QUADRATICS

By: Indroop Bassi

Table of Contents

Introduction:

- Parabolas

- Second Differences

- Transformations

Types of Equations:

- Vertex Form

- Factored Form

- Standard Form

Vertex Form

- Equation In Vertex Form

- Mapping Notation

- Determine the Equation From a Graph

Factored Form

- Multiplying Binomials

- Special Products (binomials)

- GCF and Factor by Grouping

- Simple Factoring

- Factoring Quadratic Expression/Special Cases

- Solve by Factoring

- Graphing Factored Form

Standard Form

- Quadratic Formula

- Discriminant

- Completing the square

- Graphing from standard form

Word Problems:

- Height

- Profit

- Geometry

- Number Problems

- Economic

Introduction

Second Differences

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A quadratic relation will always have a unequal first difference and equal second difference. You will know this is not a linear relation because a linear relation will always have an equal first difference. In this case, you can see the second differences that are equal which shows it is a quadratic relation.

Parabolas

When a quadratic relation is graphed a parabola is created. Below the picture are some key things that relate to the parabola.
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Vertex: where the graph changes direction. Point (x,y)
Axis of Symmetry: line that splits the parabola in half (x value)

Optimal Value: y value, can be a MAXIMUM or MINIMUM value

x intercepts: the zeroes

y intercepts: graph crosses y axis

Transformations

An equation in vertex form, each letter has a transformation.

The (-h) is the horizontal translation. It moves the parabola either to the LEFT or RIGHT. (when it's positive, moves to left. When it's negative, moves to right)


The (k) is the vertical translation. It moves the parabola UP or DOWN. When k is negative, it moves down. When k is positive, it moves up.


The (a) is the vertical stretch and is reflected in the x-axis if there is a negative sign. It also stretches the parabola.


Vertex (h,k)

Once you know the vertex, you can use the step pattern to find the rest of the points to complete the graph.

Types of Equations

1. Vertex Form: y=a(x-h)² +k

2. Standard Form: y= ax²+bx+c

3. Factored Form: y= a(x-r)(x-s)

Vertex Form

In this section I will show you how to create an equation when given the vertex and several examples to go along.

Solving Given Vertex Form

Pre-Calculus - Determining the quadratic equation given a vertex and a point

Isolating for x

Isolating for x is used when you need to find the zeroes, also known as the x intercepts.


1. Transfer -50 to the other side of the equal sign. Moving it over changes it to +50

2. Divide both sides by 2

3. Square root both sides of the equation to get rid of ²

4. Transfer -5 to the other side of the equal sign to have x by itself. (Make sure -5 changes to +5 after moving it to the other side)

5. Solve

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Graphing Vertex Form

Graphing using vertex form can be done simply using the step pattern. The video below thoroughly explains the steps on how to graph using vertex form.
Graphing a parabola in vertex form | Quadratic equations | Algebra I | Khan Academy

Factored Form

Expanding (Factored to Standard)

You use the method of expanding when you want to go from factored form to standard form. For example, if you have (x+3)(x+2), once you expand the equation you will get an answer of x²+7x+12. Below I have shown the steps of the FOIL method and two examples that use the FOIL method.

F = First Term

O = Outside Term

I = Inside Term

L = Last Term

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Simple Trinomials

The picture and video below have many examples of simple trinomials. They are called trinomials because tri means 3 and there are 3 terms in the equation.
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Simple Trinomials

Complex Trinomials

Factoring Complex Trinomials

Common Factoring

Common Factoring

Specials Cases (Perfect Square & Difference of Squares)

Example 4: Factoring quadratics as a perfect square of a difference: (a-b)^2 | Khan Academy
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Factoring by Grouping

Graphing Factored Form

Below is great video which explains how to successfully graph using a factored form equation.
Graphing Parabolas in Factored Form y=a(x-r)(x-s)

Standard Form

Solving Using Quadratic Equation

The quadratic formula is an easy way to find the x intercepts of an equation when written in standard form.


1. Find the value of a,b & c with the equation provided

2. Use the quadratic formula to sub in the values

3. Solve to find x intercepts

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Discriminant

The discriminant determines how many solutions an equation has. The picture below shows how you can tell how many solutions there are.
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Completing the Square

Completing the square is when you take an equation in standard and change it into vertex form. The video below explains the steps to successfully complete the square.
Completing the Square - Solving Quadratic Equations

Graphing Standard Form

graph parabola standard form

Word Problems

Height

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Profit

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Area

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Number Problem

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Economic

1. Get given information

2. Use revenue formula

3. Write the equation

4. Solve

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Quadratic Relation

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Connections

Factoring and Graphing

Factoring and graphing connect because you can take an equation from one of the 3 forms and find the x intercepts (zeros) and then from the equation you can find the x intercept and y intercept of the vertex. Then you can plot the points onto the graph.

Discriminant and Zeroes

The discriminant shows the amount of solutions an equation has. Before you graph the equation you can find how many solutions it has. The picture below shows how many solutions you may have depending on your number.
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Reflection

As we started the quadratics unit, I was struggling with the unit. It was stressing me out because I thought I would never be able to understand it at all. I slowly started focusing more on what I was doing and finally got the hang of it. I still struggle with some of the equations sometimes but I slow down and think before I rush into a question. I usually make small mistakes that I wouldn't normally make because I stress out and rush through questions. I have realized that it isn't the best thing to do. Coming to the end of the quadratics unit, I finally understand all of the components. I still struggle with word problems even now, but I know with more practice, I will be able to do any word problem that I get in the future. I enjoyed factoring the most because the FOIL method made it easier to do questions. The method really helped me understand what I was doing and what steps I had to follow in order to answer a question successfully. Overall, I did find this unit to be difficult at times but I am glad I put in the work to understand all of the different components in the unit.