🗊Презентация Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions

Категория: Математика
Нажмите для полного просмотра!
Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №1Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №2Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №3Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №4Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №5Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №6Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №7Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №8Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №9Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №10Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №11Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №12Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №13Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №14Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №15Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №16Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №17Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №18Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №19Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №20Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №21Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №22Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №23Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №24Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №25Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №26Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №27Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №28
Скачать
Вы можете ознакомиться и скачать презентацию на тему Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions. Доклад-сообщение содержит 28 слайдов. Презентации для любого класса можно скачать бесплатно. Если материал и наш сайт презентаций Mypresentation Вам понравились – поделитесь им с друзьями с помощью социальных кнопок и добавьте в закладки в своем браузере.

Слайды и текст этой презентации

Слайд 1


Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions, слайд №1
Описание слайда:

Слайд 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.
Описание слайда:
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.

Слайд 3


The Standard Form of a Quadratic Function The quadratic function 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.
Описание слайда:
The Standard Form of a Quadratic Function The quadratic function 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.

Слайд 4


Graphing Quadratic Functions with Equations in Standard Form 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.
Описание слайда:
Graphing Quadratic Functions with Equations in Standard Form 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.

Слайд 5


Graphing Quadratic Functions with Equations in Standard Form (continued) To graph 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.
Описание слайда:
Graphing Quadratic Functions with Equations in Standard Form (continued) To graph 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.

Слайд 6


Example: Graphing a Quadratic Function in Standard Form 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)
Описание слайда:
Example: Graphing a Quadratic Function in Standard Form 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)

Слайд 7


Example: Graphing a Quadratic Function in Standard Form (continued) Graph: Step 3 Find the x-intercepts by solving f(x) = 0. The x-intercepts are (3, 0) and (–1, 0)
Описание слайда:
Example: Graphing a Quadratic Function in Standard Form (continued) Graph: Step 3 Find the x-intercepts by solving f(x) = 0. The x-intercepts are (3, 0) and (–1, 0)

Слайд 8


Example: Graphing a Quadratic Function in Standard Form (continued) Graph: Step 4 Find the y-intercept by computing f(0). The y-intercept is (0, 3).
Описание слайда:
Example: Graphing a Quadratic Function in Standard Form (continued) Graph: Step 4 Find the y-intercept by computing f(0). The y-intercept is (0, 3).

Слайд 9


Example: Graphing a Quadratic Function in Standard Form Graph: The parabola opens downward. The x-intercepts are (3, 0) and (–1, 0). The y-intercept is (0, 3). The vertex is (1, 4).
Описание слайда:
Example: Graphing a Quadratic Function in Standard Form Graph: The parabola opens downward. The x-intercepts are (3, 0) and (–1, 0). The y-intercept is (0, 3). The vertex is (1, 4).

Слайд 10


The Vertex of a Parabola Whose Equation is Consider the parabola defined by the quadratic function The parabola’s vertex is The x-coordinate is The y-coordinate is found by substituting the x-coordinate into the parabola’s equation and evaluating the function at this value of x.
Описание слайда:
The Vertex of a Parabola Whose Equation is Consider the parabola defined by the quadratic function The parabola’s vertex is The x-coordinate is The y-coordinate is found by substituting the x-coordinate into the parabola’s equation and evaluating the function at this value of x.

Слайд 11


Graphing Quadratic Functions with Equations in the Form 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 3. Find any x-intercepts by solving f(x) = 0. The real solutions of are the x-intercepts.
Описание слайда:
Graphing Quadratic Functions with Equations in the Form 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 3. Find any x-intercepts by solving f(x) = 0. The real solutions of are the x-intercepts.

Слайд 12


Graphing Quadratic Functions with Equations in the Form (continued) To graph 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.
Описание слайда:
Graphing Quadratic Functions with Equations in the Form (continued) To graph 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.

