Graphing Quadratic Functions

1
5.1
Graphing Quadratic Functions
What you should learn
Goal
1
Graph quadratic functions.
Goal
2
Use quadratic functions to solve real-life
problems.
5.1 Graphing Quadratic Functions
2
5.1 Graphing Quadratic Functions
3
Vocabulary
A parabola is the U-shaped graph of a quadratic
function. The vertex of a parabola is the lowest
point of a parabola that opens up, and the
highest point of a parabola that opens down.
5.1 Graphing Quadratic Functions
4
Example 1
Graphing a Quadratic Function
Graph
The coefficients are a 1, b -4, c 3 Since a
gt 0, the parabola opens up. To find the
x-coordinate of the vertex, substitute 1 for a
and -4 for b in the formula
5.1 Graphing Quadratic Functions
5
To find the y-coordinate of the vertex,
substitute 2 for x in the original equation, and
solve for y.
5.1 Graphing Quadratic Functions
6
The vertex is (2, -1). Plot two points, such as
(1,0) and (0,3). Then use symmetry to plot two
more points (3,0) and (4,3). Draw the parabola.
5.1 Graphing Quadratic Functions
7
Additional Example 1
5.1 Graphing Quadratic Functions
8
Additional Example 2
5.1 Graphing Quadratic Functions
9
Example 2
Graphing a Quadratic Function in Vertex form
5.1 Graphing Quadratic Functions
10
Plot the vertex (h,k) ? (3,-4) Plot two
points, such as (2,-2) and (1,4). Use symmetry to
plot two more points, (4, -2) and (5,4). Draw the
parabola.
5.1 Graphing Quadratic Functions
11
Additional Example 1
5.1 Graphing Quadratic Functions
12
Additional Example 2
5.1 Graphing Quadratic Functions
13
Example 3
Graphing a Quadratic Function in Intercept form
The x-intercepts are (1,0) and (-3,0) The axis of
symmetry is x -1
5.1 Graphing Quadratic Functions
14
Example 3
The x-coordinate of the vertex is -1. The
y-coordinate is
Graph the parabola.
5.1 Graphing Quadratic Functions
15
5.1 Graphing Quadratic Functions
16
Additional Example 1
5.1 Graphing Quadratic Functions
17
Additional Example 2
y - (x 1)(x 3)
5.1 Graphing Quadratic Functions
18
Example 4

• Writing Quadratic Functions in Standard Form

Write y 2(x 3)(x 8) in standard form
5.1 Graphing Quadratic Functions
19
Additional Example 1
Write the quadratic function in standard form
5.1 Graphing Quadratic Functions
20
Reflection on the Section
Give an example of a quadratic equation in vertex
form. What is the vertex of the graph of this
equation?
assignment
5.1 Graphing Quadratic Functions
21
5.2
Solving Quadratic Equations by Factoring
What you should learn
Goal
1
Factor quadratic expressions and solve quadratic
equations by factoring.
Goal
2
Find zeros of quadratic functions.
5.2 Solving Quadratic Equations by Factoring
22

• A binomial is a polynomial with two terms.
• A trinomial is a polynomial with three terms.
• Factoring can be used to write a trinomial as a
  product of binomials.
• A monomial is a polynomial with only one term.

5.2 Solving Quadratic Equations by Factoring
23
We are doing the reverse of the F.O.I.L. of two
binomials. So, when we factor the trinomial, it
should be two binomials.
Example 1
Step 1 Enter x as the first term of each factor.
( x )( x )
Step 2 List pairs of factors of the constant, 8.
Factors of 8
8, 1
4, 2
-8, -1
-4, -2
5.2 Solving Quadratic Equations by Factoring
24
Step 3 Try various combinations of these factors.
Sum of Outside and Inside Products (should equal
6x)
Possible Factorizations
x 8x 9x
( x 8)( x 1)
( x 4)( x 2)
2x 4x 6x
-x - 8x - 9x
( x - 8)( x - 1)
( x - 4)( x - 2)
-2x - 4x - 6x
5.2 Solving Quadratic Equations by Factoring
25
Example 2
Step 1 Enter x as the first term of each factor.
( x )( x )
Step 2 List pairs of factors of the constant, 7.
Factors of 7
-7, -1
7, 1
Step 3 Try various combinations of these factors.
Sum of Outside and Inside Products (should equal
8x)
Possible Factorizations
x 7x 8x
( x 7)( x 1)
( x - 7)( x - 1)
-x - 7x - 8x
5.2 Solving Quadratic Equations by Factoring
26
(x )(x )
(x - )(x )
Look at the 2nd sign

• If it is positive, both signs in binomials will
  be the same. (same as the 1st sign.)
• If it is negative, the signs in binomials will be
  different.

