Mathematics
Polynomial Graphs
Polynomial graphs are curves that represent polynomial functions. These functions are made up of terms that contain variables raised to non-negative integer powers, and they can be used to model a wide range of phenomena in mathematics, science, and engineering. The degree of a polynomial function determines the shape of its graph.
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8 Key excerpts on "Polynomial Graphs"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 5 Polynomial and Rational Function Polynomial In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x 2 − 4 x + 7 is a polynomial, but x 2 − 4/ x + 7 x 3/2 is not, because its second term involves division by the variable x (4/x) and because its third term contains an exponent that is not a whole number (3/2). The term 'polynomial' indicates a simplified algebraic form such that all polynomials are similarly simple in complexity (cf. polynomial time). Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry. Overview A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) which may be multiplied by a finite number of variables (usually represented by letters). Each variable may have an exponent that is a non-negative integer, i.e., a natural number. The exponent on a variable in a term is called the degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term.
eBook - PDFCollege Algebra
Building Concepts and Connections 2E
- Revathi Narasimhan(Author)
- 2019(Publication Date)
- XYZ Textbooks(Publisher)
Example 1 Objectives ■ Define a polynomial function. ■ Determine end behavior. ■ Find x -intercepts and zeros by factoring. ■ Sketch a complete graph of a polynomial function. ■ Relate zeros, x -intercepts, and factors of a polynomial. ■ Solve applied problems using polynomials. Polynomial Functions and Their Graphs VIDEO EXAMPLES SECTION 3.2 246 Chapter 3 Polynomial and Rational Functions The rest of this section will be devoted to graphs of polynomial functions. These graphs have no breaks or holes, and no sharp corners. Figure 1 illustrates graphs of several functions, and indicates which are the graphs of polynomial functions. Polynomials of the Form x n and Their Transformations The graphs of polynomial functions can be quite varied. The simplest polynomial functions are those with just one term, x n . Consider the functions f ( x ) = x 3 and g ( x ) = x 4 . Table 1 gives a few of their function values. It is useful to examine the trend in the function value as the value of x increases to positive infinity ( x → ∞ ) or when the value of x decreases to negative infinity ( x → − ∞ ). This is known as end behavior and is summarized in Table 2. The graphs and their end behaviors are shown in Figure 2. We now summarize properties of the general function f ( x ) = x n , n a positive integer.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 256 Chapter 3 Polynomial Functions GO DIGITAL A polynomial function is in standard form when its terms are in descending order of exponents from left to right. For example, the standard form of the polynomial function f (x) = 5x 2 - 2x 7 + 4 - 2x is f (x) = -2x 7 + 5x 2 - 2x + 4. Standard form To avoid making a mistake when applying the Leading Coefficient Test, rewrite the polynomial function in standard form first, if necessary. EXAMPLE 4 Sketching the Graph of a Polynomial Function Sketch the graph of f (x) = -4x 3 + 3x 4 . Solution 1. Apply the Leading Coefficient Test. In standard form, the polynomial function is f (x) = 3x 4 - 4x 3 . Standard form The leading coefficient is positive and the degree is even, so you know that the graph eventually rises to the left and to the right (see Figure 3.11). 2. Find the Real Zeros of the Function. Find the real zeros by factoring f (x). f (x) = 3x 4 - 4x 3 = x 3 (3x - 4) Remove common monomial factor. The real zeros of f are x = 0 and x = 4 3 (both of odd multiplicity). So, the x-intercepts occur at (0, 0) and ( 4 3 , 0). Add these points to your graph, as shown in Figure 3.11. 3. Plot a Few Additional Points. To sketch the graph, find a few additional points, as shown in the table. Then plot the points (see Figure 3.12). x -1 1 2 1 3 2 f (x) 7 - 5 16 -1 27 16 4. Draw the Graph. Draw a continuous curve through the points, as shown in Figure 3.12. Both zeros are of odd multiplicity, so you know that the graph should cross the x-axis at x = 0 and x = 4 3 .
