text: 764.00K
  Категория: Математика
  
  # Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions
  
  ## 1.
  
  Chapter 3
  Polynomial and
  Rational Functions
  3.1 Quadratic Functions
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  1
  
  ## 2. Objectives:
  
  Recognize characteristics of parabolas.
  Graph parabolas.
  Determine a quadratic function’s minimum or
  maximum value.
  Solve problems involving a quadratic function’s
  minimum or maximum value.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  2
  
  ## 3. The Standard Form of a Quadratic Function
  
  The quadratic function
  f ( x ) = a ( x - h) 2 + k , a ¹ 0
  is in standard form. The graph of f is a parabola whose
  vertex is the point (h, k). The parabola is symmetric
  with respect to the line x = h. If a > 0, the parabola
  opens upward; if a < 0, the parabola opens downward.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  3
  
  ## 4. Graphing Quadratic Functions with Equations in Standard Form
  
  f ( x ) = a ( x - h) 2 + k , a ¹ 0
  To graph
  1. Determine whether the parabola opens upward or
  downward. If a > 0, it opens upward. If a < 0, it opens
  downward.
  2. Determine the vertex of the parabola. The vertex is
  (h, k).
  3. Find any x-intercepts by solving f(x) = 0. The
  function’s real zeros are the x-intercepts.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  4
  
  ## 5. Graphing Quadratic Functions with Equations in Standard Form (continued)
  
  To graph f ( x) = a ( x - h) 2 + k , a ¹ 0
  4. Find the y-intercept by computing f(0).
  5. Plot the intercepts, the vertex, and additional points
  as necessary. Connect these points with a smooth curve
  that is shaped like a bowl or an inverted bowl.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  5
  
  ## 6. Example: Graphing a Quadratic Function in Standard Form
  
  f ( x) = -( x - 1) 2 + 4
  Graph the quadratic function
  Step 1 Determine how the parabola opens.
  a = –1, a < 0; the parabola opens downward.
  Step 2 Find the vertex.
  The vertex is at (h, k). Because h = 1 and k = 4, the
  parabola has its vertex at (1, 4)
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  6
  
  ## 7. Example: Graphing a Quadratic Function in Standard Form (continued)
  
  2
  f
  (
  x
  )
  =
  (
  x
  1)
  +4
  Graph:
  Step 3 Find the x-intercepts by solving f(x) = 0.
  f ( x) = -( x - 1) 2 + 4
  0 = -( x - 1) 2 + 4
  -4 = -( x - 1) 2
  4 = ( x - 1) 2
  ( x - 1) = ±2
  x -1 = 2
  x=3
  x - 1 = -2
  x = -1
  The x-intercepts are (3, 0) and (–1, 0)
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  7
  
  ## 8. Example: Graphing a Quadratic Function in Standard Form (continued)
  
  f ( x) = -( x - 1) 2 + 4
  Graph:
  Step 4 Find the y-intercept by computing f(0).
  f ( x) = -( x - 1) 2 + 4
  f (0) = -(0 - 1) 2 + 4
  = -(-1) 2 + 4
  = -1 + 4 = 3
  The y-intercept is (0, 3).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  8
  
  ## 9. Example: Graphing a Quadratic Function in Standard Form
  
  Graph: f ( x) = -( x - 1) + 4
  The parabola opens downward.
  The x-intercepts are
  (3, 0) and (–1, 0).
  The y-intercept is (0, 3).
  The vertex is (1, 4).
  2
  ·
  ·
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  ·
  ·
  9
  
  ## 10. The Vertex of a Parabola Whose Equation is
  
  f ( x) = ax 2 + bx + c
  Consider the parabola defined by the quadratic function
  2
  f ( x) = ax + bx + c.
  The parabola’s vertex is æ b
  b öö
  æ
  ç - , f ç - ÷ ÷.
  è 2a è 2a ø ø
  b
  The x-coordinate is - 2a .
  The y-coordinate is found by substituting the
  x-coordinate into the parabola’s equation and evaluating
  the function at this value of x.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  10
  
  ## 11. Graphing Quadratic Functions with Equations in the Form
  
  f ( x) = ax 2 + bx + c
  2
  f
  (
  x
  )
  =
  ax
  + bx + c,
  To graph
  1. Determine whether the parabola opens upward or
  downward. If a > 0, it opens upward. If a < 0, it opens
  downward.
  2. Determine the vertex of the parabola. The vertex is
  b öö
  æ b
  æ
  ç- , f ç- ÷÷
  è 2a è 2a ø ø
  3. Find any x-intercepts by solving f(x) = 0. The real
  2
  ax
  + bx + c = 0 are the x-intercepts.
  solutions of
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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  ## 12. Graphing Quadratic Functions with Equations in the Form (continued)
  
