I/D 1: Unit: N- How do SRT and UC relate?

A)

Here we are dealing with a 30* angle in which the horizontal side is x radical 3, the vertical side x, hypotenuse 2x. We start of by dividing everything by 2x because we need hypotenuse to be one. After canceling out the x we are left with a vertical side of 1/2, horizontal side of radical 3/2, and hypotenuse of one. Afterwards we plot it on the first quadrant deriving the ordered pairs of (radical 3/2,1/2), (radical 3/2,0), and (0,0) as the center. 

B) 

This triangle is a 45* angle and the labeling of the sides differs than those of the 30* angle. Here both the vertical and horizontal side are radical 2/2 and the hypotenuse is 1. We the just plot it on the first quadrant and get the ordered pairs of (0,0), (radical 2/2,0), (radical 2/2, radical 2/2). 

C) 

This is a 60* angle and it is similar to the 30* angle it's just that the sides are switched around. The hypotenuse still being 2x, the vertical side x radical 3, and the horizontal side as x. We first divide everything by 2x and we get a vertical side of radical 3/2, a horizontal side of 1/2, and hypotenuse of one. Next we place the triangle on the first quadrant giving us ordered pairs of (1/2, radical 3\2), (1/2,0), and (0,0). 

This activity helps me derive the unit circle by allowing me to know the ordered pair of a 30*, 45*, and a 60* angle. Which are the basic three triangle used in the entire unit circle. By drawing triangles on the first quadrant we get the positive ordered pairs. When triangles plotted on the 2nd quadrant ordered pairs would have a negative x value. When in the 3rd quadrant both x and y would have a negative values. When in the last quadrant (4) the y value will be negative. Other bottom image shows how the triangles are just flipped around depending on the quadrant it's in. 

 

1. THE COOLEST THING I LEARNED FROM THIS ACTIVITY WAS: that the unit circle primarily consists of these three triangles a 30*, 45*, and a 60*. That they only get flipped around depending on the quadrant they are in. 
2. THIS ACTIVITY WILL HELP ME IN THIS UNIT BECAUSE: it's another way to find the ordered pairs in the unit circle. So in case I forget one of them I can use this tool  to help me find the ordered pair. 
3. SOMETHING I NEVER REALIZED BEFORE ABOUT SPECIAL RIGHT TRIANGLES AND THE UNIT CIRCLE IS:  something that I had not realized about right triangles is that they played an important part I. The unit circle. And something that the unit cicle has is that it involves same points just switched around depending on the quadrant. 

RWA1: Unit M Concepts 4-6 : Conic Sections in real life.

Parabola

1. Mathematical Definition: "Any point on a parabola is at an equal distance from a fixed point (the focus), and a fixed straight line (the directrix)" (Math is fun)."

2. Conic Section:  

A) Algebraically             

http://www.algebra-online.com/tutorials-4/algebra-formulas/articles_imgs/7976/formul92.jpg 

B) Graphically

http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/vertical_parabola.jpg

B)  Graphically                                                                                 

http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/horizontal_parabola.jpg

The parabola comes with several key features: vertex, directrix, focus, and the axis of symmetry. The vertex is the point at which the parabola starts. The directrix is a line outside the parabola, perpendicular to the axis of symmetry. The axis of symmetry is a line that breaks the graph into two equal halves. The focus is the point inside the parabola. Lastly the "p" is the distance between the directrix and the focus and the from the focus to the vertex. All of which contribute to graphing a parabola accurately. 

Each of those parts can be derived from a general equation like 2y^2+x+24y+75=0, taking the first step which would be to turn it into standard form. This being done so by completing the square, remembering to there needs to be something squared on one side and a coefficient on the other. Once the equation is in standard form the key parts can be discovered. The vertex is (h,k), the "p" is the coefficient set equal to 4p, and from looking at the "p" and the equation you can tell whether it opens up, right, down or left. If "p" is positive and the equation is x^2 goes up, if "p" is positive and equation is y^2 goes right, if "p" is negative and equation is x^2 opens down, if "p" is negative and the equation is y^2 the graph opens left. With "p" being small (closer to vertex) it makes the parabola skinny, if its bigger (farther from vertex)  the parabola is fatter. Then you find both the directrix and focus by counting away from the vertex by the value of "p". Then there's the axis of symmetry which is perpendicular to the directrix and passes through the vertex. Lastly, the eccentricity of a parabola should be equal to one. To visually see this worked out you can seek HERE for more assistance.

3. Real World Application:

http://www.jamesburgdish.org/LinkClick.aspx?link=IBA-4_elevation.JPG&tabid=524&mid=998

A Satellite dish is the perfect example of a real life application of a parabola. The actual dish capturing the actual figure of a parabola. Thus creating an effect that causes waves that come in toward the dish to reflect off of it. And as the waves are reflected they bounce their way to the sphere (receiver) which in this case is the focus point of the parabola. With that it allows for their to be better signal and efficient cable. 

The satellite itself is the parabola with a focus as the main point. That is the spot at which all the waves end up going no matter where the waves hit on the  dish. This could serve to represent the "p" show the distance between the directrix and vertex is equal to the distance from the focus and vertex. How its an  equal distance from any given point, explaining as to why the waves all end up hitting the focus. Another thing would be that in small dishes the focus would have to be closer to the vertex and in a large dish it would be farther. 

4. References:

http://www.mathsisfun.com/definitions/parabola.html

http://bit.ly/zbruh4

https://www.google.com/search?q=parabola