(x - )(x - )
(x )(x - )
5.2 Solving Quadratic Equations by Factoring
27
(x )(x )
1
6
(x - )(x )
4
16
(x - )(x - )
4
18
(x )(x - )
1
16
5.2 Solving Quadratic Equations by Factoring
28
Factor.
(x )(x )
2y
6y
(x - )(x )
4y
7y
5.2 Solving Quadratic Equations by Factoring
29
The Zero-Product Principle
If the product of two algebraic expressions is
zero, then at least one of the factors is equal
to zero.
If AB 0, then A 0 or B 0.
If, ( ???)() 0
Example)
Then either (???) is zero, or () is zero.
5.2 Solving Quadratic Equations by Factoring
30
Solve the equation.
Example 1)
According to the principle, this product can be
equal to zero if either
or
5
5
2
2
x 5
x 2
The resulting two statements indicate that the
solutions are 5 and 2.
5.2 Solving Quadratic Equations by Factoring
31
Factoring Trinomials whose Leading Coefficient
is NOT one.
Objectives
1. Factor trinomials by trial and error.
32
Factoring by the Trial-and-Error Method
How would we factor
Notice that the leading coefficient is 3, and we
cant divide it out
( 3x )( x )
33
example
Step 1 find the two First terms whose product is
.
( 3x )( x )
Step 2 Find two Last terms whose product is 28.
The number 28 has pairs of factors that are
either both positive or both negative. Because
the middle term, -20x, is negative, both factors
must be negative.
Factors of 28
-1(-28)
- 4(-7)
- 2(-14)
34
Step 3 Try various combinations of these factors.
Sum of Outside and Inside Products (should equal
-20x)
Possible Factorizations
-84x - x - 85x
( 3x - 1)( x - 28)
( 3x - 28)( x - 1)
-3x - 28x - 31x
( 3x - 2)( x - 14)
-42x - 2x - 44x
( 3x - 14)( x - 2)
-6x - 14x - 20x
-21x - 4x - 25x
( 3x - 4)( x - 7)
-12x - 7x - 19x
( 3x - 7)( x - 4)
35
example
Step 1 find the two First terms whose product is
.
( 8x )( x )
( 4x )(2 x )
Step 2 Find two Last terms whose product is -3.
Factors of -3
1(-3)
-1(3)
36
Step 3 Try various combinations of these factors.
Sum of Outside and Inside Products (should equal
-10x)
Possible Factorizations
( 8x 1)( x - 3)
-24x x - 23x
( 8x - 3)( x 1)
8x - 3x 5x
( 8x - 1)( x 3)
24x - x 23x
( 8x 3)( x - 1)
- 8x 3x - 5x
-12x 2x - 10x
( 4x 1)(2 x - 3)
( 4x - 3)( 2x 1)
4x - 6x - 2x
( 4x - 1)( 2x 3)
12x - 2x 10x
-4x 6x 2x
( 4x 3)( 2x - 1)
37
Factoring Trinomials whose Leading Coefficient
is NOT one.
Ex 1)
Ex 2)
(3x )(3x )
1
1
(2x )(2x - )
1
7
Ex 3)
Ex 4)
(2x - )(3x - )
1
2
(2x )(x - )
3
5
5.2 Solving Quadratic Equations by Factoring
38
Solve a Quadratic Equation by Factoring
Example 2)
Factor the Trinomial using the methods we know.
(2x )(x ) 0
-

