# Quadratics

### By : Azan B.

## Table of contents

**Introduction**

__What is it __

__The Algebra Of Quadratics__

__Quadratic Function__

__Analyzing Quadratics (Second Differences)__

__The Parabola__

__Transformations__

**Types of Equations**

__Factored Form a(x-r)(x-s)__

__Vertex Form a(x-h)²+k__

__Standard Form ax²+bx+c__

**Vertex form**

__Finding an equation when only given the vertex__

__Graphing vertex form__

**Factored Form**

__Factors and zeroes__

__multiplying out brackets and quadratic expressions__

__Common factoring__

__Factoring simple trinomials__

__Factoring complex trinomials__

__Perfect square__

__Factoring by grouping__

__Graphing factored form__

**Standard form**

__Completing the square __

__Quadratic formula__

__Graphing from standard form__

**Word Problems**

__Revenue__

__Numeric__

**Reflection**

## INTRODUCTION

## What is it?

## Quadratic Function

## Analyzing quadratics (Second Differences)

## Parabola

__Parts of a parabola__

vertex- is the highest or lowest point of the parabola

Line of symmetry- The very middle of the parabola through the x axes

x-intercept - Is where the parabola crosses the x axes

y-intercept - is where the parabola crosses the y axes

## Graphical Transformation Of Functions

## Scaling The Quadratic

## Types of equations

Vertex form: a(x-h)²+k

Factored form : a(x-r)(x-s)

Standard form: ax²+bx+c

## Factored form

You have to multiply everything within the brackets by each other before multiplying it by "a".

(x)(x) (x)(s) (r)(x) (r)(s)

After multiplying these you then simplify and then multiply it all by the a value

This is called expanding and simplifying

## Vertex form

For example if an equation is 0=4(x-2)3+5 the vertex will be (2+3)

How did it get from -2 to 2 well that's because the x coordinate is always changed to a positive if the number is a negative or switched to a negative if the number is already a positive.

## Standard form

## Vertex form

## Finding an equation when only vertex form is given

## Graphing from vertex form

Start with the function in vertex form:

y = a(x - h)2 + k ( y = 3(x - 2)2 - 4)

Pull out the values for h and k.

If necessary, rewrite the function so you can clearly see the h and k values.

(h, k) is the vertex of the parabola.

Plot the vertex. y = 3(x - 2)2 + (-4)

h = 2; k = -4

Vertex: (2, -4)

## Factored form

## Factors and zeros

Rewrite the factors to show the repeating factor.

( x - 3)( x - 3)(2x + 5)

Write factors in polynomial notation. The a is the leading coefficient of the polynomial

P(x) = a( x - 3)( x - 3)(2x + 5)

Multiply ( x - 3)( x - 3)

Then multiply ( x 2 - 6x + 9)(2x + 5)

For right now let an = 1

The polynomial of lowest degree with real coefficients, and factors ( x - 3)2 (2x + 5) is: P(x) = 2x 3 - 7x 2 + 8x + 45.

The main concept is simply to multiply the factors in order to determine the polynomial of lowest degree with real coefficients.

## Multiplying out of brackets and Quadratic expressions

## Common factoring

Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved

Finding the GCF of a List of Integers or Terms

1)Prime factor the numbers.

2)Identify common prime factors.

3)Take the product of all common prime factors.

If there are no common prime factors, GCF is 1

## Factoring simple trinomials

What i like to do for this type of factoring is just use mental math but others may find it difficult. To factor a simple trinomial you need to find 2 numbers that multiply to give you the C value but also add up to give you the B value

For example:

x^2 + 7x + 6

6 and 1 are factors of 6 that will equal 6 if multiplied and 7 if summed.

You will then use these 2 numbers part of the equation y=a(x-r)(x-s)

y=(x+6) (x+1)

## Factoring complex trinomials

1. You first need to multiply the A value with the C value.

2. Next, find the common factor of the product you get and find 2 numbers that will add together to give you the B value

3. After you find the 2 numbers sub them into the equation and group them.

4. Find common factors of each group and you will then use the numbers inside the brackets as well as the 2 numbers you got from grouping.

## Perfect Square

## Factoring by grouping

## Graphing factored form

A quadratic equation is a polynomial equation of degree of 2. The standard form of a quadratic equation is 0 = ax^2 + bx + c

where a, b, and c are all real numbers and a 0.

If we replace 0 with y , then we get a quadratic Function

y = ax^2 + bx + c

whose graph will be a parabola.

## Standard form

## Completing the square

## Quadtratic Formula

## Word problems

## Revenue

Calculators are sold to students for 20 dollars each. Three hundred students are willing to buy them at that price. For every 5 dollar increase in price, there are 30 fewer students willing to buy the calculator. What selling price will produce the maximum revenue and what will the maximum revenue be?

Let R represent Revenue

Let X represent increase or decrease

To solve this word problem we have to use a certain equation (Price)(People)

That gives us

R=(20+5x)(300-30x)

R=-150x² + 900 x + 600

Next we have to complete the square which gives us a Maximum Revenue of $7350 and a selling price of $35

## Numeric

The sum of the squares of two consecutive integers is 365. What are the integers?

Let one number be x. Since the two numbers are consecutive, the other number is x + 1.

The sum of the squares is (x)^2 + (x + 1)^ 2 and it is equal to 365. The equation is (x) 2 + (x + 1)^2 = 365.

If you look at the question, it does not say maximum, minimum so we do not need the vertex. What we need to do, is to solve the equation (x)^2 + (x + 1)^2 = 365, by factoring or quadratic formula.

Expand and simplify the equation into standard form: 2x^2 + 2x – 364 = 0 Factor or use the quadratic formula to find the two solutions: x = -14, x = 13.

The two solutions to the equation will give us two possible sets of integers which are answers to the question: The first set of integers is -14 and -13, and the second set of integers is 13 and 14.