The Quintessence of Quadratics
Over the course of second semester we studied quadratics. A quadratic equation is typically written as ax^2+bx+c=y. This is the standard form. We can convert this standard form into other forms to find the vertex of a parabola. We launched the project through a firework equation. We were tasked with finding the peak of a firework rocket's flight. From this we learned how to use Desmos and how to manipulate a parabolas width and location on the x and y axis'. We learned how to use Desmos to solve various math equations. Whilst studying Desmos, we also studied kinematics, motion equations. This is just a brief overview, there is quite a bit to cover.
What are Quadratics?
A quadratic equation is any equation in which x is unknown and a, b, and c represent numbers that do not equal zero. If a equals zero an equation is linear, not quadratic. There are three forms of quadratic function, standard form (y=ax^2+bx+c), vertex form, (y=a(x-h)^2+k), and intercept form (y=a(x-p)(x-q)).
Vertex Form and Parabolas
A parabola is a symmetrical open curve effected by quadratics. We can dictate a parabola using vertex form (y=a(x-h)^2+k). For example the a in vertex form (y=a(x-h)^2+k) dictates whether a parabola opens upward or downward. If it is positive the parabola opens upward, concave up, if it is negative the parabola opens downward, concave down. We can use the parabola to represent the path an object travels.
Parabolas can change very easily. All you have to do is change a few values in the equation. For example, increasing the x value makes the parabola narrower, decreasing the value makes it wider. By adding onto the x, for example x^2+1, the parabolas location on the y axis changes, this is the k value. Adding moves the parabola up, subtracting moves it down. Note: This also applies to negative values
The h value effects the parabola's location on the x axis. A positive h moves it right, a negative moves it left. Note: When the h is negative the equation changes to y=a(x+h)^2+k. This is because subtracting a negative creates a positive.
By putting all these variables together using vertex form, we can create a unique parabola.
The equation for this parabola is y=5(x-5)^2+5
Using other forms to create parabolas
You can use standard and intercept forms to graph parabolas. Here are some examples.
The red parabola is standard form, y=5x^2+5x+5
The blue parabola is vertex form, y=5(x-5)^2+5
The green parabola is intercept form y=x(x-5)(x-5)
As you can see all three react differently to the value of 5. Its quite interesting how different they all are. By making the a value x with intercept form the parabola does not stay a perfect curve.
The blue parabola is vertex form, y=5(x-5)^2+5
The green parabola is intercept form y=x(x-5)(x-5)
As you can see all three react differently to the value of 5. Its quite interesting how different they all are. By making the a value x with intercept form the parabola does not stay a perfect curve.
Converting Between Forms
We can quite simply convert each form into a different one. Here is how to convert each form.
Standard to Vertex
Lets start with y=2x^2-4x+5
Start by subtracting 5 on both sides y-5=2x^2-4x
Then factor out 2 y-5=2(x^2-2x)
Convert to a squared expression y-3=2(x-1)^2
Isolate y by adding on both sides y=2(x-1)^2+3
Standard to Vertex
Lets start with y=2x^2-4x+5
Start by subtracting 5 on both sides y-5=2x^2-4x
Then factor out 2 y-5=2(x^2-2x)
Convert to a squared expression y-3=2(x-1)^2
Isolate y by adding on both sides y=2(x-1)^2+3
Real World Application
We can use the vertex form to map an objects movement. We worked on a baseball problem where a batter hit the ball and needed to clear a fence. There were many small things such as the initial height being slightly above 0 because of the batters height and the realistic height of the fence. Say a fence is 10 feet tall and 20 feet away. By using a parabola we can see if the object clears.
Reflection
This was a very challenging course for me personally. I missed a lot of school due to health problems which prevented me from learning every bit about quadratics. The parts I did learn I worked very hard on. I now have a very dense understanding of the forms and how to manipulate parabolas using them. I hope to grasp the further concepts of quadratics during the summer and the next year. I will practice using khan academy to further my knowledge. I will keep my mind open to quadratics and how it effects other math concepts. By mastering quadratics I feel I will be able to better my knowledge in kinematics and even geometry.
If I could go back I would study a bit harder, I didn't know I would have medical complications but if I was ahead of the class I would be fine at this moment in time. I will have to put in a little more effort but I know that I will be successful. I know that with hard work I will understand quadratics entirely.
If I could go back I would study a bit harder, I didn't know I would have medical complications but if I was ahead of the class I would be fine at this moment in time. I will have to put in a little more effort but I know that I will be successful. I know that with hard work I will understand quadratics entirely.