Quadratic functions

8 Key excerpts on "Quadratic functions"

• 

  eBook - PDF

  Intermediate Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  In this section, we will explore further these types of functions and their graphs. Up to now in this chapter, we have been studying quadratic equations. If we take a quadratic equation in standard form and use it instead as the formula for a function, the result is called a quadratic function . Here is a formal definition: Any function that can be written in the form f ( x ) = a x 2 + bx + c where a , b , and c are constants with a ≠ 0, is called a quadratic function . We refer to this form as standard form . quadratic function DEFINITION The Basic Quadratic Function The simplest of all Quadratic functions results when we let a = 1 and b = c = 0. We will refer to f ( x ) = x 2 as the basic quadratic function . Table 1 gives some ordered pairs for this function, and the corresponding graph is shown in Figure 1. This graph is an example of a parabola . From the graph we can see that the domain of this function is the set of all real numbers and the range is { y | y ≥ 0}. The point (0, 0) where the parabola changes direction (the lowest point on the parabola) is called the vertex . FIGURE 1 –5 –4 –3 –2 –1 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5 Vertex y 5 x 2 2 3 -5 -4 -3 -2 -1 1 2 3 4 5 -1 1 2 3 5 6 7 8 9 x y Vertex f ( x ) = x 2 Axis of Symmetry Axis of Symmetry 4 x f ( x ) = x 2 − 3 9 − 2 4 − 1 1 0 0 1 1 2 4 3 9 TABLE 1 608 CHAPTER 8 Quadratic Equations and Functions We also observe that the graph is symmetric about the y -axis, meaning the right half of the graph is a mirror image of the left half. For this reason, the vertical line x = 0 passing through the vertex is called the axis of symmetry . All Quadratic functions have a graph that is a parabola and a domain that is the set of all real numbers. However, the shape, direction, and position of the parabola can vary as we will see in the following segment. Transformations Let's consider Quadratic functions of the form f ( x ) = a x 2 .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Elementary and Intermediate Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  We refer to this form as standard form . quadratic function DEFINITION The Basic Quadratic Function The simplest of all Quadratic functions results when we let a = 1 and b = c = 0. We will refer to f ( x ) = x 2 as the basic quadratic function . Table 1 gives some ordered pairs for this function, and the corresponding graph is shown in Figure 1. This graph is an example of a parabola . FIGURE 1 –5 –4 –3 –2 –1 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5 Vertex y 5 x 2 2 3 -5 -4 -3 -2 -1 1 2 3 4 5 -1 1 2 3 5 6 7 8 9 x y Vertex f ( x ) = x 2 Axis of Symmetry Axis of Symmetry 4 x f ( x ) = x 2 − 3 9 − 2 4 − 1 1 0 0 1 1 2 4 3 9 TABLE 1 820 Chapter 11 Quadratic Equations and Functions From the graph we can see that the domain of this function is the set of all real numbers and the range is { y | y ≥ 0}. The point (0, 0) where the parabola changes direction (the lowest point on the parabola) is called the vertex . We also observe that the graph is symmetric about the y -axis, meaning the right half of the graph is a mirror image of the left half. For this reason, the vertical line x = 0 passing through the vertex is called the axis of symmetry . All Quadratic functions have a graph that is a parabola and a domain that is the set of all real numbers. However, the shape, direction, and position of the parabola can vary as we will see in the following segment. Transformations Let's consider Quadratic functions of the form f ( x ) = a x 2 . In this case, all we are doing is taking each y -value (output) from the basic quadratic function y = x 2 and multiplying it by a factor of a . As a result, we can change the shape and direction of the basic parabola. We illustrate how this is done in the next two examples. Graph: f ( x ) = 2 x 2 . SOLUTION Because a = 2, we take each y -coordinate from the basic parabola y = x 2 and double it. Table 2 shows how this is done for several values of x .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Brief Calculus

