Cubic Function Graph

10 Key excerpts on "Cubic Function Graph"

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  Precalculus

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  B oxes can be manufactured in many shapes and sizes. A polynomial function can be used in constructing a box to meet a volume specification. See Example 10 in Section 3.2 for such an application. In this section, we study quadratic, polynomial, and rational functions. These functions have many mathematical properties that play an important role in advanced mathematics and applications. Chapter 3 Outline 10 in. 10 in. 10 – 2 x 10 – 2 x x x x x x x x x 3.1 Quadratic Functions and Their Graphs 3.2 Polynomial Functions and Their Graphs 3.3 Division of Polynomials; the Remainder and Factor Theorems 3.4 Real Zeros of Polynomials; Solutions of Equations 3.5 Complex Numbers 3.6 The Fundamental Theorem of Algebra; Complex Zeros 3.7 Rational Functions 3.8 Quadratic, Polynomial, and Rational Inequalities Polynomial and Rational Functions 3 . 1 3.1 Quadratic Functions and Their Graphs 209 In Chapter 2, you examined linear functions, which are of the form f ( x ) = mx + b . The graph of a linear function is simply a line. In this section, we examine quadratic functions , in which the independent variable x is raised to the second power. Definition of a Quadratic Function A function f is a quadratic function if it can be expressed in the form f ( x ) = ax 2 + bx + c where a , b , and c are real numbers and a ≠ 0. The domain of a quadratic function is the set of all real numbers. Throughout our discussion, a is the coefficient of the x 2 term; b is the coefficient of the x term; and c is the constant term. To better understand quadratic functions, it is helpful to look at their graphs. We study the graphs of general quadratic functions by first examining the graph of the function f ( x ) = ax 2 , a ≠ 0. We will later see that the graph of any quadratic function can be produced by a suitable combination of transformations of this graph. Consider the quadratic functions f ( x ) = x 2 , g ( x ) = − x 2 , and h ( x ) = 2 x 2 .

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  Algebra & Trig

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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. 3.1 Quadratic Functions and Models 243 GO DIGITAL The graph of a quadratic function is a U-shaped curve called a parabola. Parabolas occur in many real-life applications—including those that involve reflective properties of satellite dishes and flashlight reflectors. You will study these properties in Section 4.3. All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola. When the leading coefficient is positive, the graph of f (x) = ax 2 + bx + c is a parabola that opens upward. When the leading coefficient is negative, the graph is a parabola that opens downward. The next two figures show the axes and vertices of parabolas for cases where a > 0 and a < 0. Opens upward Vertex is lowest point Axis y f (x) = ax 2 + bx + c, a > 0 x x Opens downward Vertex is highest point Axis y f (x) = ax 2 + bx + c, a < 0 Leading coefficient is positive. Leading coefficient is negative. The simplest type of quadratic function is one in which b = c = 0. In this case, the function has the form f (x) = ax 2 . Its graph is a parabola whose vertex is (0, 0). When a > 0, the vertex is the point with the minimum y-value on the graph, and when a < 0, the vertex is the point with the maximum y-value on the graph, as shown in the figures below. x Minimum: (0, 0) - 1 - 1 - 2 - 3 - 2 - 3 1 2 3 1 2 3 y f (x) = ax 2 , a > 0 x Maximum: (0, 0) - 3 - 2 - 1 1 2 3 - 3 - 2 - 1 1 2 3 y f (x) = ax 2 , a < 0 Leading coefficient is positive.

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  Mathematics 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.

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  Algebra

  Form and Function

  • William G. McCallum, Eric Connally, Deborah Hughes-Hallett(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Chapter Three QUADRATIC FUNCTIONS © Patrick Zephyr/Patrick Zephyr Nature Photography Contents 3.1 Introduction to Quadratic Functions. . . . . . . . . 100 Creating Computer Graphics . . . . . . . . 101 3.2 Quadratic Expressions . . . . . . . . . . . . . . . . . . . . 103 Interpreting Factored Form . . . . . . . . . . . . . . . . 104 Interpreting Vertex Form . . . . . . . . . . . . . . . . . . 106 Constructing Quadratic Functions . . . . . . . . . . . 107 3.3 Converting to Factored and Vertex Form . . . . . 111 Converting to Factored Form. . . . . . . . . . . . . . . 111 How Do We Put an Expression in Vertex Form? . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Visualizing Completing the Square . . . . . . . . . . 114 3.4 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . 116 Solving Equations by Factoring . . . . . . . . . . . . . 116 Solving Equations with Perfect Squares . . . . . . 117 Solving by Completing the Square . . . . . . . . . . 119 The Quadratic Formula . . . . . . . . . . . . . . . . . . . 119 The Discriminant . . . . . . . . . . . . . . . . . . . . . . . . 121 3.5 Factoring Hidden Quadratics . . . . . . . . . . . . . . . 124 3.6 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 129 Using Complex Numbers to Solve Equations . . 129 Algebra of Complex Numbers . . . . . . . . . . . . . . 130 Addition and Subtraction of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 131 Multiplication of Complex Numbers . . 131 Division of Complex Numbers. . . . . . . 133 REVIEW PROBLEMS . . . . . . . . . . . . . . . . . . . . 134 SOLVING DRILL . . . . . . . . . . . . . . . . . . . . . . . . 138 100 Chapter 3 QUADRATIC FUNCTIONS 3.1 INTRODUCTION TO QUADRATIC FUNCTIONS The graph of a linear function is a straight line. If we want the graph to curve, we need a different sort of function. For example, Figure 3.1 shows the height of a ball thrown off the top of a building  seconds after it has been thrown.

