Mishita Dharia - Quadratics

        Over the course of the quadratic unit we discovered three types of forms. They are vertex form, factored form, and standard form. There can be many connections made between these 3 forms, since they are all interchangeable and required while solving a quadratic formula.

 

        Firstly the vertex form is written as y=a(x-h)^2+k, where the vertex is the h and k value of the equation, written as (h,k). The h-value in the equation is the axis of symmetry, while the k-value is the optimal value. In order to find the x-intercepts, you must sub in 0 for y and then isolate for x. This can then be changed into factored form, by placing the x-intercepts into the factored form. It can also be changed into standard form by expanding the equation and from there (if required) the equation can be factored and put into factored form.

 

        Secondly, an equation in factored form is written as y=a(x-r) (x-s). The "r" and "s" are the x-intercepts, and can be found by setting each bracket equal to zero. The vertex of the equation requires us to find the axis of symmetry and optimal value, since the vertex is written as (axis of symmetry, optimal value). The axis of symmetry is found by taking the average of the two x-intercepts, while the optimal value is found by replacing each x in the original equation with the axis of symmetry. Both of these methods were taught to us in the vertex form unit, and now we can use them to solve for the factored form unit. This form can then be changed into standard form by expanding, and from there is can be changed into vertex form by completing the square, so that, if needed, it can be graphed.

 

        Lastly, an equation in standard form is written as y=ax^2+bx+c. The x-intercepts can be found by subbing in the a, b, and c values into the quadratic formula. Similar to factored form, the vertex of the equation requires us to find the axis of symmetry and optimal value, since the vertex is written as (axis of symmetry, optimal value).  The axis of symmetry is the found by taking the average of the two x-intercepts, while the optimal value can be found by substituting in the axis of symmetry (or h value) into the original equation. This form can be changed to factored form, if it is possible, by factoring. Also this form can be changed into vertex form by completing the square, by doing this you are able to graph this form very easily.

 

In conclusion, all these three forms can be used to find the x-intercepts, axis of symmetry, optimal value, and vertex of a graph. They can also be used to sketch the graph of an equation. It all depends on what form you are comfortable in using or what form is required to solve/graph the equation. If you look at it at a big picture, all these units were connected to each other in some small way, which is why if you take a step back and look at these three different units as a whole, you will realize that they are only one big unit: QUADRATICS.