  Graphing Quadratic Functions with Equations in the
  Form f ( x) = ax 2 + bx + c
  (continued)
  To graph f ( x) = ax + bx + c
  4. Find the y-intercept by computing f(0). Because
  f(0) = c (the constant term in the function’s equation),
  the y-intercept is c and the parabola passes through
  (0, c).
  5. Plot the intercepts, the vertex, and additional points
  as necessary. Connect these points with a smooth curve.
  2
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  12
  
  ## 13. Example: Graphing a Quadratic Function in the Form
  
  f ( x) = ax 2 + bx + c
  2
  f
  (
  x
  )
  =
  x
  + 4x + 1
  Graph the quadratic function
  Step 1 Determine how the parabola opens.
  a = –1, a < 0, the parabola opens downward.
  Step 2 Find the vertex.
  b
  The x-coordinate of the vertex is x = - .
  2a
  a = –1, b = 4, and c = 1
  4
  b
  =x==2
  2(-1)
  2a
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  13
  
  ## 14. Example: Graphing a Quadratic Function in the Form (continued)
  
  Example: Graphing a Quadratic Function in the Form
  2
  f ( x) = ax + bx + c (continued)
  2
  f
  (
  x
  )
  =
  x
  + 4x + 1
  Graph:
  Step 2 (continued) find the vertex.
  The coordinates of the vertex are æç - b , f
  We found that x = 2 at the vertex. è 2a
  æ- b öö
  ç
  ÷÷
  è 2a ø ø
  f (2) = -(2) 2 + 4(2) + 1 = -4 + 8 + 1 = 5
  The coordinates of the vertex are (2, 5).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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  ## 15. Example: Graphing a Quadratic Function in the Form (continued)
  
  f ( x) = ax 2 + bx + c
  f ( x) = - x 2 + 4 x + 1
  Graph:
  Step 3 Find the x-intercepts by solving f(x) = 0.
  f ( x) = - x + 4 x + 1 ® 0 = - x 2 + 4 x + 1
  2
  -b ± b 2 - 4ac
  -4 ± (4) 2 - 4(-1)(1) -4 ± 16 + 4
  x=
  =
  =
  2a
  2(-1)
  -2
  -4 ± 20
  -4 + 20
  -4 - 20
  =
  x=
  » -0.2 x =
  » 4.2
  -2
  -2
  -2
  The x-intercepts are (–0.2, 0) and (4.2, 0).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  15
  
  ## 16. Example: Graphing a Quadratic Function in the Form (continued)
  
  f ( x) = ax 2 + bx + c
  Graph: f ( x) = - x + 4 x + 1
  Step 4 Find the y-intercept by computing f(0).
  2
  f (0) = -(0) 2 + 4(0) + 1 = 1
  The y-intercept is (0, 1).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  16
  
  ## 17. Example: Graphing a Quadratic Function in the Form (continued)
  
  Example: Graphing a Quadratic Function in the Form
  2
  (continued)
  f ( x) = ax + bx + c
  2
  f
  (
  x
  )
  =
  x
  + 4x + 1
  Graph:
  Step 5 Graph the parabola.
  The x-intercepts are
  (–0.2, 0) and (4.2, 0).
  The y-intercept is (0, 1).
  The vertex is (2, 5).
  The axis of symmetry is x = 2.
  ·
  ·
  ·
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  ·
  17
  
  ## 18. Minimum and Maximum: Quadratic Functions
  
  2
  f
  (
  x
  )
  =
  ax
  + bx + c.
  Consider the quadratic function
  1. If a > 0, then f has a minimum that occurs at x = b ö
  æ
  This minimum value is f ç - ÷ .
  è 2a ø
  b
  2a
  b
  2. If a < 0, then f has a maximum that occurs at x = 2a
  b ö
  æ
  This maximum value is f ç - ÷ .
  è 2a ø
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  18
  
  ## 19. Minimum and Maximum: Quadratic Functions (continued)
  
  2
  f
  (
  x
  )
  =
  ax
  + bx + c.
  Consider the quadratic function
  b
  In each case, the value of x = gives the location
  2a
  of the minimum or maximum value.
  The value of y, or
  maximum value.
  b ö
  æ
  f ç - ÷ gives that minimum or
  è 2a ø
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  19
  
