# Quadratics

### The Best Unit Ever!

## By: Elizabeth Esedafe

## Introduction To Quadratics

The origin of the term "quadratic" is Latin. It comes from the word quadratus, which is the past participle of quadrare which means "to make square."The quadratic equation is used to find the curve on a Cartesian grid. It is also used to find the curve that objects take when they fly through the air. For example a softball, tennis ball, football, etc. Other than that, the equation is used to design any object that has curves and any specific curved shape needed for a project.

## Unit Table Of Contents

*Discovering The Vertex Form*- Key Features Of Quadratic Relations (Vertex,AOS,Zeros,Optimal Value,)
- 4.2 Quadratic Relations and finite differences
- 4.3 Transformations of Quadratics
- 4.4 Graphing Quadratics in Vertex Form
- Finding x and y intercepts in Vertex Form
- Word Problems for Quadratics in Vertex Form

__Quadratics in Factored Form__

- 5.1& 5.2 Multiplying Binomials and Special Products
- 5.3 Common Factoring
- 5.4 Factoring Simple Trinomials
- 5.4 Factoring Complex Trinomials
- 5.6 Factoring Difference of Squares and Perfect Square Trinomials
- 6.2 Solving Quadratics by Factoring (finding the zeros)
- 4.5 & 6.3 Graphing Quadratics in Factored Form
- Word problems for Quadratics in Factored Form

*Quadratics in Standard Form*

- 6.1 Maximum and Minimum values (Completing the square)
- 6.4 The Quadratic Formula
- Word Problems in Standard Form (6.5)

*Connections in the Unit*

*Reflection*

## Discovering The Vertex Form

__KEY FEATURES OF QUADRATIC RELATIONS__

-The zeros and y-intercepts

-The Axis of Symmetry

-The Vertex

-The Optimal Value

## Axis Of Symmetry

## The Optimal Value

## 4.2 Quadratic Relations And Finite Differences

If the first differences are constant, then this means that the pattern is linear.Remember to have all your x-values in order!The first differences also tells you about the rate of change for the situation.If the second differences are constant,then this means the pattern is quadratic.

## 4.3 Transformations of a Parabola

The vertex form of a parabola is showed as: y= a (x-h)² +k

The basic parabola has the formula y=x² which means the value of a=1, the value of h=0, and the value of k=0.

Now, im going to show you how to find the values of a,h,and k and what transformations each value does to the parabola.

__How does the value of (a) determine the orientation and shape of a parabola?__

__How does the value of (k) determine the vertical position of the parabola?__

**How does the value of (h) determine the horizontal position of the parabola?**## Let's Put It ALL TOGETHER!

**Now that you know how a, h, and k can transform a parabola,we can figure out how to write out equations given the transformations.**

*Note: Remember if the h value of the vertex is negative to switch it to positive when you sub the value into the equation and vice versa.*

## 4.4 Graphing Quadratics in Vertex Form

__Step Pattern__

Watch the video below to further learn about graphing using the step pattern.

__Mapping Notation Formula__Mapping notation is a strategy used to accurately graph any quadratic relation.This method can be used instead of step notation. In general, to go from the graph of **y=x²** to** y=a(x-h)****²+k **, you can use the mapping formula (x.y)** ---->**(x+h, ay+k)

Here's some examples about how to use mapping notation:

We will be using the equation: y= (x-4)²+2

The first thing we need to do is find the a, h, and k value. We learned how to find these values in the previous lesson of transforming parabola's(4.3).

So we know: [a=0 h=4 and k=2]

To find the (x,y) coordinate, we just sub in the a,h,and k values into the mapping formula

(x+h, ay+k).

You should then get (x+4, y+2)

**TRY THESE QUESTIONS!**

**a) y=(x+7)²**

**b)y=5(x+2)²-3**

**c) y=1/2(x-3)²+1**

## Finding x and y intercepts in Vertex Form

**We can find the x and y intercepts from the vertex form: y=a(x-h)²+k**

## Word Problems for Quadratics in Vertex Form

**FLIGHT OF A BALL WORD PROBLEM***WATCH THE VIDEO BELOW TO LEARN HOW TO DO THIS TYPE OF WORD PROBLEM*

__BUSINESS APPLICATION (PROFIT QUESTION)__Profits(revenue) can be maximized and this relation can be modeled by a quadratic relation.

- The relationship between selling price and profit can be modeled by a quadratic

relation.

- The relationship between selling price and the number of units sold can be modeled by

a quadratic relation

*Ex.1* **Maple is selling sunglasses. Let P represent the profit and let s represent the selling ****price. The profit can be modelled by the equation:**

**P= -4(s-35)²+2500**

**a)What is the selling price that will maximize profits?**

We know that the max profit $2500, so the selling price that will maximize profits is 35$.

**b)What is the selling price when profits equal zero (also known as the break-even point)**

***Solution Below***

## Quadratics In Factored Form

__Special Cases__

**Perfect Squares :Squaring a Binomial**A perfect square is a number that can be expressed as the product of two equal integers.

The formula used for this concept is:

** (a+b)²=a²+2ab+b² OR (a-b)****²=a²-2ab+b²**

Example 1: (2x+3)²= 4x²+12x+9

Example 2: (3x-5)²=9x²-30x+25

__More special cases and differences of squares__

**5.3 Common Factoring**

__5.4 Factoring Simple Trinomials__

__Factoring by Grouping__

__5.4 Factoring Complex Trinomials__

__5.6 Factoring Difference of Squares and Perfect Square Trinomials__

__6.2 Solving Quadratics by Factoring (finding the zeros)__

__4.5 & 6.3 Graphing Quadratics In Factored Form__

**Note: I found the vertex from the axis of symmetry value(h-value of vertex) and optimal value(k-value of vertex)*## Word Problems for Quadratics in Factored Form

__Dimensions of a rectangle given the area__

__Flight of an Object__

__Shaded region of a shape__## Quadratics in Standard Form

**6.1 Maximum and Minimum values (Completing the square)**

__6.4 The Quadratic Formula__## Word Problems In Standard form (6.5)

## |Connections between Topics|

Quadratics in vertex form, standard form and factored form can all be easily be graphed. For example, when graphing in standard form, you still have to convert the equation into a factored equation. Also an equation can tell you all the different aspects of the parabola without even graphing it. For example if there is a negative a value you can figure out that the parabola will be reflected into the x-axis. Also many topics we learned can convert from one form to another. For example you can go from factored form, to standard form, to vertex form easily. When we learned simple trinomials and complex trinomials I discovered how similar they are. The only difference between them is the number in front of "a". When finding roots, finding the x intercepts in vertex form and standard form are very similar. In both units you could get 2 x-intercepts as well as you had to make sure to use the plus and minus sign in front of the square root symbol. Therefore this unit is extremely interconnected.