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.
7 Key excerpts on "Graphs of Common Functions"
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...PART II FUNCTIONS, GRAPHS, AND LIMITS Chapter 2 Analysis of Graphs I. ANALYSIS OF GRAPHS A. Basic Functions—you need to know how to graph the following functions and any of their transformations by hand. 1. Polynomials, absolute value, square root functions 2. Trigonometric functions 3. Inverse trigonometric functions and their domain and range 4. Exponential and Natural Logarithmic functions 5. Rational functions 6. Piecewise functions 7. Circle Equations i. Upper semicircle with radius a and center at the origin:. This is a function. For example, ii. Lower semicircle with radius a and center at the origin:. This is a function. For example, iii. Circle with radius a and center at the origin: x 2 + y 2 = a 2. This is not a function since some x -values correspond to more than one y -value. For example, x 2 + y 2 = 9 iv. Circle with radius a and center at (b, c): (x – b) 2 + (y – c) 2 = a 2. This is not a function either. For example, (x – 2) 2 + (y + 3) 2 = 9 8. Summary of Basic Transformations of Functions A. Making changes to the equation of y = f (x) will result in changes in its graph. The following transformations occur most often. B. For trigonometric functions, f (x) = a sin(bx + c) + d or f (x) = a cos(bx + c) + d, a is the amplitude (half the height of the function), b is the frequency (the number of times that a full cycle occurs in a domain interval of 2 π units), is the horizontal shift and d is the vertical shift. Keep in Mind.... The reciprocal of sin(x),, is equivalent to csc(x), whereas sin –1 (x) is the inverse of sin(x), which is the reflection of sin(x) in the line y = x. When changing a function by adding a positive constant to x, the graph will shift to the left, not the right. The graph shifts to the right a units when a is subtracted from x. When graphing a function on the calculator (TI-83 or TI-84), make sure that all the plots are turned off; otherwise you risk getting an error and not being able to graph...
eBook - ePub- Thomas Pfaff(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...2 Functions and Their Graphs We define and use functions throughout this book, and in this chapter we focus on functions and their associated graphs. We provide examples of commonly used mathematical functions: piecewise, step, parametric, and polar. We broaden the use of function by using the geometric definition to obtain a parabola, discuss functions that return functions, and create a function that returns Pythagorean triples. In each case we graph the function; we keep our graphs basic leaving chapter 3 for further details on graphing, except for our last example where we create a checkerboard graph. The command for creating a function is function() {}, where the variable(s) are listed inside the parenthesis and the function is defined within the braces. In our first example, we define the function f to be x 2 sin (x) and evaluate it at 3 with f(3). There are a number of predefined functions such as abs, sqrt, the trigonometric functions, hyperbolic functions, log for the natural log, log10, log2, and the exponential function exp. So, for example, sin(x) is available to use in our definition of f. Note that * must be used for multiplication as we cannot simply juxtapose objects. R Code > f=function(x){x ^ 2*sin(x)} > f(3) [1] 1.27008 We can plot our function with curve. The first three arguments must be the function, the lower value for the independent variable, and the upper value for the dependent variable. The default range for the dependent variable is selected based on the minimum and maximum of the function on the given interval. There are numerous options, such as ylim for the y limits, lwd for the width (i.e., thickness) of the curve, col for the color of the curve, xlab and ylab for labeling the axis, and lty for the type of line (e.g., dashed, dotted). R Code > curve(f,-5,5) Our next two examples illustrate functions of two and three variables. The first returns the area of a rectangle given the length and width...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The slope of a parabola changes along the curve. Linear and quadratic functions are special types of algebraic functions. When an algebraic function takes the form of an equation, the equation is typically written in the explicit form y = … in which the right side of the equation is obtained by performing some combination of basic algebraic operations on x. These operations include addition, multiplication, division, raising to a power, and taking the n th root. LESSONS IN CHAPTER 4 • Lesson 4-1: Function Concepts • Lesson 4-2: Quadratic Functions and Their Graphs • Lesson 4-3: Solving Quadratic Equations • Lesson 4-4: Solving a Linear-Quadratic System • Lesson 4-5: Applying Quadratic Equations • Lesson 4-6: Solving Quadratic Inequalities Lesson 4-1: Function Concepts KEY IDEAS Functions arise whenever one quantity depends on another. If your grade on your next math test depends on the number of hours you study, your test grade is a function of the number of hours studied. In mathematics, however, the term function has a narrower meaning. A function is a relationship between two variables, say x and y, such that each possible value of x is associated with exactly one value of y. AN EXAMPLE OF A FUNCTION Let set X consist of five teenagers and set y consist of their possible ages: If each teenager in set X is associated with his or her present age in set Y, the result can be written as a set of ordered pairs: {(Alice, 17), (Barbara, 13), (Chris, 16), (Dennis, 19), (Enid, 15)}. Since each teenager from set X is associated with exactly one age from set Y, the set of ordered pairs is called a function. DEFINITION OF A FUNCTION A function is a set of ordered pairs in which no two ordered pairs of the form (x,y) have the same x -value but different y -values. If f = {(1,1), (2,3), (3,7), (4,5)} then f is a function...