# Quadratics

### By: Sonali Thakkar

## Table of Contents

- Features
- Parabolas
- Ways to represent Parabola's
- Vertex Form
- Factored Form
- Standard Form

Vertex form

- Investigating
- Graphing
- Transformations
- How to find a equation using a graph

Mini Test #1

Expanding and Factoring:

- Multiplying Binomials
- Special Products
- Common Factoring
- Factoring simple Trinomials
- Factoring Complex Trinomials
- Differences of square Trinomials
- How to find a equation using a graph

Mini Test #2

- Finding the Maxima and Minima
- Graphing using x-Intercept
- The Quadratic Formula

Test #3

Word Problems

## Introduction to Quadratics

## Key Features of a Quadratic Realtion

## Description:

- the opening of the parabola is up
- the axis of symmetry is the x value of the vertex
- the optimal value is the y value of the vertex
- the zeros are when the parabola touches or goes through the x axis
- y- intercept is when the the parabola goes through the y axis
- the vertex is the minimum or maximum point of the parabola

## Ways To Represent Parabolas

## Table of Values

## Graph

## Equation

## 3 Types of Equations

- Standard Form
- Vertex Form
- Factored Form

## Standard Form

## Vertex Form

where h and k represent the vertex. h represents the x value of the vertex. If h is positive in the equation, it is negative on the graph. If h is negative in the equation then, it is positive on the graph.

k represents the y value of the vertex. The value of k does not have any special rules, as you see it on the equation is where you place it on the graph.

## Factored Form

## Vertex Form

## Investigation

## How to Graph with Vertex Form

## Basic Transformations

The four transformation are:

- if your parabola has an opening up or down
- if your parabola is vertically stretches or compressed
- how many units is it transforming left or right
- how many units it is transforming up or down

For example:

y= -2(x-4)-5

The transformations for that equation would be:

- it has opening down
- the parabola has a vertical stretch of 2
- the parabola if transformed 4 units to the right
- the parabola is transformed 5 units down

## How to Find a Equation Using a Graph

## Unit Test 1

## Expanding and Factoring

## Expanding Binomials

For example, (3x+2)(3x-3)

To expand this binomial, you have to:

- First, you multiply the outer term with inner term. so you would multiply (3x)(3x)= 9x^2
- Second, you multiply the outer term with the inner term. (3x)(-3)= -9x
- Third, you multiply the two inner terms. (2)(3x)= 6x
- Fourth, you multiply the inner term with the outer term. (2)(-3)= -6
- Fifth, you group the like terms.

9x^2-3x-6

- Finally, you have you standard form equation 9x^2-3x-6

## Special Products

(x+4)^2

x is the value a and 4 is the value of b, so now you plug that into the formula:

x^2+ 2(x)(4)+ 4^2

x^2+8x+16

## Common Factoring

1) Monomial Common Factor

2) Binomial Common Factor

3) Factor by Grouping

Monomial Common Factor:

i) Find the GCF of coefficients and variables

ii) Divide each term by GCF

Examples:

5c+10d

- the GCF is 5 therefore you divide both terms by 5.
- you would rewrite that as

5(c+2d)

8x^2-7x

- the GCF is x therefore you divide both terms by x.
- you would rewrite this as

Binomial Common Factor:

i) if there are two binomials that are exactly same, consider that as a binomial common factor.

Examples:

- x(x-2)+2(x-2)
- x-2 are the same therefore they are the common factor and you would rewrite it as

(x-2)(x+2)

- the common binomial goes first and what is left comes after.

Factor by Grouping:

i) factor groups of two terms with a common factor to produce a binomial common factor.

Example:

- ax+ay+2x+2y -ax and ay have the common factor a and 2x and 2y have the common factor 2.
- (ax+ay)+(2x+2y) - then you add brackets around common terms.
- a(x+y)+2(x+y) - then you take the common factor outside the brackets. At this you know you are correct if both of the terms inside the brackets are the same.
- then you set them into a binomial common factor

## Simple Trinomial Factoring

x^2+bx+c = (x+r)(x+s)

There are 3 steps to factor a simple trinomial.