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  Algebra and Trigonometry

  • Sheldon Axler(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  For example, consider the function f such that Distinguishing be- tween f and f (x) is usually worthwhile and helps lead to better understand- ing. Use f to de- note a function and f (x) to denote the value of a function f at a number x. f (x) = 3x for every real number x. There is no particular x here. The symbol x is simply a placeholder to indicate that f associates any number with 3 times that number. We could have defined the same function f by using the formula f (y) = 3y for every real number y . Or we could have used the formula f (t) = 3t for every real number t . All these formulas show that f (2) = 6 and that f (2w + 5) = 3(2w + 5). section 3.1 Functions 121 The Graph of a Function A function can be visualized by considering its graph. The graph of a function The graph of a function f is the set of points of the form ( x, f (x) ) as x varies over the domain of f . Thus in the xy -plane, the graph of a function f is the set of points (x, y) satisfying the equation y = f (x). example 4 Suppose f is the function with domain [0, 3] defined by f (x) = x 2 - 2x + 3. Sketch the graph of f . solution The graph of f is the set of points (x, y) in the xy -plane such that x is in the interval [0, 3] and y = x 2 - 2x + 3. To sketch the parabola defined by the equation above, we complete the square: 1 2 3 x 2 4 6 y The graph of the function f with domain [0, 3] defined by f (x) = x 2 - 2x + 3. y = x 2 - 2x + 3 = (x - 1) 2 - 1 + 3 = (x - 1) 2 + 2. Thus this parabola has vertex at the point (1, 2). We also see that if x = 0, then y = 3, producing the point (0, 3), and if x = 3, then y = 6, producing the point (3, 6). This information leads to the sketch shown in the margin. Sketching the graph of a complicated function usually requires the aid of a computer or calculator. The next example uses Wolfram|Alpha in the solution, but you could use a graphing calculator or any other technology instead.

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  College Algebra

  • James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  This allows us to see how changing a value in the definition of the functions affects the shape of its graph. In the next example we apply this principle to a family of third-degree polynomials. EXAMPLE 10 ■ A Family of Polynomials Sketch the family of polynomials P 1 x 2  x 3  cx 2 for c  0, 1, 2, and 3. How does changing the value of c affect the graph? SOLUTION The polynomials P 0 1 x 2  x 3 P 1 1 x 2  x 3  x 2 P 2 1 x 2  x 3  2 x 2 P 3 1 x 2  x 3  3 x 2 are graphed in Figure 14. We see that increasing the value of c causes the graph to develop an increasingly deep “valley” to the right of the y -axis, creating a local maxi-mum at the origin and a local minimum at a point in Quadrant IV. This local mini-mum moves lower and farther to the right as c increases. To see why this happens, factor P 1 x 2  x 2 1 x  c 2 . The polynomial P has zeros at 0 and c , and the larger c gets, the farther to the right the minimum between 0 and c will be. Now Try Exercise 75 ■ 10 _10 _2 4 c=0 c=1 c=2 c=3 FIGURE 14 A family of polynomials P 1 x 2  x 3  cx 2 CONCEPTS 1. Only one of the following graphs could be the graph of a polynomial function. Which one? Why are the others not graphs of polynomials? I y x II y x III y x IV y x 2. Describe the end behavior of each polynomial. (a) y  x 3  8 x 2  2 x  15 End behavior: y S as x S  y S as x S  3.2 EXERCISES Copyright 2016 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. 302 CHAPTER 3 ■ Polynomial and Rational Functions (b) y   2 x 4  12 x  100 End behavior: y S as x S  y S as x S  3.