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...CHAPTER 3 Tables, Graphs, and Functions The relations between quantities, how one depends on another, are at the core of science. The main use of elementary mathematics is to describe these relations. If we are to discuss them, we must be able to display them, and it helps if we can refer to them by name. The relations can be displayed in several ways: we can list values in a table, we can draw a graph, we might be able to use a formula, or we might be content just to give a verbal description. Functions provide a sort of packaging that allows us to encapsulate certain relations and refer to them by name. Functions will be central to the material presented in most of the rest of the book. After illustrating how functions can be used with simple examples, two new functions will be introduced. Exponentials and logarithms arise frequently whenever we need to consider change, for example in descriptions of the growth of bacterial populations or the decline of drug concentrations between doses. Logarithms are part of the very definitions of pH, used to indicate H + ion concentrations, and of decibels, used to measure sound intensities. Logarithms are also employed to aid the presentation of any data that are spread over a large range of values, for example the more than 1000-fold range of concentrations often employed when measuring drug or hormone binding. 3.1 Tables Experimental results are usually reported in tables or graphs. Tables are particularly useful when the data are qualitative (for an example see Section 3.3) or when the precise values listed will be wanted for subsequent use. In addition, they are often the starting point for preparing a graph. The entries in a table are organized into rows and columns. Each column refers to a particular property stated in a column heading. Everything below the heading should be a value of that property. A row represents a group of the different properties for a particular experiment, individual, or object...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...12 FUNCTIONS AND GRAPHING WHAT YOU WILL LEARN • How to produce a table, an equation, and a graph of a linear function; • How to identify the slope of a line as positive or negative from a graph; • How to determine the slope of a linear function, given an equation not necessarily in slope-intercept form; • How to interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; • How to compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions); • A function is a rule that assigns to each input exactly one output; • About domain and range, and which is allowed to have numbers repeat; • How to use the vertical line test to determine a function; • How to determine if functions are linear or non-linear; • How to identify linear functions from their equation, table, and graph. SECTIONS IN THIS CHAPTER • What Is Coordinate Geometry? • How Do We Graph a Line from a Table of Values? • What Is Slope? • How Do We Graph a Line Using Slope? • What Is a. Function? • What Is the Rule of Four? • How Can We Tell If a Function Is Linear or Non-linear? DEFINITIONS Axis A horizontal or vertical line used in the Cartesian coordinate system to locate a point on the coordinate graph. Axes The horizontal and vertical lines dividing a coordinate plane into four quadrants. Cartesian (coordinate) plane The plane formed by a horizontal axis and a vertical axis, often labeled the x -axis and y -axis, respectively...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 5 Functions and Their Graphs CHAPTER 5 FUNCTIONS AND THEIR GRAPHS ELEMENTARY FUNCTIONS A function is any process that assigns a single value of y to each number of x. Because the value of x determines the value of y, y is called the dependent variable and x is called the independent variable. The set of all the values of x for which the function is defined is called the domain of the function. The set of corresponding values of y is called the range of the function. PROBLEM Is y 2 = x a function? SOLUTION Graph the equation. Note that x can have two values of y. Therefore, y 2 = x is not a function. PROBLEM Find the domain and range for y = 5 – x 2. SOLUTION First determine if there are any values that would make the function undefined (i.e., division by 0). There are none. Thus, the domain is the set of real numbers. The range can be found by substituting some corresponding values for x in the equation. The range is the set of real numbers less than or equal to 5. PROBLEM Evaluate f (1) for y = f (x) = 5 x + 2. SOLUTION f (x) = 5 x + 2 f (1) = 5(1) + 2 = 5 + 2 = 7 OPERATIONS ON FUNCTIONS Functions can be added, subtracted, multiplied, or divided to form new functions. a. (f + g) (x) = f (x) + g (x) b. (f – g) (x) = f (x) – g (x) c. (f × g) (x). = f (x) g (x) d. PROBLEM Let f (x) = 2 x 2 – 1 and g (x) = 5 x + 3. Determine the following functions: 1. f + g 2. f – g 3. f × g 4. SOLUTION COMPOSITE FUNCTION The composite function f ° g is defined (f ° g)(x) = f (g (x)). PROBLEM Given f (x) = 3 x and g (x) = 4 x + 2. Find (f ° g) (x) and (g ° f) (x). SOLUTION (f ° g) (x) = f (g (x)) =. 3(4 x + 2) = 12 x + 6 (g ° f) (x) = g (f (x)) = 4(3 x) + 2 = 12 x + 2 Note that (f ° g) (x) (g ° f) (x). PROBLEM Find (f ° g) (2) if f (x) = x 2 – 3 and g (x) = 3 x + 1. SOLUTION Substitute the value of x = 2 in g (x): Substitute the value of g (2) in f (x): f (7) = (7) 2 – 3 = 49 – 3 =...
eBook - ePubDifferentiating Instruction in Algebra 1
Ready-to-Use Activities for All Students (Grades 7-10)
- Kelli Jurek(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...Unit 1 IntRoduction to Functions and Relationships DOI: 10.4324/9781003234180-2 Beginning algebra students need to take their understanding of linear equations and solving simple one-variable equations and apply it to studying special relations of data called functions. Functions describe everyday situations where one specific quantity determines the value of another. Students must learn to write and evaluate functions because they describe a unique relationship between two quantities and are frequently used to model everyday situations. This unit begins with a preassessment and three real-life applications of functions that can be discussed in small groups and then as a larger group. Many of the activities will offer the students an opportunity to choose learning activities according to their learning style, personal interests, and readiness level. What Do We Want Students to Know? Common Core State Standards Addressed: • 8.F.1, 4 • A.CED.1, 2 • F.IF.1, 2, 5 • F.BF.1c Big Ideas • Not all relationships are functions. • A function denotes a special relationship between independent and dependent variables. • A function must pass the vertical line test. Essential Questions • What makes a relationship a function? • How is the vertical line test used to determine if a relationship represents a function? • Does it matter that a set of data does not represent a function? Critical Vocabulary Domain Function notation Dependent variable Output Range Relation Function Independent variable Vertical line test Input Unit Objectives As a result of this unit, students will know: ➤ a function is a special type of relation, ➤ the difference between a relation and a function, ➤ all functions are relations but not all relations are functions, ➤ the difference between an independent and dependent variable, and ➤ f x) is read...
