Download Quadratic Functions

Algebra
Notes: Quadratic Functions
Quadratic Functions
There are two different forms of quadratic functions that we will study.
Standard/Expanded Form
Factored Form
Graphs of Quadratic Functions
The graph of a quadratic function is called a …
The vertex of a parabola is …
The roots of a parabola are …
Roots are also called …
Unit Goals
The goals of this unit are as follows:
1. To combine polynomials by addition, subtraction, and multiplication.
2. To translate quadratic expressions from expanded form to factored form and vice-versa.
3. To solve quadratic equations.
4. To graph quadratic functions.
5. To solve application problems involving quadratic functions.
Quadratic Function Notes
Page # 1
Algebra
Notes: Quadratic Functions
Polynomials
Equations like
are called polynomials.
A Quadratic Polynomial is …
Adding and Subtracting Polynomials
When you combine two polynomials by addition or subtraction, you can only combine
_________________.
Like terms are …
(2x3 + 3x2 – 4x + 7) + ( 7x4 – 8x3 + 3x2 + 9)
Adding Polynomials
(5x2 – 7x + 12) + (- 8x2 – 3x + - 8)
Quadratic Function Notes
Page # 2
Algebra
Notes: Quadratic Functions
Subtracting Polynomials
(10x2 – 9x - 18) - (6x2 – 13x - 8)
Another Example
(3.63x2 – 2.15x – 9) – (- 5.21x2 + 6.01x – 3.14)
Multiplying Binomials
Three techniques that can be used to multiply binomials are…
1.
2.
3.
FOIL
(x + 2)(x + 4)
Quadratic Function Notes
Page # 3
Algebra
Notes: Quadratic Functions
Punnett Square
(2x - 7)(-3x + 6)
Vertical Multiplication
(2x – 3)(3x + 4)
Factoring
Factoring is…
(x + 4)(x + 1) =
The Process of Factoring
Write x2 + 14x + 24 in factored form
A Little Tougher One
Write x2 - 5x - 24 in factored form
Quadratic Function Notes
Page # 4
Algebra
Notes: Quadratic Functions
You Try These
Write each of the following quadratic expressions in factored form.
x2 + 8x + 15 =
x2 - 12x + 35 =
x2 + 8x - 20 =
FACTORING ALGORITHM 1

Original Expression:
+
1 ac = ____, ____ = b
2
3
____ + ____ + ____ + ____
4
___ ( ___ + ___ ) + ___ ( ___ + ___ )
5
( ___ + ___ ) ( ___ + ___ )
FACTORING ALGORITHM 2
Original Expression:

+
ac
=
____,
____
=b
1
2
3
____ + ____ + ____ + ____
4
___ ( ___ + ___ ) + ___ ( ___ + ___ )
5
( ___ + ___ ) ( ___ + ___ )
FACTORING ALGORITHM 3