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  NCV4 Mathematics

  • J Daniels, N Solomon, J Daniels, N Solomon(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  Sketching graphs of cubic functions • x -intercept: where y = 0 • y -intercept: where x = 0 • Maximum turning point, minimum turning point and point of inflection. Maximum turning point dy dx = 0 d y dx 2 2 < 0 Minimum turning point dy dx = 0 d y dx 2 2 > 0 Point of inflection d y dx 2 2 = 0 12. Integration • & dx : Integrate in respect of x . • & dy : Integrate in respect of y . 160 Mathematics: Hands-On Training • Integration rules Standard integrals & x n dx = x n n + + 1 1 + c , where n ≠ – 1 & ax n dx = ax n n + + 1 1 = c , where n ≠ – 1 & ax n dx = a & x n dx = ax n n + + 1 1 + c , where n ≠ – 1 & dx = & 1 x 0 dx = x + c & a x dx = a ln x + c & e x dx = e x + c & ae kx dx = ae k kx + c & a sin kx dx = − a kx k cos + c & a cos kx dx = a kx k sin + c & a sec 2 kx dx = a kx k tan + c & [ f ( x ) ± g ( x )] dx = & f ( x ) dx ± & g ( x ) dx • Definite integrals & a b f ( x ) dx = [ g ( x ) + c ] a b = [ g ( b ) + c ] – [ g ( a ) + c ] • Substitute x = b (upper limit) in first = g ( b ) – g ( a ) bracket. Subtract the second bracket after x = a (lower limit) has been substituted. • Calculating areas between a curve and an axis Area above the x -axis Area below the x -axis Area = & a b f ( x ) dx or Area = & a b y dx Area = – & a b f ( x ) dx or Area = – & a b y dx • Steps to follow when calculating the area under a curve Step 1 Sketch the graph. Step 2 Show the representative strip that you will use to calculate the area. Step 3 Write down the area of the strip: Δ A = y Δ x Step 4 Write down the sum of all such areas, which is the definite integral, and calculate the area. & a b f ( x ) dx 161 Chapter 2 Functions and algebra Summative assessment 1: Chapter 2 • Remainder and factor theorem • Factorising of a third-degree trinomial • Inverse functions • Linear programming Mark allocation: 60 Time allocation: 1 hour 45 minutes Instructions and information 1. Answer all the questions. 2. Read all the questions carefully.

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  Calculus

  Resequenced for Students in STEM

  • David Dwyer, Mark Gruenwald(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  . ., a n are real numbers (called coefficients), and a n 6= 0. Note that a polynomial function of degree 1 has the form f (x) = a 1 x+a 0 , and its graph is a line with slope a 1 and y-intercept a 0 . Thus, we see that linear functions are also polynomial functions. The simplest nonlinear polynomial functions are the quadratic functions, which have the form f (x) = a 2 x 2 + a 1 x + a 0 and are of degree 2. The graphs of quadratic functions are parabolas. Cubic functions have the form f (x) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 and are of degree 3. Two useful examples of quadratic and cubic polynomials are y = x 2 and y = x 3 , the graphs of which are shown in Figures 1.15 and 1.16. -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 y = x 2 Figure 1.15 -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 y = x 3 Figure 1.16 Polynomial functions are defined for all real numbers, and so their domain is (-∞, ∞). Their graphs have no breaks and are “smooth” in the sense that there are no abrupt changes in direction. Because of their simplicity, their variety, and the smoothness of their graphs, polynomials are often used to construct mathematical models of real-world phenomena. Rational Functions When two polynomial are added, subtracted, or multiplied, the result is again a polynomial. However, when one polynomial is divided by another, the result is generally not a polynomial. Thus, quotients of polynomials form a new class of functions, called the rational functions. We make the following definition. 1.2. LIBRARY OF FUNCTIONS 17 Definition of Rational Function A rational function is a function of the form f (x) = p(x) q(x) where p(x) and q(x) are both polynomials. Note that by taking q(x) to be the constant function q(x) = 1, we see that a polynomial function can also be viewed as a rational function. More generally, if a rational function has a denominator that is never zero, the function will have some of the same nice properties that polynomials have.

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  College Algebra

  • 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.

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  Pre-Calculus All-in-One For Dummies

  Book + Chapter Quizzes Online

  • Mary Jane Sterling(Author)
  • 2023(Publication Date)
  • For Dummies

    (Publisher)

  2 Getting the Grip on Graphing IN THIS UNIT . . . Applying your graphing skills to polynomial functions. Dealing with exponential and logarithmic functions. Piecing together piece-wise functions. Grappling with greatest integer functions. CHAPTER 5 Graphing Polynomial Functions 95 Graphing Polynomial Functions E ver since those bygone days of algebra, variables have been standing in for unknowns in equations. You’re probably very comfortable with using variables by now, so you’re ready to move on and find out how to deal with equations that use multiple terms and figure out how to graph them. When variables and constants start multiplying, the result of a variable times a constant is a monomial, which means “one term.” Examples of monomials include 3y, x 2 , and 4ab 3 c 2 . When you start creating expressions by adding and subtracting distinct monomials, you get poly- nomials, because you create something with one or more terms. Usually, monomial refers to a polynomial with one term only, binomial refers to two terms, trinomial refers to three, and the word polynomial is reserved for four or more. Think of a polynomial as the umbrella under which are binomials and trinomials. Each part of a polynomial that’s added or subtracted is a term; so, for example, the polynomial 2 3 x has two terms: 2x and 3. Part of the official definition of a polynomial is that it can never have a variable in the denomi- nator of a fraction, it can’t have negative exponents, and it can’t have fractional exponents. The exponents on the variables have to be non-negative integers. A general format of a polynomial is a x a x a x a x a n n n n n n 1 1 2 2 1 1 0  . The a n factors represent the coefficients, and the n’s are always non-negative integers.

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