Quadratic Equations, Inequalities and Functions Module 1 Lesson 1 Quadratic Functions

2 LAS # 2: STANDARD Form of a Quadratic Function The function defined by the second degree equation f(x) = ax 2 + bx + c where a, b, and c are real numbers and a ≠ 0, is a quadratic function in x. This function can also be written as y = ax 2 + bx + c, where y = f(x). Example: 1. y = 2x 2 – 3x y = 3x 2 + 5x 3. y = 4x y = x 2 abcabc y = -5x 2 – 3x Copyright © by Mr. Florben G. Mendoza

3 STANDARD FORM of a Quadratic Function Example: Write the following equations in STANDARD FORM. 1. y – 12 = - 5x + 3x 2 y = – 5x + 3x y = (x + 5) 2 – 2(x + 17) + 55 y = x x + 25 y = x 2 + 8x y = 7 – (x – 3) (x + 3) y = 7 – (x 2 – 9) y = 7 – x y = -x f (x) = 3x 2 – 5x + 12 (a + b) 2 = a 2 + 2ab + b 2 (x + 5) 2 = x 2 + 2(x)(5) (x + 5) 2 = x x + 25 – 2(x + 17)= -2x - 34 y = 3x 2 – 5x + 12 (a + b) ( a – b) = a 2 - b 2 (x – 3) (x + 3)= x f(x) = x 2 + 8x + 46 or f(x) = -x – 2x – f(x) = ax 2 + bx + c Copyright © by Mr. Florben G. Mendoza

4 4. y = x – 2(3x – 1) 2 - 5(a – b) 2 = a 2 – 2ab +b 2 (3x – 1) 2 = (3x) 2 – 2(3x)(1) +1 2 (3x – 1) 2 = 9x 2 – 6x +1 y = x – 2(9x 2 – 6x +1) - 5 y = x – 18x x y = – 18x x -7 f(x) = – 18x x -7 Copyright © by Mr. Florben G. Mendoza

5 Equal Differences Method x-2012 y nd Difference 1 st Difference x f(x ) st Difference The ordered pairs represents a QUADRATIC FUNCTION 1) (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4) Direction: Using the equal differences method, determine which of the following ordered pairs represent a quadratic function. 2) (-6, -7), (-4, -3), (-2, 1), (0, 5), (2, 9), (4, 13), (6, 17) 3) (-4, -73), (-2, -13), (0, -1), (2, 11), (4, 71) x f(x ) st Difference 2 nd Difference 3 rd Difference NOT A QUADRATIC FUNCTION – LINEAR FUNCTION NOT A QUADRATIC FUNCTION – CUBIC FUNCTION Copyright © by Mr. Florben G. Mendoza

6 LAS # 2 Activity # 1 1. y – 5 = - 2x + 4x 2 2. y - 6 = (x + 3) 2 – 2(x + 12) 3. y = 9 – (x – 4) (x + 4) 4. y + 5 = x – 3(2x – 1) 2 5. y = 5 + x(x - 3) + (x – 5) 2 I. Write each of these quadratic function in general form, then identify the real numbers a, b, and c. General Formabc Copyright © by Mr. Florben G. Mendoza

7 II. Using the equal differences method, name if the given ordered pairs represents the following: (a)Linear Function(c) Cubic Function(e) Quintic Function (b)Quadratic Function (d) Quartic Function 5. {(-3, 83), (-2, 18), (-1, 3), (0, 2), (1, 3), (2, 18), (3, 83)} 2. {(-5, 60), (-2, 15), (1, 6), (4, 33), (7, 96), (10, 195)} 3. {(-3, -243), (-2, -32), (-1, -1), (0, 0), (1, 1), (2, 32), (3, 243)} 4. {(-3, -49), (-2, -19), (-1, -5), (0, -1), (1, -1), (2, 1), (3, 11)} 1. {(-6, -23), (-4, -17), (-2, -11), (0, -5), (2, 1), (4, 7), (6, 13)} Copyright © by Mr. Florben G. Mendoza

8 LAS # 3: Graphs of Quadratic Functions The graph of a quadratic function offers an interpretation of the nature of its zeros, its symmetry and other characteristics. The usual method of graphing by plotting points can be used for this purpose. Consider the following functions and their graph. 1. f(x) = x 2 or y = x 2 x y Parabola Vertex: The axis of symmetry intersects the parabola at a point called the vertex. (highest or lowest point of the graph) Axis of Symmetry: The axis of symmetry is the line that divide the graph into two halves Axis of Symmetry: x = 0 Vertex: (0, 0) Copyright © by Mr. Florben G. Mendoza Symmetric Points: Points that has equal distance from the axis of symmetry.

9 2. f(x) = -x 2 or y = -x 2 x y f(x) = x 2 or y = x 2 x y Graphs of Quadratic Functions: (y = ax 2 ) When a is positive, the graph opens upward. When a is negative, the graph opens downward. The graph of f(x)= x 2 and f(x) = -x 2 are reflections of each other about the x-axis. Copyright © by Mr. Florben G. Mendoza

10 1. f(x) = x 2 x y Graphs of Quadratic Functions: (y = ax 2 ) 2. f(x) = 1 2 x2x2 x y f(x) = 2x 2 x y As |a| increases the graph becomes narrower and closer to the y-axis. Copyright © by Mr. Florben G. Mendoza

11 GivenVertexAxis of Symmetry Direction of opening Symmetric Points Illustration (Graph) 1. y = 2x 2 2. y = ½ x 2 3. y = -3x 2 (0, 0) x = 0 Upward (1, 2) (-1, 2) (0, 0) x = 0 Upward (1, ½ ) (-1, ½ ) (0, 0)x = 0Downward (1, -3) (-1, -3) Copyright © by Mr. Florben G. Mendoza y = ax 2 Vertex: (0,0)Axis of Symmetry: x = 0