  An Applied Approach

  • Michael Sullivan(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Pol  omial nctions are arguably the simplest expressions in algebra. For this reason, they are often used to approximate other, more complicated nctions. Rational nctions are simply ratios of pol  o- mial nctions. We begin with a discussion of quadratic nctions, a type of polynomial nction. PRERING FOR THIS SECTION Bre getting started, review the following: > Completing the Square (Chapter 0. Section 0.4 . pp. 45-46) > Quadratic Equations (Chapter 0. Section 0.4. pp. 43-49) > Square Root Method (Chapter 0. Section 0.4. pp. 44-45) > Geometry Review (Chapter 0. Section 0.7. pp. 71-74) OBJECTIVES Locate the vertex and axis of symmetry of a quadratic function Graph Quadratic functions Find the maximum or the minimum value of a quadratic function Use the maximum or the minimum value of a quadratic function to solve applied problems A quadtic function is a nction that is defined by a second-degree polynomial in one variable. A quadratic nction is a nction of the rm I f(x) = ax 2 + bx + c I (1) where a, b, and care real numbers and a # 0. T he domain of a quadratic nction is the set of all real numbers. Many applications require a knowledge of the properties of the graph of a quadratic nction. For example, suppose that Texas Instruments collects the data shown in Table I that relate the number of calculators sold at the price p per calculator. Since the price of a product determines the quantity that will be purchased, we treat price as the independent variable. TABLE 1 Price per Number of Calculator. Calculators. p (Dollars) X 60 11,100 65 10,115 70 9,652 75 8,731 80 8,087 85 7,205 90 6,439 Quadratic functions 165 A linear relationship between the number of calculators and the price p per calcula- tor may be given by the equation X = 21,000 - 150p Then the revenue R derived om selling x calculators at the price p per calculator is R = xp R(p) = (21,000 - 150p)p = -150p 2 + 21,000p So the revenue R is a quadratic nction of the price p.

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  Mathematics for Business

  • Gary Bronson, Richard Bronson, Maureen Kieff(Authors)
  • 2021(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  Quadratic functions , and have the form:

  f (x ) = a 2 x 2 + a 1 x + a 0

  where a 2 ≠ 0. If we replace the constants a 2 , a 1 , and a 0 , by a , b , and c , respectively, this second-degree polynomial function is written in its more conventional form as:

  y = ax 2 bx + c (Eq. 3.8)

  In Equation 3.8, the variable that is squared is referred to as the quadratic variable , which in this case is x . Note that what determines if an equation is a function are not the symbols used in the equation, but whether the equation, domain, and range satisfy the definition of a function provided in Section 3.1 .

  Example 1 Determine which of the following functions are Quadratic functions. For those that are, state their coefficients, a, b, and c.

  a. y = 2x 2 − ½

  b. y = 3x − x 2

  c. n 2 = 2p + 4

  Solution

  a. This is a quadratic function in the variable x with a = 2, b = 0, and c = − 1/2.

  b. Rewriting this equation as y = − x 2 + 3x , we see that this is a quadratic function in the variable x with a = − 1, b = 3, and c = 0.

  c. Rewriting this equation as f (p ) = 1/2 n 2 − 2, we see that it is a quadratic function in the variable n , with a = ½, b = 0, and c = − 2.

  As in the case of linear equations and in part (c) of this example, the letters y and x

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  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.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Functions Modeling Change

  A Preparation for Calculus

  • Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  CONTENTS 3.1 Introduction to the Family of Quadratic functions . . . . . . . . . . . . . . . . . . 106 Finding the Zeros of a Quadratic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Concavity and Rates of Change for Quadratic functions . . . . . . . . . . . . . . . . . . . 107 Finding a Formula From the Zeros and Vertical Intercept . . . . . . . . . . . . . . . . . . . . . . . . . 109 Formulas for Quadratic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Summary for Section 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.2 The Vertex of a Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 The Vertex Form of a Quadratic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Finding a Formula Given the Vertex and Another Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Modeling with Quadratic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Summary for Section 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 STRENGTHEN YOUR UNDERSTANDING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Chapter 3 Quadratic functions 106 Chapter 3 Quadratic functions 3.1 INTRODUCTION TO THE FAMILY OF Quadratic functions A baseball is “popped” straight up by a batter. The height of the ball above the ground is given by the function  =  () = −16 2 + 47 + 3, where  is time in seconds after the ball leaves the bat and  is in feet. See Figure 3.1. Note that the path of the ball is straight up and down, although the graph of height against time is a curve. The ball goes up fast at first and then more slowly because of gravity; thus the graph of its height as a function of time is concave down.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Precalculus