  ## 20. Example: Obtaining Information about a Quadratic Function from Its Equation
  
  Consider the quadratic function f ( x) = 4 x 2 - 16 x + 1000
  Determine, without graphing, whether the function has a
  minimum value or a maximum value.
  a = 4; a > 0.
  The function has a minimum value.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  20
  
  ## 21. Example: Obtaining Information about a Quadratic Function from Its Equation (continued)
  
  Consider the quadratic function f ( x) = 4 x 2 - 16 x + 1000
  Find the minimum or maximum value and determine
  where it occurs.
  b
  -16 16
  x=- ==
  =2
  a = 4, b = –16, c = 1000
  2a
  2(4) 8
  b ö = f (2)
  2
  æ
  =
  4(2)
  - 16(2) + 1000 = 16 - 32 + 1000
  f ç- ÷
  è 2a ø
  = 984
  The minimum value of f is 984.
  This value occurs at x = 2.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  21
  
  ## 22. Example: Obtaining Information about a Quadratic Function from Its Equation
  
  Consider the quadratic function f ( x) = 4 x 2 - 16 x + 1000
  Identify the function’s domain and range (without
  graphing).
  Like all quadratic functions, the domain is (-¥, ¥).
  We found that the vertex is at (2, 984).
  a > 0, the function has a minimum value at the vertex.
  The range of the function is (984, ¥).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  22
  
  ## 23. Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions
  
  1. Read the problem carefully and decide which
  quantity is to be maximized or minimized.
  2. Use the conditions of the problem to express the
  quantity as a function in one variable.
  2
  f
  (
  x
  )
  =
  ax
  + bx + c.
  3. Rewrite the function in the form
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  23
  
  ## 24. Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions (continued)
  
  b
  b
  4. Calculate - . If a > 0, f has a minimum at x = 2a
  2a
  b ö
  æ
  This minimum value is f ç - ÷ . If a < 0, f has a
  è 2a ø
  b
  maximum at x = - . This maximum value is f
  2a
  æ - b ö.
  ç
  ÷
  è 2a ø
  5. Answer the question posed in the problem.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  24
  
  ## 25. Example: Maximizing Area
  
  You have 120 feet of fencing to enclose a rectangular
  region. Find the dimensions of the rectangle that
  maximize the enclosed area. What is the maximum
  area?
  Step 1 Decide what must be maximized or
  minimized
  We must maximize area.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  25
  
  ## 26. Example: Maximizing Area (continued)
  
  Step 2 Express this quantity as a function in one
  variable.
  We must maximize the area of the rectangle, A = xy.
  We have 120 feet of fencing, the perimeter of the
  rectangle is 120. 2x + 2y = 120
  Solve this equation for y:
  A = xy
  2 y = 120 - 2 x
  A = x(60 - x)
  120 - 2 x = 60 - x
  A = 60 x - x 2
  y=
  2
  A = - x 2 + 60 x
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  26
  
  ## 27. Example: Maximizing Area (continued)
  
  Step 3 Write the function in the form
  f ( x) = ax 2 + bx + c
  A( x) = - x 2 + 60 x
  b
  b
  60
  Step 4 Calculate x=- == 30
  2a
  2a
  2(-1)
  a < 0, so the function has a maximum at this value.
  This means that the area, A(x), of a rectangle with
  perimeter 120 feet is a maximum when one of the
  rectangle’s dimensions, x, is 30 feet.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  27
  
  ## 28. Example: Maximizing Area (continued)
  
  Step 5 Answer the question posed by the problem.
  You have 120 feet of fencing to enclose a rectangular
  region. Find the dimensions of the rectangle that
  maximize the enclosed area. What is the maximum
  area?
  The rectangle that gives the maximum square area has
  dimensions 30 ft by 30 ft.
  A( x) = - x 2 + 60 x ® A(30) = -(30) 2 + 60 x = 900
  The maximum area is 900 square feet.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  28
explanation: The document is a detailed educational material on quadratic functions, including their standard form, graphing techniques, and examples. It contains multiple examples and explanations on how to graph quadratic functions and solve problems related to their minimum and maximum values. The document also includes a step-by-step example of maximizing the area of a rectangular region given a fixed perimeter.
  