Слайд 13


Example: Graphing a Quadratic Function in the Form 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 x-coordinate of the vertex is a = –1, b = 4, and c = 1
Описание слайда:
Example: Graphing a Quadratic Function in the Form 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 x-coordinate of the vertex is a = –1, b = 4, and c = 1

Слайд 14


Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 2 (continued) find the vertex. The coordinates of the vertex are We found that x = 2 at the vertex. The coordinates of the vertex are (2, 5).
Описание слайда:
Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 2 (continued) find the vertex. The coordinates of the vertex are We found that x = 2 at the vertex. The coordinates of the vertex are (2, 5).

Слайд 15


Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 3 Find the x-intercepts by solving f(x) = 0. The x-intercepts are (–0.2, 0) and (4.2, 0).
Описание слайда:
Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 3 Find the x-intercepts by solving f(x) = 0. The x-intercepts are (–0.2, 0) and (4.2, 0).

Слайд 16


Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 4 Find the y-intercept by computing f(0). The y-intercept is (0, 1).
Описание слайда:
Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 4 Find the y-intercept by computing f(0). The y-intercept is (0, 1).

Слайд 17


Example: Graphing a Quadratic Function in the Form (continued) 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.
Описание слайда:
Example: Graphing a Quadratic Function in the Form (continued) 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.

Слайд 18


Minimum and Maximum: Quadratic Functions Consider the quadratic function 1. If a > 0, then f has a minimum that occurs at This minimum value is 2. If a < 0, then f has a maximum that occurs at This maximum value is
Описание слайда:
Minimum and Maximum: Quadratic Functions Consider the quadratic function 1. If a > 0, then f has a minimum that occurs at This minimum value is 2. If a < 0, then f has a maximum that occurs at This maximum value is

Слайд 19


Minimum and Maximum: Quadratic Functions (continued) Consider the quadratic function In each case, the value of gives the location of the minimum or maximum value. The value of y, or gives that minimum or maximum value.
Описание слайда:
Minimum and Maximum: Quadratic Functions (continued) Consider the quadratic function In each case, the value of gives the location of the minimum or maximum value. The value of y, or gives that minimum or maximum value.

Слайд 20


Example: Obtaining Information about a Quadratic Function from Its Equation Consider the quadratic function Determine, without graphing, whether the function has a minimum value or a maximum value. a = 4; a > 0. The function has a minimum value.
Описание слайда:
Example: Obtaining Information about a Quadratic Function from Its Equation Consider the quadratic function Determine, without graphing, whether the function has a minimum value or a maximum value. a = 4; a > 0. The function has a minimum value.

Слайд 21


Example: Obtaining Information about a Quadratic Function from Its Equation (continued) Consider the quadratic function Find the minimum or maximum value and determine where it occurs. a = 4, b = –16, c = 1000 The minimum value of f is 984. This value occurs at x = 2.
Описание слайда:
Example: Obtaining Information about a Quadratic Function from Its Equation (continued) Consider the quadratic function Find the minimum or maximum value and determine where it occurs. a = 4, b = –16, c = 1000 The minimum value of f is 984. This value occurs at x = 2.

Слайд 22


Example: Obtaining Information about a Quadratic Function from Its Equation Consider the quadratic function 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
Описание слайда:
Example: Obtaining Information about a Quadratic Function from Its Equation Consider the quadratic function 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

Слайд 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. 3. Rewrite the function in the form
Описание слайда:
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. 3. Rewrite the function in the form

Слайд 24


Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions (continued) 4. Calculate If a > 0, f has a minimum at This minimum value is If a < 0, f has a maximum at This maximum value is 5. Answer the question posed in the problem.
Описание слайда:
Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions (continued) 4. Calculate If a > 0, f has a minimum at This minimum value is If a < 0, f has a maximum at This maximum value is 5. Answer the question posed in the problem.

Слайд 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.
Описание слайда:
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.

Слайд 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:
Описание слайда:
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:

Слайд 27


Example: Maximizing Area (continued) Step 3 Write the function in the form Step 4 Calculate 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.
Описание слайда:
Example: Maximizing Area (continued) Step 3 Write the function in the form Step 4 Calculate 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.

Слайд 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. The maximum area is 900 square feet.
Описание слайда:
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. The maximum area is 900 square feet.

Скачать

Похожие презентации
Mypresentation.ru
Загрузить презентацию