Forms of Quadratic Functions

10 Key excerpts on "Forms of 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

  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 - ePub

  Differentiating Instruction in Algebra 1

  Ready-to-Use Activities for All Students (Grades 7-10)

  • Kelli Jurek(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  Directions: Choose three problems in a row, in a column, or on a diagonal. Solve the problems and show your work on a separate sheet of paper. Circle the problems you choose on this page and then reference them on your work paper with the number from the square.

  Name:────────── Hour/Block:───── Date:─────

  Lesson 2: Characteristics of Quadratics

  Forms of Quadratic Equations

  General/Standard Form	Vertex Form	Factored Form
  y = ax + bx2 + c, where c is the starting value (y-intercept) of the function	y = a(x - h)2 + k where (h, k) is the vertex of the parabola	y = a(x - r1 )(x - r2 ), where r1 and r2 are the zeros of the quadratic function

  Directions: Identify the form for each of the following quadratic equations. If the function is written in general/standard form, identify the starting value. For all equations, find the vertex (as a coordinate pair), write the equation for the axis of symmetry (in the form x=), and then find the zeros. There is one sample problem. Round decimals to two places if necessary.

  Equation	Form	Starting Value	Vertex	Axis of Symmetry	Zeros
  y = 2(x + 2)2 - 4	vertex form	—	(–2,–4)	x = –2	–3.41,–0.59
  y = 2(x-2) +5
  y = 3(x-2)(x-4)
  y = -6(x-3) +6
  y = 2x2 + 4x - 6
  y = x2 - 3x + 7

  When you have completed the problems, check your answers. Each box is worth 1 point.

  My score is ───────.

  If you scored 0-17 points: You need additional practice and will complete the hexagon puzzle labeled with a moon.

  If you scored 18-22 points: You are working on level but need a little more practice. You will complete the hexagon puzzle labeled with a sun.

  If you scored 23-25 points: You are ready to be challenged and will complete the hexagon puzzle labeled with a star.

  Name:────────── Hour/Block:───── Date:─────

  Lesson 2: Characteristics of Quadratics

  Hexagon Puzzle

  Directions

  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 - ePub

  Introduction To Linear Algebra

  Computation, Application, and Theory

  • Mark J. DeBonis(Author)
  • 2022(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  CHAPTER 7 Quadratic Forms

  DOI: 10.1201/9781003217794-7

  I n this chapter , we introduce the notion of a quadratic form. Quadratic forms are prevalent in linear algebra and have many applications. In Section 7.1 , we introduce the notion of a quadratic form and the associated definitions of positive and negative definite and semi-definite and indefinite. In Section 7.2 , we derive the first test for determining if a quadratic form is positive or negative, definite or semi-definite, or indefinite called the Principal Minor Criterion. In Section 7.3 , we derive the second test for determining if a quadratic form is positive or negative, definite or semi-definite, or indefinite called the Eigenvalue Criterion. In Section 7.4 , we apply the criteria developed in Sections 7.2 and 7.3 to analyze critical points to determine if they are extrema for a multivariate function. In Section 7.5 we generalize the notion of a quadratic form.

  7.1INTRODUCTION TO QUADRATIC FORMS

  In this section, we introduce the notion of a quadratic form and the associated definitions of positive and negative definite and semi-definite.

  Definition 7.1. Let A be a symmetric

  n × n

  matrix. The quadratic form associated with A, written QA , is a map from

  ℝ n

  to R defined by

  Q A

  ( x ) =

  x T

  A x

  for

  x ∈

  ℝ n

  .

