A-Z of Quadratics
Learning Quadratics the EASY Way!
The Struggle Is Real?
Quadratics was a long unit and we learned all these different types of equations and how to graph curved lines:
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
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:
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.
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:
Standard Form to Vertex Form Converssion
Factored Form To Standard Form!
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
Solving By Graphing From Vertex Form!
Solving By Graphing From Factored Form!
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
Factored Form:Area Question
Flight problems were what we worked on all unit. However we focused on geometry questions like below also:
My Personal Reflection of Quadratics
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.