  The document contains multiple questions and answers, particularly in the form of examples and problem-solving steps.
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  Категория: Математика
  
  # Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions
  
  ## 1.
  
  Chapter 3
  Polynomial and
  Rational Functions
  3.1 Quadratic Functions
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  1
  
  ## 2. Objectives:
  
  Recognize characteristics of parabolas.
  Graph parabolas.
  Determine a quadratic function’s minimum or
  maximum value.
  Solve problems involving a quadratic function’s
  minimum or maximum value.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  2
  
  ## 3. The Standard Form of a Quadratic Function
  
  The quadratic function
  f ( x ) = a ( x - h) 2 + k , a ¹ 0
  is in standard form. The graph of f is a parabola whose
  vertex is the point (h, k). The parabola is symmetric
  with respect to the line x = h. If a > 0, the parabola
  opens upward; if a < 0, the parabola opens downward.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  3
  
  ## 4. Graphing Quadratic Functions with Equations in Standard Form
  
  f ( x ) = a ( x - h) 2 + k , a ¹ 0
  To graph
  1. Determine whether the parabola opens upward or
  downward. If a > 0, it opens upward. If a < 0, it opens
  downward.
  2. Determine the vertex of the parabola. The vertex is
  (h, k).
  3. Find any x-intercepts by solving f(x) = 0. The
  function’s real zeros are the x-intercepts.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  4
  
  ## 5. Graphing Quadratic Functions with Equations in Standard Form (continued)
  
  To graph f ( x) = a ( x - h) 2 + k , a ¹ 0
  4. Find the y-intercept by computing f(0).
  5. Plot the intercepts, the vertex, and additional points
  as necessary. Connect these points with a smooth curve
  that is shaped like a bowl or an inverted bowl.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  5
  
  ## 6. Example: Graphing a Quadratic Function in Standard Form
  
  f ( x) = -( x - 1) 2 + 4
  Graph the quadratic function
  Step 1 Determine how the parabola opens.
  a = –1, a < 0; the parabola opens downward.
  Step 2 Find the vertex.
  The vertex is at (h, k). Because h = 1 and k = 4, the
  parabola has its vertex at (1, 4)
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  6
  
  ## 7. Example: Graphing a Quadratic Function in Standard Form (continued)
  
  2
  f
  (
  x
  )
  =
  (
  x
  1)
  +4
  Graph:
  Step 3 Find the x-intercepts by solving f(x) = 0.
  f ( x) = -( x - 1) 2 + 4
  0 = -( x - 1) 2 + 4
  -4 = -( x - 1) 2
  4 = ( x - 1) 2
  ( x - 1) = ±2
  x -1 = 2
  x=3
  x - 1 = -2
  x = -1
  The x-intercepts are (3, 0) and (–1, 0)
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  7
  
  ## 8. Example: Graphing a Quadratic Function in Standard Form (continued)
  
  f ( x) = -( x - 1) 2 + 4
  Graph:
  Step 4 Find the y-intercept by computing f(0).
  f ( x) = -( x - 1) 2 + 4
  f (0) = -(0 - 1) 2 + 4
  = -(-1) 2 + 4
  = -1 + 4 = 3
  The y-intercept is (0, 3).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  8
  
  ## 9. Example: Graphing a Quadratic Function in Standard Form
  
  Graph: f ( x) = -( x - 1) + 4
  The parabola opens downward.
  The x-intercepts are
  (3, 0) and (–1, 0).
  The y-intercept is (0, 3).
  The vertex is (1, 4).
  2
  ·
  ·
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  ·
  ·
  9
  
  ## 10. The Vertex of a Parabola Whose Equation is
  
  f ( x) = ax 2 + bx + c
  Consider the parabola defined by the quadratic function
  2
  f ( x) = ax + bx + c.
  The parabola’s vertex is æ b
  b öö
  æ
  ç - , f ç - ÷ ÷.
  è 2a è 2a ø ø
  b
  The x-coordinate is - 2a .
  The y-coordinate is found by substituting the
  x-coordinate into the parabola’s equation and evaluating
  the function at this value of x.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  10
  
  ## 11. Graphing Quadratic Functions with Equations in the Form
  
  f ( x) = ax 2 + bx + c
  2
  f
  (
  x
  )
  =
  ax
  + bx + c,
  To graph
  1. Determine whether the parabola opens upward or
  downward. If a > 0, it opens upward. If a < 0, it opens
  downward.
  2. Determine the vertex of the parabola. The vertex is
  b öö
  æ b
  æ
  ç- , f ç- ÷÷
  è 2a è 2a ø ø
  3. Find any x-intercepts by solving f(x) = 0. The real
  2
  ax
  + bx + c = 0 are the x-intercepts.
  solutions of
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  11
  