  Example 7.1. Let

  A =

  1

  − 2

  − 2

  5

  , then

  Q A

  (

  x 1

  ,

  x 2

  ) =

  x 1

  x 2

  1

  − 2

  − 2

  5

  x 1

  x 2

  =

  x 1

  x 2

  x 1

  − 2

  x 2

  − 2

  x 1

  + 5

  x 2

  =

  x 1

  (

  x 1

  − 2

  x 2

  ) +

  x 2

  ( − 2

  x 1

  + 5

  x 2

  ) =

  x 1 2

  − 4

  x 1

  x 2

  + 5

  x 2 2

  A couple of things to notice about Q

  A

  : First, the formula consists of a sum of terms each of which is degree two. Secondly the coefficients of the terms can quickly be obtained from the entries in the matrix A

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Introduction to Linear Algebra, 2nd edition

  • Thomas A Whitelaw, T.A. Whitelaw(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER TEN QUADRATIC FORMS 74.   Introduction As in earlier chapters, F will denote an arbitrary field of scalars, except that in this chapter, where virtually everything rests on the possibility of dividing by 2, we impose throughout the restriction char F ≠ 2 (cf. the final remark in §11). Once again, readers for whom such technicalities are unexplored territory can be assured that they will lose nothing if they simply regard F as standing for a familiar number system like ℝ or ℂ; and indeed some of the most interesting things in the chapter relate exclusively to the case F = ℝ. A quadratic form of order n over F means a mapping— q, say—from F n to F specifiable by a formula of the. form q (x 1, x 2, …, x n) = α 11 x 1 2 + α 22 x 2 2 + … + α n n x n 2 + 2 α 12 x 1 x 3 + 2 α 13 x 1 x 3 + … + 2 α 1 n x 1 x n + 2 α 23 x 2 x 3 + … + 2 α 2 n x 2 x n + …[--=P. LGO-SEPARATOR=--]+ 2 α n − 1, n x n − 1 x n (x 1, …, x n ∈ F) } (1) or (the same thing tidied up by using. ∑-notation) q (x 1, x 2, …, x n) = ∑ i = 1 n α i i x i 2 + ∑ i, k = 1 i < k n 2 α i k x i x k (x 1, …, x n ∈ F), (2) each of the coefficients α 11, etc. being a constant in F (constant in the sense of independent of x 1, …, x n). The cardinal thing about the defining expression for q (x 1, x 2, …, x n) is that it is a polynomial in x 1, x 2, …, x n in which every term is of degree 2, being either a “square term” of the form α i i x i 2 or a “mixed product term” of the form 2 α ik x i x k with i ≠ k. The order of the quadratic form q refers to the dimension of its domain, F n. To give a specific example, the mapping q 1 : ℝ 3 → ℝ defined by q (x 1, x 2, x 3) = x 1 2 − 4 x 2 2 + 5 x 3 2 + 2 x 1 x 2 − 6 x 1 x 3 + 3 x 2 x 3 (x 1, x 2, x 3 ∈ ℝ) (3) is a quadratic form of order 3 over ℝ. Quadratic forms appear in many places in mathematics and its applications

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Explorations in College Algebra

  • Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  8.2 Visualizing Quadratics: The Vertex Form 463 Domain and range Domain: The domain for the general quadratic y = ax 2 + bx + c is all real numbers; that is, all x in the interval (−∞, +∞). Range: If the quadratic is written in vertex form, y = a(x − h) 2 + k, it’s easy to define the range, since the maximum (or minimum) of its graph is at the vertex (h, k). So if a > 0, the parabola opens up, so the range is restricted to all y in the interval [k, +∞). a < 0, the parabola opens down, so the range is restricted to all y in the interval (−∞, k]. EXAMPLE 6 Finding the Function From Its Graph Figure 8.19 shows the graph of f (x) = 2x 2 transformed into three new parabolas. Assume that each of the three new graphs retains the overall shape of f (x). a. Estimate the coordinates of the vertex for all four parabolas. b. Use your estimates from part (a) to write equations for each new parabola in Figure 8.19 in vertex form. c. Identify the domain and range of each parabola. –10 Graph A Graph B Graph C 10 f (x) = x 2 6 –6 y x –10 10 6 –6 y x –10 10 6 –6 y x –2f (x + 3) = –2(x + 3) 2 –2f (x + 3) + 5 = –2(x + 3) 2 + 5 or g(x) f (x + 3) = (x + 3) 2 FIGURE 8.18 The graph of f(x) = x 2 in Graph A is shifted horizontally to the left three units, then in Graph B stretched vertically by a factor of 2 and reflected across the x‐axis, and finally in Graph C shifted up vertically five units to generate g(x) = −2(x + 3) 2 + 5. –10 10 f (x) g(x) h(x) k(x) 5 –5 y x FIGURE 8.19 Three transformations of f (x) = 2x 2 Solution a. Vertex for: f (x) is at (0, 0); g(x) is at (3, 4); h(x) is at (3, −1); k(x) is at (−1, −4) b. g(x) = 2(x − 3) 2 + 4; h(x) = −2(x − 3) 2 − 1; k(x) = −2(x + 1) 2 − 4 c. The domain for all four functions is (−∞, +∞). The range for: f (x) is [0, +∞); g(x) is [4, +∞); h(x) is (−∞, −1]; k(x) is (−∞, −4]. 464 CHAPTER 8 Quadratics and the Mathematics of Motion Algebra Aerobics 8.2b 6. Create new functions by performing the following transformations on f (x) = x 2 .

