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Chapter 8 Quadratic Functions and Equations

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Chapter 8

Quadratic Functions and Equations

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

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

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

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)

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]

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

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

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

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.

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

y = x2 + 25 y = 4x2 – 20x + 25 y = 3x2 + 11x - 20

No Solution One Solution Two Solutions

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

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

The solutions to ax 2 + bx + c = 0 with a = 0 are given by - b + b2– 4ac

X =

2a

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