Transcript MAT 1033/Ch 8

Chapter 8
Quadratic Functions and
Equations
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 =
Axis of symmetry
0.
(0, 2)
-2
1
2
1
(0, 0)
0
-2
-1 0 1 2
-1
(2, -1)
Vertex
x=2
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
Example 1 (pg 578)
Graph the equation f(x) = x2 -1 whether it is
increasing or decreasing and Identify the vertex and
axis of symmetry
vertex
x
y = x2 -1
-2
3
-1
0
0
-1
1
0
2
3
Equal
3
2
1
0
-1
Vertex
The graph is decreasing when x < 0
And Increasing when x > 0
Example 1(c) pg 578
x
-5
-4
-3
Vertex -2
-1
0
1
y = x2 + 4x + 3
8
3
0
-1
0
3
Axis of symmetry
Equal
x = -2
8
Vertex (-2, -1)
Find the vertex of a parabola
f(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]
Example 7 (Pg 583)
Maximizing Revenue
The regular price of a hotel room is $ 80, Each room
rented the price decreases by $2
900
800
700
600
500
400
(20, 800)
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
8.2 Vertical and Horizontal Translations
Translated upward and downward
y2 = x2 + 1
y1=
y1= x2
y1= (x-1)2
x2
y3 = x2 - 2
Translated
horizontally to the
right 1 unit
y1= x2
y2= (x + 2 )2
Translated horizontally to the left 2 units
Vertical and Horizontal Translations Of
Parabolas (pg 591)
Let h , k be positive numbers.
To graph
y = x2 + k
y = x2 – k
y = (x – k)2
y = (x +k)2
shift the graph of y = x2 by k units
upward
downward
right
left
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.
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
Quadratic Equations and Solutions
y = x2 + 25
No Solution
y = 4x2 – 20x + 25
One Solution
y = 3x2 + 11x - 20
Two Solutions
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
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 + 0.238 x = 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
8.4 Quadratic Formula
The solutions to ax 2 + bx + c = 0 with a = 0
are given by - b + b2 – 4ac
X=
2a
The Discriminant and Quadratic Equation
To determine the number of solutions to
ax2 + bx + c = 0 , evaluate the discriminant
b
If
If
If
2
2
– 4ac
> 0,
b – 4ac > 0, there are two real solutions
2
b – 4ac = 0, there is one real solution
2
b – 4ac
< 0, there are no real solutions , but two complex
solution