12 Graphs of Quadratic Functions: y = x 2 + k y = x 2 y = x y = x In general, if f(x) = x 2 + k, the vertex of the parabola lies on the y-axis, but shifted vertically |k| units upward if k > 0 and downward if k < 0. Observe that the graph have different vertices though all of them are located along the y-axis. 1. y = x y = x y = x Vertex (0, 4) (0, -5) (0, 1) Copyright © by Mr. Florben G. Mendoza y = x 2 + k

13 GivenVertexAxis of Symmetry Direction of opening Symmetric Points Illustration (Graph) 1. y = x y = x y = -x (0, 1) x = 0 Upward (1, 2) (-1, 2) (0, -3) x = 0 Upward (-1, -2 ) (1, -2) (0, -2) x = 0 Downward (1, -3) (-1, -3) Copyright © by Mr. Florben G. Mendoza y = x 2 + k Axis of Symmetry: x = 0Vertex: (0,k)

14 Graphs of Quadratic Functions: y = (x - h) 2 y = x 2 y =(x - 3) 2 y =(x + 2) 2 Notice that the only difference in the graphs is the position of the vertex of each parabola. All vertices are on the x-axis, but translated horizontally. In general, if f(x) = (x - h) 2, the vertex of the parabola is at (h, 0) 3. y = (x + 5) 2 Vertex 2. y = (x - 3) 2 1. y = (x - 4) 2 (3, 0) (-5, 0) (4, 0) Copyright © by Mr. Florben G. Mendoza y = (x-h) 2

15 GivenVertexAxis of Symmetry Direction of opening Symmetric Points Illustration (Graph) 1. y = (x + 1) 2 2. y = -(x – 3) 2 3. y = (x + 2) 2 (-1, 0) x = -1 Upward (0, 1) (-2, 1) (3, 0) x = 3 Downward (2, -1 ) (4, -1) (-2, 0) x = -2 Upward (-1, 1) (-3, 1) Copyright © by Mr. Florben G. Mendoza y = (x - h) 2 Vertex: (h,0)Axis of Symmetry: x = h

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 GivenVertexAxis of Symmetry Direction of opening Symmetric Points Illustration (Graph) 1.y = x 2 + 8x + 16 Check: 16 = 4 x 2 = 8 y = (x + 4) 2 (-4, 0) x = -4 Upward (-3, 1) (-5, 1)

17 Graphs of Quadratic Functions y = (x – 3) x y Now consider the graph of f(x) = a (x-h) 2 + k, a ≠ 0. Observe that the vertex is at (h, k) and the axis of symmetry is at x = h. Copyright © by Mr. Florben G. Mendoza y = a(x-h) 2 + k Vertex Form

18 1. y = x 2 – 4x + 5 Solution: Copyright © by Mr. Florben G. Mendoza Graphs of Quadratic Functions: y = a(x-h) 2 + k y = ( x 2 – 4x) + 5 y = [( x 2 – 4x + 4) – 4] + 5 y = [( x – 2) 2 – 4] + 5 y = ( x – 2) 2 – Vertex:(2, 1) Axis of Symmetry: Direction: x = 2 upward Symmetric Points: (1, 2) (3, 2) y = ( x – 2) 2 + 1

19 Graphs of Quadratic Functions: y = a(x-h) 2 + k 2. y = 2x 2 – 4x - 5 Solution: y = 2 ( x 2 – 2x) - 5 y = 2 [( x 2 – 2x + 1) – 1] - 5 y = 2 [( x – 1) 2 – 1] - 5 y = 2 ( x – 1) 2 – y = 2 [( x – 1) 2 – 1] - 5 y = 2 ( x – 1) 2 – 7 Vertex:(1, -7) Axis of Symmetry: Direction: x = 1 upward Symmetric Points: (0, -5) (2, -5) Copyright © by Mr. Florben G. Mendoza

20 3. y = -x 2 - 2x - 3 Solution: Copyright © by Mr. Florben G. Mendoza y = - (x 2 + 2x) - 3 y = - [(x + 1) 2 – 1] - 3 y = - ( x + 1) 2 -2 Vertex:(-1, -2) Axis of Symmetry: Direction: y = - (x + 1) (0, -3) (-2, -3) x = -1 Symmetric Points: downward y = - [(x 2 + 2x +1) – 1]- 3 Graphs of Quadratic Functions: y = a(x-h) 2 + k

21 Graphs of Quadratic Functions: y = a(x-h) 2 + k 4. y = - 3x x - 23 Solution: y = - 3(x 2 – 6x) - 23 y = - 3[(x 2 – 6x +9) – 9]- 23 y = -3[(x – 3) 2 – 9] - 23 y = - 3( x – 3) Vertex:(3, 4) Axis of Symmetry: Direction: y = - 3(x – 3) (2, 1) (4, 1) x = 3 Symmetric Points: downward Copyright © by Mr. Florben G. Mendoza

22 Copyright © by Mr. Florben G. Mendoza LAS # 4: Activity # 2 I. Complete the table below. GivenVertexAxis of Symmetry Direction of opening Symmetric Points Illustration (Graph) 1. y = 2x 2 2. y = -3x y = 5x y = (x – 7) 2 3. y = 2(x + 5) 2 4. y = x 2 – 6x y = x x y = -x 2 - 8x y = (x – 1) y = -2(x + 2) 2 - 3

23 Copyright © by Mr. Florben G. Mendoza LAS #5: Activity # 3 II. Complete the table below. GivenVertex Form VertexAxis of Symmetry Direction of opening Symmetric Points Illustration (Graph) 1. y = x 2 – 6x y = x 2 – 8x y = x x y = 3x x y = -2x 2 + 8x - 5