  ## 12. Graphing Quadratic Functions with Equations in the Form (continued)
  
  Graphing Quadratic Functions with Equations in the
  Form f ( x) = ax 2 + bx + c
  (continued)
  To graph f ( x) = ax + bx + c
  4. Find the y-intercept by computing f(0). Because
  f(0) = c (the constant term in the function’s equation),
  the y-intercept is c and the parabola passes through
  (0, c).
  5. Plot the intercepts, the vertex, and additional points
  as necessary. Connect these points with a smooth curve.
  2
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  12
  
  ## 13. Example: Graphing a Quadratic Function in the Form
  
  f ( x) = ax 2 + bx + c
  2
  f
  (
  x
  )
  =
  x
  + 4x + 1
  Graph the quadratic function
  Step 1 Determine how the parabola opens.
  a = –1, a < 0, the parabola opens downward.
  Step 2 Find the vertex.
  b
  The x-coordinate of the vertex is x = - .
  2a
  a = –1, b = 4, and c = 1
  4
  b
  =x==2
  2(-1)
  2a
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  13
  
  ## 14. Example: Graphing a Quadratic Function in the Form (continued)
  
  Example: Graphing a Quadratic Function in the Form
  2
  f ( x) = ax + bx + c (continued)
  2
  f
  (
  x
  )
  =
  x
  + 4x + 1
  Graph:
  Step 2 (continued) find the vertex.
  The coordinates of the vertex are æç - b , f
  We found that x = 2 at the vertex. è 2a
  æ- b öö
  ç
  ÷÷
  è 2a ø ø
  f (2) = -(2) 2 + 4(2) + 1 = -4 + 8 + 1 = 5
  The coordinates of the vertex are (2, 5).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  14
  
  ## 15. Example: Graphing a Quadratic Function in the Form (continued)
  
  f ( x) = ax 2 + bx + c
  f ( x) = - x 2 + 4 x + 1
  Graph:
  Step 3 Find the x-intercepts by solving f(x) = 0.
  f ( x) = - x + 4 x + 1 ® 0 = - x 2 + 4 x + 1
  2
  -b ± b 2 - 4ac
  -4 ± (4) 2 - 4(-1)(1) -4 ± 16 + 4
  x=
  =
  =
  2a
  2(-1)
  -2
  -4 ± 20
  -4 + 20
  -4 - 20
  =
  x=
  » -0.2 x =
  » 4.2
  -2
  -2
  -2
  The x-intercepts are (–0.2, 0) and (4.2, 0).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  15
  
  ## 16. Example: Graphing a Quadratic Function in the Form (continued)
  
  f ( x) = ax 2 + bx + c
  Graph: f ( x) = - x + 4 x + 1
  Step 4 Find the y-intercept by computing f(0).
  2
  f (0) = -(0) 2 + 4(0) + 1 = 1
  The y-intercept is (0, 1).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  16
  
  ## 17. Example: Graphing a Quadratic Function in the Form (continued)
  
  Example: Graphing a Quadratic Function in the Form
  2
  (continued)
  f ( x) = ax + bx + c
  2
  f
  (
  x
  )
  =
  x
  + 4x + 1
  Graph:
  Step 5 Graph the parabola.
  The x-intercepts are
  (–0.2, 0) and (4.2, 0).
  The y-intercept is (0, 1).
  The vertex is (2, 5).
  The axis of symmetry is x = 2.
  ·
  ·
  ·
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  ·
  17
  
  ## 18. Minimum and Maximum: Quadratic Functions
  
  2
  f
  (
  x
  )
  =
  ax
  + bx + c.
  Consider the quadratic function
  1. If a > 0, then f has a minimum that occurs at x = b ö
  æ
  This minimum value is f ç - ÷ .
  è 2a ø
  b
  2a
  b
  2. If a < 0, then f has a maximum that occurs at x = 2a
  b ö
  æ
  This maximum value is f ç - ÷ .
  è 2a ø
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  18
  
  ## 19. Minimum and Maximum: Quadratic Functions (continued)
  
  2
  f
  (
  x
  )
  =
  ax
  + bx + c.
  Consider the quadratic function
  b
  In each case, the value of x = gives the location
  2a
  of the minimum or maximum value.
  The value of y, or
  maximum value.
  b ö
  æ
  f ç - ÷ gives that minimum or
  è 2a ø
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  19
  
