Overview and BeginningThe idea and essence behind the ‘Quintessence of Quadratics’ project was to undergo a series of challenges and problems in order to gain a proper understanding of parabolas, their most common forms, and to give us a start in understanding quadratics as a whole. We began the process with a simple problem regarding the trajectory metrics of a rocket. Seeing as it was unaffected by wind and other environmental occurrences, its path could easily be mapped with a simple parabola. We were presented with the example equation in standard form; y=160+92x+-16x^2. The goal; to use this equation to find various attributes of our rocket’s path such as it’s flight time and max height. Here you can see a graphic representation of that problem:
The "a" Variable and ParabolasThe “a” variable, as I explain in the next section, is a multiplier that affects the sharpness of a parabola’s curve. In Vertex form, “h” and “k” define the parabola’s vertex’s “x” and “y” coordinates respectively. A quadratic equation, regardless of form, when provided with any “x” or “y” value, will return a value for its opposite. This creates an infinite set a of points, that when connected by an ideological line, creates a parabola.
|
Variable ExplanationsDr. Drew supplied us with a plentiful amount of worksheets and engaging problems outlining the various forms for a parabola. Vertex, Standard, and Factored forms are among the forms we’d learned in class. With each form came their respective variables, but present in all equations was “a”. This variable acted as a multiplier, affecting the extremity of the parabola’s curve, as seen in the below graphic.
Vertex form (y=a(x-h)^2+k) is pretty straightforward, with “a” aside, we are left with variables “h” and “k”. In Vertex form, regardless of the other variables, “h” and “k” will represent the vertex’s “x” and “y”, respectively.
Standard Form is simply an expanded version of Vertex form, formatted as (y=ax^2+bx+c). Our example problem at the top of the page contains an equation of this format. If you are confused about these new “b” and “c” variables, no fear. We came up with a tidy conversion formula: (y=a(x+(b/2a))^2+c-(b/2a)^2). This might seem like a bit, but it simply means that “h” = b/2a, and k= c-(b/2a)^2. The final form we learned about was Factored form (which was really an unsimplified version of Vertex form.) This form was formatted as (y=a(x-r1)(x-r2)), the “1” and “2” displaying that the two “r”’s were in fact, different. Dr. Drew explained this using an area diagram, shown below: |
Conversion and Variations
Using my earlier-mentioned Vertex to Standard form equation (y=a(x+b/2a)^2+c-(b/2a)^2), you can easily plug in variables from the standard form equation and get the vertex equivalent. Converting from vertex to standard is just the inverse, using the equation: (y=a(x+b/2a)^2+c-b^2/4a). If you are looking to convert from factored into standard form, you only need to execute: (y=x^2+r1x+r2x+r1*r2).
Future Uses
Contrary to the popular teen belief that what we learn won’t be used, quadratic equations prove to be quite useful in a variety of real-world applications. Firstly, mapping the motion/time relationship in a moving object. This was demonstrated in the earlier-mentioned rocket trajectory problem, where it was used to find various attributes of the rocket’s path.
Quadratics is also very useful in finding optimum shape dimensions for maximum area in a shape. This has potential large/small construction uses and is a useful thing to know. If you are running a business and have survey data on how much customers value your products, you can compare cost and demand with a quadratic function, provided you have enough data.
|