1
4
or
1
1
- 4
- 4
2x 1
x - 4
x 1/2
The resulting two statements indicate that the
solutions are 1/2 and - 4.
5.2 Solving Quadratic Equations by Factoring
39
Solve a Quadratic Equation by Factoring
Example 3)
Move all terms to one side with zero on the
other. Then factor.
(x )(x ) 0
-
-
3
3
The trinomial is a perfect square, so we only
need to solve once.
3
3
x 3
The resulting two statements indicate that the
solutions are 3.
5.2 Solving Quadratic Equations by Factoring
40
Factoring out the greatest common factor.
But, before we do thatdo you remember the
Distributive Property?
When factoring out the GCF, what we are going to
do is UN-Distribute.
5.2 Solving Quadratic Equations by Factoring
41
What I mean is that when you use the Distributive
Property, you are multiplying. But when you are
factoring, you use division.
Factor
example
1st determine the GCF of all the terms.
5
2nd pull 5 out, and divide both terms by 5.
5.2 Solving Quadratic Equations by Factoring
42
Factor each polynomial using the GCF.
ex)
ex)
ex)
5.2 Solving Quadratic Equations by Factoring
43
Sometimes polynomials can be factored using more
than one technique. When the Leading Coefficient
is not one. Always begin by trying to factor
out the GCF.
Example 1
factor out 3x
3x(x )(x )
7
-
2

5.2 Solving Quadratic Equations by Factoring
44
Factor.
Example 2
Example 3
3( )
( )
(a - )(a - )
2
9
(x )(x - )
3
16
5.2 Solving Quadratic Equations by Factoring
45
Example 4
Factoring GCF First
Step 1) GCF
5.2 Solving Quadratic Equations by Factoring
46
The Difference of Two Squares If A and B are real
numbers, variables, or algebraic expressions,
then In words The difference of the squares
of two terms is factored as the product of the
sum and the difference of those terms.
5.2 Solving Quadratic Equations by Factoring
47
Factoring the Difference of Two Squares
Example 1)
1.) Difference of the Two Squares,
2.) or you could look at this as the trinomial
5.2 Solving Quadratic Equations by Factoring
48
Difference of the Two Squares,
Example 2
We must express each term as the square of some
monomial. Then use the formula for factoring
1.)
You can check it by using FOIL on the binomial.
2.) or you could look at this as the trinomial
(x )(x )
-
4
4

5.2 Solving Quadratic Equations by Factoring
49
Factoring out the GCF and then factoring the
Difference of two Squares.
Example 1)
Whats the GCF?
5.2 Solving Quadratic Equations by Factoring
50
Factoring out the GCF and then factoring the
Difference of two Squares.
Example 2)
Whats the GCF?
5.2 Solving Quadratic Equations by Factoring
51
Additional Examples
Example 3
5.2 Solving Quadratic Equations by Factoring
52
Factoring Perfect Square Trinomials
Example 4
(x )(x )

3
3

Since both binomials are the same you can say
5.2 Solving Quadratic Equations by Factoring
53
Factoring Perfect Square Trinomials
Example 5
(x )(x )
-
5
5
-
Since both binomials are the same you can say
5.2 Solving Quadratic Equations by Factoring
54
Example 6
5.2 Solving Quadratic Equations by Factoring
55
Reflection on the Section
What must be true about a quadratic equation
before you can solve it using the zero product
property?
assignment
5.2 Solving Quadratic Equations by Factoring
56
www.animationfactory.com
Backdrops - These are full sized backdrops, just
scale them up! - Can be Copy-Pasted out of
Templates for use anywhere!
Title Backdrop
Slide Backdrop
Print Backdrop
Transitional Backdrop

• Additional Graphics
• Scale them up or down!
• .GIF clipart is animated.
• .JPG clipart can be scaled up and take up little
  file space.
• .PNG clipart can be scaled unusually large
  without distortion.