  ## 20. Example: Obtaining Information about a Quadratic Function from Its Equation
  
  Consider the quadratic function f ( x) = 4 x 2 - 16 x + 1000
  Determine, without graphing, whether the function has a
  minimum value or a maximum value.
  a = 4; a > 0.
  The function has a minimum value.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  20
  
  ## 21. Example: Obtaining Information about a Quadratic Function from Its Equation (continued)
  
  Consider the quadratic function f ( x) = 4 x 2 - 16 x + 1000
  Find the minimum or maximum value and determine
  where it occurs.
  b
  -16 16
  x=- ==
  =2
  a = 4, b = –16, c = 1000
  2a
  2(4) 8
  b ö = f (2)
  2
  æ
  =
  4(2)
  - 16(2) + 1000 = 16 - 32 + 1000
  f ç- ÷
  è 2a ø
  = 984
  The minimum value of f is 984.
  This value occurs at x = 2.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  21
  
  ## 22. Example: Obtaining Information about a Quadratic Function from Its Equation
  
  Consider the quadratic function f ( x) = 4 x 2 - 16 x + 1000
  Identify the function’s domain and range (without
  graphing).
  Like all quadratic functions, the domain is (-¥, ¥).
  We found that the vertex is at (2, 984).
  a > 0, the function has a minimum value at the vertex.
  The range of the function is (984, ¥).
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  22
  
  ## 23. Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions
  
  1. Read the problem carefully and decide which
  quantity is to be maximized or minimized.
  2. Use the conditions of the problem to express the
  quantity as a function in one variable.
  2
  f
  (
  x
  )
  =
  ax
  + bx + c.
  3. Rewrite the function in the form
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  23
  
  ## 24. Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions (continued)
  
  b
  b
  4. Calculate - . If a > 0, f has a minimum at x = 2a
  2a
  b ö
  æ
  This minimum value is f ç - ÷ . If a < 0, f has a
  è 2a ø
  b
  maximum at x = - . This maximum value is f
  2a
  æ - b ö.
  ç
  ÷
  è 2a ø
  5. Answer the question posed in the problem.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  24
  
  ## 25. Example: Maximizing Area
  
  You have 120 feet of fencing to enclose a rectangular
  region. Find the dimensions of the rectangle that
  maximize the enclosed area. What is the maximum
  area?
  Step 1 Decide what must be maximized or
  minimized
  We must maximize area.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  25
  
  ## 26. Example: Maximizing Area (continued)
  
  Step 2 Express this quantity as a function in one
  variable.
  We must maximize the area of the rectangle, A = xy.
  We have 120 feet of fencing, the perimeter of the
  rectangle is 120. 2x + 2y = 120
  Solve this equation for y:
  A = xy
  2 y = 120 - 2 x
  A = x(60 - x)
  120 - 2 x = 60 - x
  A = 60 x - x 2
  y=
  2
  A = - x 2 + 60 x
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  26
  
  ## 27. Example: Maximizing Area (continued)
  
  Step 3 Write the function in the form
  f ( x) = ax 2 + bx + c
  A( x) = - x 2 + 60 x
  b
  b
  60
  Step 4 Calculate x=- == 30
  2a
  2a
  2(-1)
  a < 0, so the function has a maximum at this value.
  This means that the area, A(x), of a rectangle with
  perimeter 120 feet is a maximum when one of the
  rectangle’s dimensions, x, is 30 feet.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  27
  
  ## 28. Example: Maximizing Area (continued)
  
  Step 5 Answer the question posed by the problem.
  You have 120 feet of fencing to enclose a rectangular
  region. Find the dimensions of the rectangle that
  maximize the enclosed area. What is the maximum
  area?
  The rectangle that gives the maximum square area has
  dimensions 30 ft by 30 ft.
  A( x) = - x 2 + 60 x ® A(30) = -(30) 2 + 60 x = 900
  The maximum area is 900 square feet.
  Copyright © 2014, 2010, 2007 Pearson Education, Inc.
  28
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a detailed educational material on quadratic functions, including their standard form, graphing techniques, and examples. It contains multiple examples and explanations on how to graph quadratic functions and solve problems related to their minimum and maximum values. The document also includes a step-by-step example of maximizing the area of a rectangular region given a fixed perimeter.
  
  The document contains multiple questions and answers, particularly in the form of examples and problem-solving steps.
  
  Label: multiple-question-and-answer