# A-Z of Quadratics

## The Struggle Is Real?

Have YOU been struggling on Quadratics during class? Have YOU been receiving poor marks on assessments and tests lately? Don't worry no more, this website will explain everything in Quadratics so well you'll become the Queen of Quadratics! Just get some paper and a pencil and ready to learn!

## Unit Review!

Quadratics was a long unit and we learned all these different types of equations and how to graph curved lines:

-Vertex Form

-Factored Form

-Standard Form

We started from vertex form; the equation for vertex form looks like: y=(x-h)^2+k. We then moved on to factored form; the equation for factored form looks like: y=a(x-r)(x-s). Then we learned standard form; the equation for standard form looks like: ax^2+bx+c=0. We started by learning the definitions such as; axis of symmetry, optimal value, zeroes, x-intercepts, and vertex. We then moved on to vertex form and learned the equation and what each part of the equation represents. Then we learned how to graph using vertex form!

## How to find Linear and Quadratic Relations?

To find whether the relation is quadratic or linear we can find out by calculating the first and second differences from a table of values, if it is given. We can find if the relation is linear if we subtract the first x coordinate from the second and we keep doing it. If the numbers are the same each time then the relation is linear, if not then the it is non linear. For second differences, we would complete the same process, but instead we do it for the first difference. Image on the side shows process:

## Solving In Vertex Form

In class we, learned that vertex form was the hardest form for quadratics. It was the equation that would give us the vertex and we would have to solve by using two methods. The two methods are the step pattern and mapping notation.

The step pattern teaches you that the main equation for graphing is y=x^2 and that the main numbers are 1,3,5,7. To find the step pattern you would add the (a) value to these numbers and get the second difference. The next method that we learned about was mapping notation, mapping notation has a equation that is used to solve for the (a) value.

The equation for vertex form: y=a(x-h)^2+k

-We learned that (a) means the vertical stretch.

-(h) means horizontal translation

-(k) means vertical translation

We also learned that (h,k) are the points for the vertex. These are the transformations that you would list if you are explaining about vertex form.

To solve from vertex form, there is a process by which you square root to find the x-intercepts from vertex form. You can square root or you can change the equation into factored form then solve for the x-intercepts. Here are some examples I made!

## Solving Using Factored Form!

We learned factored form after the vertex form.

-Factored form is used to convert standard form into factored so we can easily determine the x-intercepts

-We were taught that the equation for factored form is y=(x-r)(x-s)

-The similarity for factored form is that the (a) value does not change. Also, this equation is used differently because we will not be able to graph the parabola using the vertex

-An easy way to find the x-intercepts is to set each bracket equal to zero, then we will be able to find what (x) equals if we move the number over to the other side and divide both numbers by the (x) coefficient

Here is an example:

## Factored Form!

There are three main equations for factored which the answer will show, they are the perfect square trinomials and the difference of squares. These three equations are the main ways that factored form will appear in. Here is a picture of a great example:

## Solving Using Standard Form!

The equation for standard form is ax^2+bx+c=0. There are several ways to solving this equation, one of the ways are using the quadratic formula. The quadratic formula is the easiest way to solve from standard form and find the x-intercepts. Also, once you find the x-intercepts you can find the axis of symmetry by adding the two numbers and dividing them by how many there are, which is two. Then, you can sub in the axis of symmetry, into the equation and solve for the optimal value. Once, you get the optimal value, you will have your vertex and you can also graph the parabola.

Quadratic Formula Question

## Standard Form Into Factored Form Converssion

To convert standard form into factored form there are several methods to do this

-There is one method that we learned in class which: trial and error. I found that method a bit difficult because the table was confusing.

-Yet there is another method called decomposition which was much easier. In decomposition you multiply the first and last term together and the number that you get should add up to the middle number. I found this method the easiest because if was very easy to understand and was not confusing.

Here is an example of decomposition:

Factoring by Decomposition ( MyMathStudyGuides.com )

## Standard Form to Vertex Form Converssion

Converting from standard form to vertex form is quite simple! To convert standard form into vertex for we can complete a process called completing the square. This process can be used to find vertex form from standard form. Below is a picture that explains how completing the square can be done:

## Discriminant

Discriminant ''D'' is used to determine how many solutions there are in the equation. When we use the quadratic formula we should figure out how many x-intercepts there are before actually finding the x-intercepts. The discriminant could be found by using the formula ''D=b^2-4ac'', this formula is in the quadratic equation and we can use this to find the number of solutions to the equation. If ''D'' is positive then there will be two x-intercepts, if ''D'' is negative then there will be no x-intercepts because we cannot square root a negative number, and if ''D'' is zero then there will be one x-intercept. As shown below:
Discriminant of Quadratic Equations

## How to Convert?

Converting factored to standard form is very simple and quick! We just and expand and simplify like below:

## Graphing Tecnhinques and Info

There are several different apps and programs you can use to help graph parabola's for equations, no matter in what form. Desmos is used to graph word problems.

## Solving By Graphing From Vertex Form!

We worked with flight problems throughout the entire unit of quadratics. To figure out the two x-intercepts and the axis of symmetry. Since the equation was in vertex form, the vertex would already be given to us, therefore we would have to graph the parabola and list the transformations. When we graph, we should use the step pattern. Step Pattern can also be used by second differences!

## Solving By Graphing From Factored Form!

In factored form, we were given the x-intercepts, so we had to find the axis of symmetry and optimal value to get the vertex, and graph the parabola. The factored form equation is y=(x-r)(x-s). For this equation we do not list the transformations because they do not apply for it, yet we can list them if we convert factored form in to vertex then it will be acceptable. To find the x-int we must set each bracket equal to zero and then solve for (x). After finding the x-int we can find the axis of symmetry by adding the two numbers and dividing them by two. After we have found the axis of symmetry we can solve for (y) and find the optimal value. once we have found both the Axis of Symmetry and the optimal value we can plot the graph.

## Vertex Form Word Problems!

Word Problems are the most fun part of the entire unit! Just Kidding :p

Here are some excellent examples:

## Factored Form Word Problems!

These questions were intense and complicated. These generally were focused on area/geometry based questions/

Here are examples below:

## Flight Word Problem

Here is a great flight word problem!

## Standard Form!

Flight problems were what we worked on all unit. However we focused on geometry questions like below also:

## My Personal Reflection of Quadratics

I really enjoyed this unit and thank Ms. Molyneux for teaching it so well. I really enjoyed learning about the quadratic form formula and factoring but didn't enjoy certain word problems in this huge unit. On quizzes early in the unit, I was struggling but really did well in unit tests, especially on communication questions. On mini test 2 I got perfect marks for the communication question because I can really easily express my thoughts and feelings on open response questions. I studied really hard and puts lots of effort before a mini test but should have done more extra review questions for further clarification.

## Final Refrences!

https://www.edmodo.com/

https://www.mathsisfun.com/

http://www.purplemath.com/

http://calculus.nipissingu.ca/tutorials/quadratics.html

I originally made most of the pictures and the video. Other videos were taken from YouTube and khan academy meanwhile images just of google images.