57
Solving Quadratic Equations by Finding Square
Roots
5.3
What you should learn
Goal
1
Solve quadratic equations by finding square roots.
Goal
2
Use quadratic equations to solve real-life
problems.
5.3 Solving Quadratic Equations by Finding Square
Roots
58
Properties of Square Roots (a gt 0, b gt 0)
Product Property
Quotient Property
Example)
Example)
5.3 Solving Quadratic Equations by Finding Square
Roots
59
Simplify the expression.
Example 1)
Example 2)
Example 3)
Example 4)
5.3 Solving Quadratic Equations by Finding Square
Roots
60
Rationalizing the denominator eliminate a
radical as denominator by multiplying.
Simplify the expression.
Which means No radicals (square roots) in
the denominator.
Example 5)
Example 6)
Example 7)
5.3 Solving Quadratic Equations by Finding Square
Roots
61
Simplify the expression.
Example 8)
Example 9)
Example 10)
5.3 Solving Quadratic Equations by Finding Square
Roots
62
Solve the Quadratic Equation.
Example 1)
Example 2)
Example 3)
5.3 Solving Quadratic Equations by Finding Square
Roots
63
Solve the equation Quadratic Equation.
Example 4)
-1
-1
2
2
5.3 Solving Quadratic Equations by Finding Square
Roots
64
Solve the equation Quadratic Equation.
Example 5)
1
1
5.3 Solving Quadratic Equations by Finding Square
Roots
65
Solve the equation Quadratic Equation.
3
Example 6)
-5
-5
5.3 Solving Quadratic Equations by Finding Square
Roots
66
Reflection on the Section
For what purpose would you use the product or
quotient properties of square roots when solving
quadratic equations using square roots?
assignment
5.3 Solving Quadratic Equations by Finding Square
Roots
67
5.4
Complex Numbers
What you should learn
Goal
1
Solve quadratic equations with complex solutions
and perform operations with complex numbers.
Goal
2
Apply complex numbers to fractal geometry.
5.4 Complex Numbers
68
Imaginary numbers i , defined as
Note that
The imaginary number i can be used to write the
square root of any negative number.
5.4 Complex Numbers
69
Simplify the expression.
Example 1)
Example 2)
Example 3)
Example 4)
5.4 Complex Numbers
70
Solve the equation Quadratic Equation.
Example 1)
-1
-1
2
2
5.4 Complex Numbers
71
Adding and Subtracting Complex Numbers
Example 1)
Example 2)
5.4 Complex Numbers
72
Reflection on the Section
Describe the procedure for each of the four basic
operations on complex numbers.
assignment
Page 277 1 10, 17- 28, 37 - 46
5.4 Complex Numbers
73
5.5
Completing the Square
What you should learn
Goal
1
Solve quadratic equations by completing the
square.
Goal
2
Use completing the square to write quadratic
functions in vertex.
5.5 Completing the Square
74
Reflection on the Section
Why was completing the square used to find the
maximum value of a function?
assignment
5.5 Completing the Square
75
5.6
The Quadratic Formula and the Discriminate
What you should learn
Goal
1
Solve quadratic equations using the quadratic
formula.
Goal
2
Use quadratic formula to solve real-life
situations.
5.6 The Quadratic Formula and the Discriminant
76
Pre-Stuff Solve for x.
5.6 The Quadratic Formula and the Discriminant
77
Pre-Stuff Solve for x.
Ex3)
Factor out GCF
Ex1)
Ex2)
5.6 The Quadratic Formula and the Discriminant
78
Quadratic Formula When solving a quadratic
equation like
use
5.6 The Quadratic Formula and the Discriminant
79
Solve the Quadratic Equation.
Example 1)
Split this.
Put these into calculator
5.6 The Quadratic Formula and the Discriminant
80
Solve the Quadratic Equation.
Example 2)
Split this.
5.6 The Quadratic Formula and the Discriminant
81
Reflection on the Section
Describe how to use a discriminant to determine
the number of solutions of a quadratic equation.
discriminant
if , then 2 real solutions.
if , then 1 real solutions.
if , then 2 imaginary
solutions.
Page 295 17 - 45
assignment
5.6 The Quadratic Formula and the Discriminant
82
(No Transcript)
83
5.7
Graphing and Solving Quadratic Inequalities
What you should learn
Goal
1
Graph quadratic inequalities in two variables.
Goal
2
Solve quadratic inequalities in one variable.
5.7 Graphing and Solving Quadratic Inequalities
84
Reflection on the Section
What is the procedure used to solve quadratic
inequality in two variables?
assignment
5.7 Graphing and Solving Quadratic Inequalities
85
5.8
Modeling with Quadratic Functions
What you should learn
Goal
1
Write quadratic functions given characteristics
of their graphs.
Goal
2
Use technology to find quadratic models for data.
5.8 Modeling with Quadratic Functions
86
Reflection on the Section
Give four ways to find a quadratic model for a
set of data points.
assignment
5.8 Modeling with Quadratic Functions
87
(No Transcript)