Quadratics

By: Jaspreet Chahal

Table of Contents

Introduction

Analyzing Quadratics

Quadratic Terms

Step Pattern

Parabola Transformations

Types of Equations

Quadratic Relations

Factored Form

Vertex Form

Quadratic Expressions

Expanding

Common Factors

Factoring Simple Trinomials

Factoring Complex Trinomials

Perfect Squares

Difference of Squares

Quadratic Equations

Quadratic Equations

Quadratic Formula

Quadratic Equation Problems

Geometry

Revenue

Reflection

Introduction

Analyzing Quadratics (Using Table of Values)

Quadratics. Before we move on, we need to know how to spot a quadratic relation. You will be able to spot a quadratic relation if the second differences are constant and the first differences are not.

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This shows that the equation y=x² is a quadratic relation because the second differences are constant.

Quadratic Terms

Here are some terms and a diagram that will help you understand parabolas more:

Vertex: The maximum or minimum point on the graph. It is the point where the graph changes direction.

Optimal Value (Maximum or Minimum Value): The highest/lowest point on the parabola.

Axis of Symmetry: A line which that goes down the middle of the parabola.

Y-intercept: The point where your parabola touches the y-axis.

X-intercepts: The point where your parabola touches the x-axis.

Zeroes/Roots: The point where the parabola will cross the x-axis.

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Step Pattern

For example, if your equation was y=x², then your base step pattern would be to start from the vertex and move over 1 and up 1, then over 2 and up 4, and so on. If your A was a number other than 1, then it would affect the vertical stretch of the parabola. The step pattern would change. Now you would move over 1, then multiply move up 1 (the vertical stretch), with the number that is your A, and then move that many numbers up to get your second point.

Parabola Transformations

a = stretches the parabola. It multiplies the vertical stretch. (When a is negative, the parabola opens down, when a is positive, the parabola opens up)

h = moves the vertex of the parabola left or right. (When h is negative, the vertex moves right, when h is positive, the vertex moves left)

k = moves the vertex of the parabola up or down. (When k is negative, the vertex moves up, when k is positive, the vertex moves down)

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Types of Equations

1. Standard Form y=ax²+bx+c

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

3. Vertex Form y=a(x+h²)+k

Quadratic Relations

Factored Form

Equation: y=a(x-r)(x-s)


In the example I will teach you how to make a parabola using the factored form equation.

Before that, here are the terms that are going to be used:

Zeroes: found by setting each factor equal to zero

Axis of Symmetry: midpoint of zeroes

Optimal Value: found by subbing axis of symmetry value into the equation

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With these points, you would now plot them onto your graph and make your parabola.

Vertex Form

Equation: y=a(x+h²)+k


Finding the Equation Given the Vertex

With the vertex) and the other point you were given, you must sub that information into the equation. With this equation you must now find the value of a. When you find the value of a, you will be able to write the complete equation.

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Quadratic Expressions

Expanding

Expanding is very simple. In the next 2 examples I will show you how to expand with algebra tiles, and without algebra tiles.

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Common Factoring

Common Factoring is the opposite of expanding. You are taking an expanded equation, and putting it back into its un-expanded form. With two brackets beside each other, and terms inside these two brackets (factored form), you would multiply everything in the first bracket by everything in the second bracket. In the next 2 videos, I will show you 2 different examples on how to factor.
http://youtu.be/U6buHoGXl3A

Factoring Simple Trinomials

Factoring: ax²+bx+c


b,c= different numbers

a=a variable, in this case, 1


In the next video I will show you how to factor simple trinomials.

http://youtu.be/t_5MAXoN_2E

Factoring Complex Trinomials

Factoring: ax²+bx+c


b,c= different numbers

a= a variable, in this case, any number, negative or positive


a multiplied by c should equal the product, and a+c should equal b.


In the example I will show you how to factor complex trinomials.

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Perfect Squares

Here are the perfect square equations:


(a²+b²) = (a+b) (a+b)

(a+b)² = a²+2ab+b²


Here is an example of solving a perfect square.

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Difference of Squares

Here are the difference of squares equation:


(a²-b²) = (a+b) (a-b)

(a-b)² = a²-2ab+b²


Here is an example of solving a difference of squares.

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Quadratic Equations

Quadratic Equations

Here is an example on how to solve a quadratic equation.
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Quadratic Formula

Here is a video on how to solve a quadratic equation, and the picture is a continuation from the video.
Quadratic Formula
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Quadratic Equation Problems

Word Problem - Geometry

Word Problem - Revenue

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Reflection

Overall, the quadratics unit was very long and a bit challenging for me. I did not do so well on mini test 1, I some what improved on mini test 2, and I hope to do well on mini test 3.

We do not realize that we see parabolas everywhere, but now that I have learned about it, every time I see a parabola I ask myself, did they use quadratics to build that?!

And it's obvious that they did, and it's amazing how much work they had to put in to build it.

Throughout this unit I have learned many different ways on how to make a parabola and I hope to use these skills next year.