  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

  Intermediate Algebra

  Concepts with Applications

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

    (Publisher)

  We will continue this discussion of the relationship between graphs of functions and solutions to equations in the Using Technology material in the next section. FIGURE 1 0 2.0 0 50 FIGURE 2 0 350 0 3500 Y 1 Y 2 Y 3 EXERCISE SET 7.2 602 Chapter 7 Quadratic Equations and Functions SCAN TO ACCESS Vocabulary Review Choose the correct words to fill in the blanks below. rational quadratic standard form factored 1. An equation in , ax 2 + bx + c = 0, has two solutions that are always given by the formula x = −b ± √ — b 2 − 4ac __________ 2a . 2. The formula x = −b ± √ — b 2 − 4ac __________ 2a is called the formula. 3. If solutions to a quadratic equation turn out to be numbers, then the original equation could have been solved by factoring. 4. Using the quadratic formula may be a faster way to finding a solution, even if the equation can be . Problems A Solve each equation. Use factoring or the quadratic formula, whichever is appropriate. (Try factoring first. If you have any difficulty factoring, then try the quadratic formula.) 1. x 2 + 5x + 6 = 0 2. x 2 + 5x − 6 = 0 3. a 2 − 4a + 1 = 0 4. a 2 + 4a + 1 = 0 5. 1 __ 6 x 2 − 1 __ 2 x + 1 __ 3 = 0 6. 1 __ 6 x 2 + 1 __ 2 x + 1 __ 3 = 0 7. x 2 __ 2 + 1 = 2x __ 3 8. x 2 __ 2 + 2 __ 3 = − 2x __ 3 9. y 2 − 5y = 0 10. 2y 2 + 10y = 0 11. 30x 2 + 40x = 0 12. 50x 2 − 20x = 0 13. 2t 2 ___ 3 − t = − 1 __ 6 14. t 2 __ 3 − t _ 2 = − 3 __ 2 15. 0.01x 2 + 0.06x − 0.08 = 0 16. 0.02x 2 − 0.03x + 0.05 = 0 17. 2x + 3 = −2x 2 18. 2x − 3 = 3x 2 19. 100x 2 − 200x + 100 = 0 20. 100x 2 − 600x + 900 = 0 21. 1 __ 2 r 2 = 1 __ 6 r − 2 __ 3 7.2 Exercise Set 603 22. 1 __ 4 r 2 = 2 __ 5 r + 1 __ 10 23. (x − 3)(x − 5) = 1 24. (x − 3)(x + 1) = −6 25. (x + 3) 2 + (x − 8)(x − 1) = 16 26. (x − 4) 2 + (x + 2)(x + 1) = 9 27. x 2 __ 3 − 5x __ 6 = 1 __ 2 28. x 2 __ 6 + 5 __ 6 = − x __ 3 29. √ — x = x − 1 30. √ — x = x − 2 Multiply both sides of each equation by its LCD. Then solve the resulting equation. 31. 1 _____ x + 1 − 1 __ x = 1 __ 2 32.

  Sign up to read

  Learn more about book