1 / 16

Chapter 8 Quadratic Functions and Equations

Chapter 8 Quadratic Functions and Equations. QuadraticFunction. A quadratic equation is an equation that can be written as  f(x) = ax 2 + bx + c , where a, b, c are real numbers, with a = 0. Axis of symmetry. (0, 2). -2 1 -1. 2 1 0. (0, 0). -2 -1 0 1 2. (2, -1).

kenneth-smith
Télécharger la présentation

Chapter 8 Quadratic Functions and Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 8 Quadratic Functions and Equations

  2. QuadraticFunction A quadratic equation is an equation that can be written as  f(x) = ax2 + bx + c , where a, b, c are real numbers, with a = 0. Axis of symmetry (0, 2) -2 1 -1 2 1 0 (0, 0) -2 -1 0 1 2 (2, -1) Vertex x = 2

  3. Vertex Formula The x-coordinate of the vertex of the graph of y = ax2 +bx +c, a = 0, is given by   x = -b/2a  To find the y-coordinate of the vertex, substitute this x-value into the equation

  4. Example 1 (pg 578)Graph the equation f(x) = x2 -1whether it is increasing or decreasing and Identify the vertex and axis of symmetry x y = x2 -1 • -2 3 • -1 0 • 0 -1 • 0 • 2 3 vertex Equal 3 2 1 0 -1 Vertex The graph is decreasing when x < 0 And Increasing when x > 0

  5. Example 1(c) pg 578 x y = x2 + 4x + 3 -5 8 -4 3 -3 0 -2 -1 -1 0 0 3 1 8 Axis of symmetry Equal Vertex x = -2 Vertex (-2, -1)

  6. Find the vertex of a parabolaf(x) = 2x2 - 4x + 1 Symbolically f(x) = 2x2 – 4x + 1 a = 2 , b = - 4 x = -b/2a = - (-4)/2.2 = 4/4 = 1 To find the y-value of the vertex, Substitute x = 1 in the given formula f(1) = 2. 12 - 4.1 + 1= -1 The vertex is (1, -1) Graphically [ -4.7, 4.7, 1] , [-3.1, 3.1, 1]

  7. Example 7 (Pg 583) Maximizing Revenue The regular price of a hotel room is $ 80, Each room rented the price decreases by $2 (20, 800) 900 800 700 600 500 400 Maximum revenue 0 5 10 15 20 25 30 35 40 If x rooms are rented then the price of each room is 80 – 2x The revenue equals the number of rooms rented times the price of each room. Thus f(x) = x(80 – 2x) = 80x - 2x2 = -2x2 + 80x The x-coordinate of the vertex x = - b/2a = - 80/ 2(-2) = 20 Y coordinate f(20) = -2(20)2 + 80 (20) = 800

  8. 8.2 Vertical and Horizontal Translations Translated upward and downward y2 = x2 + 1 y1= x2 y3 = x2 - 2 y1= x2 y1= (x-1)2 Translated horizontally to the right 1 unit y1= x2 y2= (x + 2 )2 Translated horizontally to the left 2 units

  9. Vertical and Horizontal Translations Of Parabolas (pg 591) Let h , k be positive numbers. To graph shift the graph of y = x2 by k units y = x2 + k upward y = x2 – k downward y = (x – k)2 right y = (x +k)2 left

  10. Vertex Form of a Parabola (Pg 592) The vertex form of a parabola with vertex (h, k) is y = a (x – h)2 + k, where a = 0 is a constant. If a > 0, the parabola opens upward; if a < 0, the parabola opens downward.

  11. Ch 8.3 Quadratic Equations A quadratic equation is an equation that can be written as ax2 +bx +c= 0, where a, b, c are real numbers with a = 0

  12. Quadratic Equations and Solutions y = x2 + 25 y = 4x2 – 20x + 25 y = 3x2 + 11x - 20 No Solution One Solution Two Solutions

  13. Ch 8.4 Quadratic Formula The solutions of the quadratic equation ax2 + bx + c = 0, where a, b, c are real numbers with a = 0 No x intercepts One x – intercepts Two x - intercepts Ex 1

  14. Modeling Internet Users Use of the Internet in Western Europe has increased dramatically shows a scatter plot of online users in Western Europe with function f given by f(x) = 0.976 x2 - 4.643x + 0.238x = 6 corresponds to 1996 and so on until x = 12 represents 2002 90 80 70 60 50 40 30 20 10 0 f(10) = 0.976(10) 2 - 4.643(10) + 0.238 = 51.4 6 7 8 9 10 11 12 13

  15. 8.4 Quadratic Formula The solutions to ax 2 + bx + c = 0 with a = 0 are given by - b + b2– 4ac X = 2a

  16. The Discriminant and Quadratic Equation To determine the number of solutions to ax2 + bx + c = 0 , evaluate the discriminant b 2 – 4ac > 0, If b 2 – 4ac > 0,there are two real solutions If b 2 – 4ac= 0,there is one real solution If b 2 – 4ac < 0,there are no real solutions , but two complex solution

More Related