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
5 MODULE Algebraic graphs On completion of this module, you should be able to: 5.1 Represent ordered number pairs (coordinates) on a Cartesian plane 5.2 Perform function notation: write a linear equation of the form y = mx + c in the form f ( x ) = mx + c and a rectangular hyperbola of the form y = c x in the form f ( x ) = c x 5.3 Draw a linear graph in the form y = mx + c with the aid of a table 5.3.1 Gradient-intercept form of a straight line: y = mx + c 5.3.2 Sketching a straight-line graph in standard form y = mx + c 5.4 Determine the gradient and y -intercept from a graph. 5.5 Draw a rectangular hyperbola: 5.5.1 Sketch a rectangular hyperbola in the form y = c x or xy = c , where c ≠ 0 5.5.2 Explain the concepts direct and inverse relation . 140 Module 5 • Algebraic graphs Introduction A graph is a diagram showing either the relationship between some variable quantities or the connections that exist between a set of points. Information can be represented in graph form, for example: • in medicine and pharmacy to work out the correct strength of drugs • to analyse or predict future markets and opportunities • to work out whether or not you are the correct healthy weight for your body height, for example the BMI (body mass index) graph • to show how the population of a country increased over a certain period, as shown on the graphs below. 1950 2000 Population 1950 2000 Population The population increased steadily over the 50 years. The population growth started off slow, but then increased rapidly over the 50 years. Different types of graph Linear graph Rectangular hyperbola 3 2 y x y = – x + 3 3 2 1 3 –1 –3 xy = 1 or y = y x 1 x Bar graph Parabola 0 2 4 Blue Brown Eye colour Frequency Green y x 1 1 –1 y = x 2 y x 141 Introductory Mathematics| Hands-On Pre-knowledge • Real numbers can be represented on a number line. –3 –2 –1 0 1 2 3 x = 3 This number line shows x = 3.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
5 MODULE Algebraic graphs On completion of this module, you should be able to: 5.1 Represent ordered number pairs (coordinates) on a Cartesian plane 5.2 Perform function notation: write a linear equation of the form y = mx + c in the form f ( x ) = mx + c and a rectangular hyperbola of the form y = c x in the form f ( x ) = c x 5.3 Draw a linear graph in the form y = mx + c with the aid of a table 5.3.1 Gradient-intercept form of a straight line: y = mx + c 5.3.2 Sketching a straight-line graph in standard form y = mx + c 5.4 Determine the gradient and y -intercept from a graph. 5.5 Draw a rectangular hyperbola: 5.5.1 Sketch a rectangular hyperbola in the form y = c x or xy = c , where c ≠ 0 5.5.2 Explain the concepts direct and inverse relation . 140 Module 5 • Algebraic graphs Introduction A graph is a diagram showing either the relationship between some variable quantities or the connections that exist between a set of points. Information can be represented in graph form, for example: • in medicine and pharmacy to work out the correct strength of drugs • to analyse or predict future markets and opportunities • to work out whether or not you are the correct healthy weight for your body height, for example the BMI (body mass index) graph • to show how the population of a country increased over a certain period, as shown on the graphs below. 1950 2000 Population 1950 2000 Population The population increased steadily over the 50 years. The population growth started off slow, but then increased rapidly over the 50 years. Different types of graph Linear graph Rectangular hyperbola 3 2 y x y = – x + 3 3 2 1 3 –1 –3 xy = 1 or y = y x 1 x Bar graph Parabola 0 2 4 Blue Brown Eye colour Frequency Green y x 1 1 –1 y = x 2 y x 141 N1 Mathematics| Hands-On! Pre-knowledge • Real numbers can be represented on a number line. –3 –2 –1 0 1 2 3 x = 3 This number line shows x = 3.
eBook - PDFPre-Calculus All-in-One For Dummies
Book + Chapter Quizzes Online
- Mary Jane Sterling(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
The calculator will not only identify the zeros, but also show you the maximum and minimum values of the graph so that you can draw the best possible representation. Graphing when all the roots are real numbers When graphing a polynomial function, you want to use the following steps: 1. Plot the zeros (x-intercepts) on the coordinate plane and the y-intercept. 2. Determine which way the ends of the graph point. 3. Figure out what happens between the zeros by picking any value to the left and right of each intercept and plugging it into the function. 4. Plot the graph. Q. Plot the graph of the polynomial f x x x x x 2 9 21 88 48 4 3 2 . This is an example that appears earlier. The zeros are x 3 , x 1 2 , and x 4 . A. Use the following steps. 1. Plot the zeros (x-intercepts) on the coordinate plane. Mark the zeros that you found previously: x 3 , x 1 2 , and x 4 . Now plot the y-intercept of the polynomial. The y-intercept is always the constant term of the polynomial — in this case, y 48. If no constant term is written, the y-intercept is 0. 2. Determine which way the ends of the graph point. You can use a handy test called the leading coefficient test, which helps you figure out how the polynomial begins and ends. The degree and leading coefficient of a polyno- mial always explain the end behavior of its graph (see the section, “Understanding Degrees and Roots,” for more on finding degree): CHAPTER 5 Graphing Polynomial Functions 119 • If the degree of the polynomial is even and the leading coefficient is positive, both ends of the graph point up. • If the degree is even and the leading coefficient is negative, both ends of the graph point down. • If the degree is odd and the leading coefficient is positive, the left side of the graph points down and the right side points up. • If the degree is odd and the leading coefficient is negative, the left side of the graph points up and the right side points down.
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 - 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?
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