ ■ Certain times of the day are start times for sporting events at a university. They all describe a particular correspondence between two groups. A relation is a correspondence between two sets. The first set is called the domain and the corresponding second set is called the range. Members of these sets are called elements. 1.1.1 SKILL Determine whether a relation is a function. 1.1.1 CONCEPTUAL Understand that all functions are relations, but not all relations are functions.
eBook - PDFMathematics NQF2 SB
TVET FIRST
- M Van Rensburg, I Mapaling A Thorne(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
58 Module 3 Topic 2: Functions and algebra Graphs of functions Module 3 Learning Outcomes This module will show you how to do the following: • Unit 3.1: Generate graphs by means of point-by-point plotting using, or supported by, available technology. • Unit 3.2: Define functions. • Unit 3.2: Identify characteristics of functions. • Units 3.3 to 3.7: Generalise the effects of the parameters a and q on the generated graphs of functions. • Units 3.3 to 3.7: Use the generated graphs to make and test conjectures. • Units 3.3 to 3.7: Sketch graphs and find equations of graphs for certain functions. Unit 3.1: Introduction to graphs A graph is a useful way to represent data visually and it enables us to easily see the relationship between the variables we are considering. A graph is drawn on the Cartesian plane , which is also known as a coordinate plane. –5 –4 –3 –2 –1 1 2 3 4 5 5 4 3 2 1 –1 –2 –3 –4 –5 Quadrant I Quadrant II Quadrant IV Quadrant III y x 0 – y – x Figure 3.1: The Cartesian plane This plane consists of a horizontal and vertical number line, with a positive and negative section that cross each other at zero. This point of intersection is called the origin . When the two axes cross each other, they form four quadrants , as shown in Figure 3.1. These are numbered I, II, III and IV in an anti-clockwise direction. The independent variable (usually x ) is plotted on the horizontal axis and the dependent variable (usually y ) is plotted on the vertical axis. It is important that the units of each axis are spaced at equal distances and marked off according to a scale when plotting graphs to ensure accuracy.
eBook - PDFMathematics N4 Student's Book
TVET FIRST
- SA Chuturgoon(Author)
- 2022(Publication Date)
- Macmillan(Publisher)
4 Module 109 Functions and graphs TVET FIRST Functions and graphs Overview of Module 4 Graphs are used as important tools to visually illustrate relationships between two variables with one variable being dependent on the other. For example, in Electrical Engineering alternating voltages and currents are represented by sinusoidal waves. In a purely inductive circuit, the current lags the voltage by 90°. These waveforms (graphs) help us to draw vector diagrams. In this module, we will learn how to represent different functions and relations on the Cartesian plane and determine the various characteristics of these functions and relations. Note The full Learning Outcomes for each module are listed in the table at the back of the book. Functions and graphs Sketching of algebraic graphs: ● Straight line ● Circle ● Semicircle ● Ellipse ● Rectangular hyperbola ● Non-rectangular hyperbola ● Exponential graph ● Logarithmic graph ● Parabola 4.2 Algebraic graphs Domain and range Relations and functions Dependent and independent variables Inverse functions and relations Continuous and discontinuous graphs Symmetry 4.1 Introduction to functions and graphs Sketching of trigonometric graphs: ● Sine ( y = a sin (bx + c) + d) ● Cosine ( y = a cos (bx + c) + d) ● Tangent ( y = a tan (bx + c) + d) ● Secant ( y = sec x) ● Cosecant ( y = cosec x) ● Cotangent ( y = cot x) 4.3 Trigonometric graphs Introduction to the cubic function: y = ax 3 + bx 2 + cx + d The factor theorem 4.4 The cubic function Figure 4.1: Graphs illustrate the relationship between two variables Starter activity Discuss the following in groups: 1. What do you think is the difference between a function and a relation? 2. Can you think of a method that you can use to establish whether a particular graph represents a function or a relation? 3. If tan 90° and tan 270° are undefined, can we include these values as part of the domain used to sketch the tan graph?
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
Commonly Used Parent Functions The graphs below represent the most commonly used functions in algebra. Familiarity with the characteristics of these graphs will help you analyze more complicated graphs obtained from these graphs by the transformations studied in the next section. Figure 2.26 x - 3 - 4 - 5 - 4 - 2 2 1 4 3 5 4 - 1 6 1 - 3 - 5 - 6 y 6 3 y = 2x + 3 y = - x + 4 (a) Identity function (b) Absolute value function (c) Square root function (d) Squaring function (e) Cubic function (f) Reciprocal function x 1 1 - 1 - 1 - 2 - 2 2 y 2 f (x) = x x 1 2 1 - 1 - 1 - 2 - 2 2 y f (x) = | x | x 1 3 1 2 3 y 2 f (x) = x x 1 1 - 1 - 2 2 3 4 y 2 f (x) = x 2 x 1 2 1 - 1 - 1 - 2 - 2 2 y f (x) = x 3 x 1 2 3 1 3 2 y f (x) = 1 x Summarize (Section 2.4) 1. Explain how to identify and graph linear and squaring functions (pages 198 and 199). For an example involving a linear function, see Example 1. 2. Explain how to identify and graph cubic, square root, and reciprocal functions (page 200). 3. Explain how to identify and graph step and other piecewise-defined functions (page 201). For examples involving these functions, see Examples 2 and 3. 4. Identify and sketch the graphs of parent functions (page 202). 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. 2.4 A Library of Parent Functions 203 2.4 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. GO DIGITAL Vocabulary and Concept Check In Exercises 1–9, write the most specific name of the function. 1. f (x) = ⟨x⟩ 2. f (x) = x 3. f (x) = 1x 4. f (x) = x 2 5.
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 - PDF- David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
— P rofessor Donald E. Knuth (one of the world’s preeminent computer scientists) in Mathematical People (Boston: Birkhäuser, 1985) When the domain and range of a function are sets of real numbers, we can graph the function in the same way that we graphed equations in Chapters 1 and 2. In graphing functions, the usual practice is to reserve the horizontal axis for the independent vari-able (the inputs) and the vertical axis for the dependent variable (the outputs). In terms of the familiar x -y coordinate system, for each x -coordinate or input in the domain of the function, the function or rule tells you the corresponding y -coordinate or output in the range of the function. These ideas are summarized in the following definition. 3.2 The Graph of a Function 141 Definition Graph of a Function The graph of a function f in the x -y plane consists of those points ( x , y ) such that x is in the domain of f and y f ( x ). See Figure 1. y x Graph of f {a, f(a)} f(a) a EXAMPLE 1 Using the Definition of the Graph of a Function In Figures 2(a) and 2(b), specify the y -coordinates of the points P and Q , respec-tively. In each case, give an exact expression and also a calculator approximation rounded to two decimal places. (We’ll look at the graphs of these two functions in more detail later in this section.) Figure 2 What are the y -coordinates of the points P and Q ? (a) ƒ=œx „ (b) ©=˛ P 2 ƒ=œ„ x x y ©=˛ x y Q _ œ2 „ 3.2 Figure 1 Copyright 201 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.
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