  A Prelude to Calculus

  • Sheldon Axler(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Parabolas can be defined geometrically, but for our purposes it is simpler to define a parabola algebraically. Parabola A parabola is the graph of a quadratic function. For example, the graph of the quadratic function f defined by f ( x) = x 2 is the familiar parabola shown here. This parabola is symmetric about the vertical axis, x The graph of f (x) = x 2 on the interval [−1, 1]. meaning that the parabola is unchanged if it is flipped across the vertical axis. Note that this line of symmetry intersects this parabola at the origin, which is the lowest point on this parabola. Every parabola is symmetric about some line. The point where this line of symmetry intersects the parabola is sufficiently important to deserve a name. The vertex of the parabola shown above is the origin. Vertex The vertex of a parabola is the point where the line of symmetry of the parabola intersects the parabola. Example 5 Suppose f ( x) = x 2 + 6x + 11. The ancient Greeks discovered that the intersection of a cone and an appropriately positioned plane is a parabola. (a) For what value of x does f ( x) attain its minimum value? (b) What is the minimum value of f ( x)? (c) Sketch the graph of f . (d) Find the vertex of the graph of f . solution (a) First complete the square, as follows: f ( x) = x 2 + 6x + 11 = ( x + 3) 2 − 9 + 11 = ( x + 3) 2 + 2. Because ( x + 3) 2 equals 0 when x = −3 and is positive for all other values of x, the last expression shows that f ( x) takes on its minimum value when x = −3. Section 2.2 Quadratic functions and Conics 139 (b) From the expression above, we see that f (−3) = 2. Thus the minimum value of f ( x) is 2. (c) The expression found for f ( x) in part (a) shows that the graph of f is obtained x The graph of f (x) = x 2 + 6x + 11 on the interval [−5, −1]. by shifting the graph of x 2 left 3 units and up 2 units, giving the graph shown here. The line of symmetry for this graph is x = −3, which is shown in blue.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introductory Algebra

  Concepts with Applications

  • Charles P. McKeague(Author)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  e. The fact that every landing point can come from two different paths makes us think that the equations that give us the landing points must be what type of equations? Find the Mistake Each sentence below contains a mistake. Circle the mistake and write the correct word(s) or expression on the line provided. 1. All equations of the form ax 3 = 0 have parabolas for graphs. 2. The graph of the function y = ax 2 + bx + c crosses the x–axis where y = 0. 3. The vertex for the graph y = 4x 2 + 6x + 7 occurs when y = − 4 __ 12 = − 1 _ 3 . 4. The graph of y = ax 2 + bx + c will be concave down when a is positive. 0 0 20 40 60 80 100 120 140 160 180 10 20 30 40 50 60 70 80 90 Vertical Distance in Feet Vertical Distance in Feet Horizontal Distance in Feet Horizontal Distance in Feet 0 0 20 40 60 80 100 120 140 160 180 10 20 30 40 50 60 70 80 90 Initial Angle: 80° 70° 60° 50° 40° 30° 20° Vertical Distance in Feet Vertical Distance in Feet Horizontal Distance in Feet Horizontal Distance in Feet 9.5 673 9.5 Introduction to Functions Before the harvest each fall, a Nebraskan farmer climbs onto his riding lawn mower and rides through acres of corn. He uses the mower to carve an intricate maze in his cornfield. One of his most impressive designs took the shape of a Star Wars starship. He is known to fly in a plane over his freshly cut maze to ensure its perfection. Thousands flock to Grandpa John’s Amazing Maze; each visitor paying $5.00 to walk through it. In this section, we will introduce you to functions. Using Grandpa John’s Amazing Maze as an example, the amount of money the maze earns for the farmer each season depends on the number of people who visit it. In mathematics, we say that his seasonal earnings are a function of how many people visit the maze. Relations Before we begin our discussion of functions, let's examine what happens when a coordinate from a set of inputs is paired with one or more elements from a set of outputs.

  Sign up to read

  Learn more about book