### By: Ahmed Mohamed

Quadratics is the study of Quadratic relation. The difference between a quadratic relation and a linear relation is that a linear relation only has a first difference in its table of values. as opposed to a Quadratic relation has a constant second difference in its relation.

## The components of a parabola

There are 5 components of a parabola, they go as follows:

1)The curve:this is the curve of the parabola

2)The axis of symmetry:This is a line that goes through the center of the parabola

3)The X-intercept: This is where the parabola intersects the x axis. there can be 1, 2 or 0 depending on the position.

4)The y-intercept: This is where the parabola intersects the y axis. there can be 1 or none depending on the position

5)The Vertex: This is either the highest or lowest point of the parabola, depending on the direction of the opening.

## 1.How to find the vertex in vertex form

This is done by taking the second number in the bracket or the h value in

y=a(x-h)2+k, and taking the opposite of it and making that the x-value. Then you take the k value in y=a(x-h)2+k, and make that the y-value. For example for the relation y=2(x-5)2+7, the vertex of the parabola would be (5,7).

## Locating the AOS

The axis of symmetry is the opposite of the number in the bracket. For example if the equation was y=2(x-5)2+7, the AOS would be 5.

## How to calculate the step pattern of a graph

The step pattern of a basic parabola(y=x2) is over one, up one to get the first point on the graph from the vertex. Then it is over two, up four to get to the second point from the vertex.To calculate the step pattern of a relation is to take the stretch factor, or the a value in y=a(x-h)2+k and multiply the second number by it. For example, if the relation is y=2(x-5)2+7. you would take the 2 and multiply it by the one from the basic parabola's step pattern giving you over one, up two. Then you take the four from the basic parabola's step pattern, and multiply it by 2 giving you over two, up eight.

## The four transformations of a parabola

Their are four transformations of a parabola. They are the a, h, the -sign, and k values of the basic vertex form equation(y=a(x-h)2+k). The a value determines the stretch of the Parabola. We previously explored this. The second is the h value this determines the horizontal translation. You take the opposite of the h value, and move the parabola to the side by that number. For example in the quadratic relation y=2(x-5)2+7 the -5 is changed to a 5, and then the parabola is moved to the right by 5. The third is the k value. This determines the vertical translation. For example in the parabola

y=2(x-5)2+7 the parabola is moved upwards by 7 and if was -7 it would be lowered by 7. The fourth and final is the - sign. This determines if the parabola has a max or min vertex(if it faces up or down).

## The second Quadratic relation form

The second form is y=a(x-r) (x-s). this can also be called factored form, because it is similar to a factored simple trinomial. This form can be used to display the x-intercepts. the r and s values are the x-intercepts.

## The types of factoring

1)Common factoring

2)simple Trinomials

3)Complex Trinomials

4)Perfect Squares

5)Difference of squares

## Common Factoring

Common factoring can be used to factor out a number or variable evenly. A common mistake that a lot of students make is leaving out said number/variable instead of putting it on the outside of the parenthesis.
Factoring trinomials with a common factor

## Simple Trinomials

For easier understanding we will name the first number in a standard from relation a, the second b, and the third c. When common factoring a simple trinomial you would set up 2 sets of parenthesis, and put your a value in both which is usually an x or other variable. Then find two factors of the c value that can be added to give you the b value. Below is a video of me demonstrating such.

## Factoring Complex Trinomials

Factoring trinomials with a non-1 leading coefficient by grouping

## Perfect Sqaures

Factoring perfect square trinomials

## Difference of Squares

Factoring difference of squares

## Graphing factored form

To graph factored form you need to locate the x-intercepts. This can be done by taking the r and s values of y=a(x-r) (x-s) and taking the opposite of it and placing it on the graph. For example if the relation you were graphing was y=2(x-4) (x+12). You would take -4 and make it 4, and take 12 and make it -12. You would then plot them on the graph. You would then take the x-intercepts and add them together, then divide it by 2. For example 4+12=16, and 16/2= 8. Now you have the x coordinate of your parabola then you sub in your x coordiante into your relation as x value. so in this case it would become y=2(8-4) (8+12) and you solve. You would get 160. Therefore the vertex would be (8,160). The final step would be to draw the curve.
Graphing Parabolas in Factored Form y=a(x-r)(x-s)

## Standard Form

The third quadratics form is standard form. It looks like this: y=ax2+bx+c. You can go from this form to factored form by factoring. You can go back by expanding. Here is a video explaining how to do so.
Converting Factored to Standard Form
Standard Form to Factored Form:Finding a Pattern

## From standard form to vertex form

This can be done by a process known a completing the square. This is shown below.

## How to find the AOS

The quadratic equation can be used to find the x-intercepts. This is used in place of factoring when the numbers aren't easily factor able. Below is an example of me finding the x-intercepts.

## Discriminants

Within the Quadratic equation, their is a part that can tell you if their is a x-intercept or not. This equation is D=b2-4ac if the answer is greater than 0 it has 2 x-intercepts, if it is 0 it has 1 x-intercept, and if it is less than 0 it has no x-intercepts

## Connections

Their are a few connections between topics. For example all three forms are capable of being graphed, and they are all interchangeable. Even though Quadratics is taught in separate units it is all one strand of math.

## Reflection

Overall all Quadratics wasn't too bad. I found Quadratics 1 to be much easier than the rest, and I particularly disliked quadratics 2. I felt this way due to the fact that their was too many different types of factoring. I found my self not knowing which one to use. So in conclusion it wasn't the best nor the worst part of the grade 10 math curriculum.

## How to answer an optimization word problem

Calculus - Optimization - General Steps for Optimization Word Problems Intuitive Math Help