Original Expression:
+
1 ac = ____, ____ = b
2
3
4
5
____ + ____ + ____ + ____
___ ( ___ + ___ ) + ___ ( ___ + ___ )
Quadratic Function Notes
Page # 5
( ___ + ___ ) ( ___ + ___ )
Algebra
Notes: Quadratic Functions
Flow Map Time!
Let’s Try Some
3x2 - 13x + 12 =
12x2 + 24x – 15 =
Quadratic Function Notes
Page # 6
Algebra
Notes: Quadratic Functions
The Five Types of Problems That I Have to Solve in Quadratic Applications
The equation y =-11.5x2 + 1009.825 x – 9095 describes the profit, y, that a company makes
when it sells its product for a price of x dollars. Answer the following questions.
1. How should the company set its price in order to make the maximum profit? What is the
maximum profit that the company can make?
2. How much profit will the company make if they set the price at $30?
3. What are the “break even” prices for the company?
4. How much money would the company lose if they gave their product away for free?
5. If the company needed their profit to be exactly $11,500 for tax purposes, what could
they set their price at?
Problem #1: Find the Vertex
1. How should the company set its price in order to make the maximum profit? What is the
maximum profit that the company can make?
Record the Keystrokes that you need to use to
find the VERTEX on your calculator
Problem #2: Find y, Given x
2. How much profit will the company make if they set the price at $30?
Quadratic Function Notes
Page # 7
Algebra
Notes: Quadratic Functions
Problem #3: Find the Roots
3. What are the “break even” prices for the company?
Record the Keystrokes that you need to
use to find the ROOTS on your calculator
Problem #4: Find the Y-Intercept
4. How much money would the company lose if they gave their product away for free?
Problem #5: Find x, Given y
5. If the company needed their profit to be exactly $11,500 for tax purposes, what could
they set their price at?
Quadratic Function Notes
Page # 8
Algebra
Notes: Quadratic Functions
Three Methods for Solving Quadratic Equations
1.
2
3.
Inverse Operations
An Important Analogy
We solve the equation 3x – 7 = 14 by using _______________ to _____________ the variable x.
First, we “undo” the -7 by …
Then, we “undo” the 3 by dividing …
___________________ undoes _______________________
___________________ undoes _______________________
#1: Using Square Roots
An Example
Solve the following equation:
3 x 2  7  14
Quadratic Function Notes
Page # 9
Algebra
Notes: Quadratic Functions
Try These
3 2
x 47
2
5 x 2  8  33
9 x 2  5  2 x 2  18
#2: Factored Form and Solving Equations
If a·b=0, what must be true?
This is called …
Factored Form and Solving Equations (cont.)
If (x+2)(x+4)=0, what must be true?
Solve These Equations
(x + 5)(x + 1) = 0
(x - 3)(x + 4) = 0
Quadratic Function Notes
Page # 10
(2x - 3)(3x + 5) = 0
Algebra
Notes: Quadratic Functions
Solving By Factoring #1
Solve the equation x2 + 10x = - 21 by factoring.
Step #1: Set the equation equal to 0.
Step #2: Factor
Step #3: Apply the Zero Product Property
Solving By Factoring #2
Solve the equation 2x2 = 17x + 19 by factoring.
Step #1: Set the equation equal to 0.
Step #2: Factor

2
3
+
1 ac = ____, ____ = b
Original Expression:
____ + ____ + ____ + ____
4
___ ( ___ + ___ ) + ___ ( ___ + ___ )
5
( ___ + ___ ) ( ___ + ___ )
Step #3: Apply the Zero Product Property
Quadratic Function Notes
Page # 11
Algebra
Notes: Quadratic Functions
Solve These Equations
x2 + 7x + 12 = 0
2x2 + 5x = 12
5x2 - 4x - 4 = 2x2 - 15x
#3: The Quadratic Formula
An equation of the form _____________________________ can always be solved by using the
quadratic formula.
Example #1
Solve the equation 3x2 – 4x – 5 = 0
Step #1: Identify a, b, and c.
Step #2: Plug a, b, and c into the formula
Quadratic Function Notes
Page # 12
Algebra
Notes: Quadratic Functions
Step #3: Simplify the expression
Step #4: Evaluate the two answers
Make sure to do a quick check using your calculator!!!!!
Example #2
Solve the equation -4x2 + 3x = -6
Try These
2x2 + 5x - 12 = 0
5x2 - 4x - 4 = 2x2 - 15x
Quadratic Function Notes
Page # 13
Algebra
Notes: Quadratic Functions
Now that you have completed the investigating parabolas chart, take some time to analyze your
results and answer the following questions.
1. How can you tell if a parabola “opens up” or “opens down” simply by looking at the
equation?
2. How can you determine the location of the roots of a parabola before you actually graph
the parabola?
3. In terms of the roots, where will the vertex of the parabola always be located? What
other pattern do you notice related to the vertex?
4. How can you determine the location of the y-intercept of the parabola before you actually
graph the parabola?
Quadratic Function Notes
Page # 14