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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).

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Chapter 8 Quadratic Functions and Equations

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  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

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