Step 1: Find the product and sum

- Find two numbers whose product is c
- Find two numbers whose sum is b

For example,

x^2+12x+27

- two numbers whose product is 27 and also have the sum 12 are 3 and 9.

Step 2: Look at the signs of b and c in the given expression

- If b and c are positive, both r and s are positive.

Both b and c are positive therefore, r and s are positive and the answer is

(x+4)(x+3)

- If b is negative and c is positive, both r and s are negative.

b is negative and c is positive therefore, both r and s are negative.

(x-28)(x-1)

Step 3:

- If c is negative, one of r or s is negative.

For example:

x^2+3X-18

c is negative therefore either one of r or s is negative.

(x-3)(x+6)

## Factoring Complex Trinomials

Step 2: To factor ax^2+bx+c, find two integers whose product is ac and whose sum is b.

Step 3: The check up the middle term and factor by grouping

Example: 3x^2 +8x+4

- Two integers whose product is (3)(4) and sum is 8

- Break up the middle term

- Factor by grouping

(3x+2)(x+2)

## Differences of Squares

Example: x^2-16

x^2-4^2

(x+4)(x-4)

## Unit Test 2

## Finding the Maxima and Minima

Completing the squares turns a standard form equation (x^2+bx+c) into a vertex form equation (y=a(x+k)^2-h).

Step one: you add brackets around your a and b

Step 2: divide your b value by 2 and square your answer

Step 3: put a positive square in the bracket and a negative square outside the bracket

Step 4: you now have a perfect square trinomial in the brackets, so we turn that into a binomial.

Step 5: solve the values outside the bracket

For example:

y=x^2+8x+3

y=(x^2+8x)+3

y=(x^2+8x+16)-16+3

y=(x+4)^2-16+3

y=(x+4)^2-13

Therefore, the vertex of this equation is (4,-13).

## Graphing Using the x-intercepts

y=x^2-6x+8

-4,-2

y=x^2-4x-2x+8

y=(x^2-4x)(-2x+8)

y=x(x-4)-2(x-4)

y= (x-4)(x-2)

x-4=0 x-2=0

x=4 x=2

(4,0) (2,0)

Therefore, the x-intercepts are (4,0) and (2,0)

4+2/2=6/2=3

the AOS is 3. Now you plug in your x value.

y= (x-4)(x-2)

y= (3-4)(3-2)

y=(-1)(1)

y=-1

Therefore the vertex is (3,-1)

Now we can graph this because we know our x-intercepts, our vertex, and the y-intercept. The y-intercept is the value of c in the original equation.

## The Quadratic Formula

- When given your standard form equation you determine you a value, your b value and c value so there is no confusion.
- You then plug in all of you variables.
- First, you square your b and multiply your -4ac
- Second, multiply your 2a
- Third, solve inside the square root
- Fourth, square root that answer
- Now the equation goes two ways; one you add your -b with your square root divided by your 2a; two you subtract your -b with your square root and divide it with your 2a
- The two answers you get are your x-intercepts.

## Revenue Word Problems

a) Create an algebraic equation to model Bryce's total sales revenue

Let (18.50+0.50x) represent the cost

Let (216-8x) represent the total number of people

R= (18.50+0.50x)(216-8x)

b) Determine the maximum revenue and the price at which this maximum revenue will occur

R= (18.50+0.50x)(216-8x)

18.50+0.50x=0 216-8x=0

0.50x= -18.50/0.50 -8x= -216/-8

x= -37 x= 27

-37+27/2= -10/2 = -5

R= (18.50+0.50x)(216-8x)

R= (18.50+0.50(-5))(216-8(-5))

R=(18.50-2.5)(216+40)

R=(16)(256)

R= 4096

Therefore the maximum Revenue $4096 will occur at the maximum price of $16