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.

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)

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)

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

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

FACTORING ALGORITHM 2

FACTORING ALGORITHM 3

Flow Map Time!

Let’s Try Some

3x2 - 13x + 12 =

12x2 + 24x – 15 =

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?

Problem #2: Find y, Given x

2. How much profit will the company make if they set the price at $30?

Problem #3: Find the Roots

3. What are the “break even” prices for the company?

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?

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:

[pic]

Try These

[pic] [pic]

[pic]

#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 (2x - 3)(3x + 5) = 0

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

Step #3: Apply the Zero Product Property

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

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

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?

-----------------------

4

___ ( ___ +actoring Algorithm Template__ ___word. Can you help you out again ___ ) + ___ ( ___ + ___ )

5

( ___ + ___ ) ( ___ + ___ )

____ + ____ + ctoring Algorithm Template__ ___word. Can you help you out again ____ + ____

3

2

1

+

(

a(c = ____, ____ = b

Original Expression:

( ___ + ___ ) ( ___ + ___ )

5

___ ( ___ +actoring Algorithm Template__ ___word. Can you help you out again ___ ) + ___ ( ___ + ___ )

4

____ + ____ + ctoring Algorithm Template__ ___word. Can you help you out again ____ + ____

3

2

1

+

(

a(c = ____, ____ = b

Original Expression:

Original Expression:

a(c = ____, ____ = b

(

+

1

2

3

____ + ____ + ctoring Algorithm Template__ ___word. Can you help you out again ____ + ____

4

___ ( ___ +actoring Algorithm Template__ ___word. Can you help you out again ___ ) + ___ ( ___ + ___ )

5

( ___ + ___ ) ( ___ + ___ )

Original Expression:

a(c = ____, ____ = b

(

+

1

2

3

____ + ____ + ctoring Algorithm Template__ ___word. Can you help you out again ____ + ____

4

___ ( ___ +actoring Algorithm Template__ ___word. Can you help you out again ___ ) + ___ ( ___ + ___ )

5

( ___ + ___ ) ( ___ + ___ )

Record the Keystrokes that you need to use to find the VERTEX on your calculator

Record the Keystrokes that you need to use to find the ROOTS on your calculator

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