I/D#1: Unit N: Concept 7: Unit Circle and Special Right Triangles.

Inquiry Activity Summary:
      In this activity regarding unit circles, we labeled 3 special right triangles according to the Special Right Triangle Rule which will tie in to our derivation of the unit circle.

1. In this picture, we see a 30 degree triangle. To go by this problem step, our numbers in circles are chronological, which means they are in order of what to solve for first. For number 1, we label the triangles according to the aforementioned rule, so our hypotenuse value is 2x, our x value is x radical 3, and our y value is x. We got these values by looking them up on google and applying them here in this problem. For step number 2, we first follow instructions by setting the hypotenuse's value to 1, then divide the value of every other side by the value of the hypotenuse, which is in this case, 2x. Therefore, the y value becomes 1/2, and the x value becomes radical 3/2. We now see the relations of the unit circle to this activity as the values of the sides are identical to the ordered pairs in the unit circle. Step 3,4, and 5 has us simply labeling the r,x, and y value, which are respectively shown in the picture. Step 6 has us draw the coordinate grid of this triangle, and this is where we see how this triangle is literally identical to what it is on the unit circle.

In this picture, the 45 degree triangle is represented. We will go step by step just like the last picture. For step 1, the labeling of this triangle is thanks to the internet, and the hypotenuse value is x radical 2 with the x and y values being x. How this is so is because the 45 degree triangle has two sides which are equal due to this triangle being an angle that is in the middle of 90 degrees. Step 2 has us again set the hypotenuse to 1. We then divide the x and y values by the hypotenuse value, which is x radical 2. We therefore get radical 2/2 for the x and y value because they are the same values. Steps 3,4, 5 are shown as the labeling of the sides and step 6 has the graph of this triangle, and as we can see, the angle is representative of the unit circle.

In this picture, we have a 60 degree triangle, which is basically the same as the 30 degree triangle but the x values and the y values are reversed. The labeling is the same, so step 1 remains unchanged, except that the labeling for x and y are of course different from the first picture in that they are now respectives of each other now. Step 2 has the hypotenuse as 2x, the x value as  1/2, and the y value as radical 3 over 2. We derived these numbers from the work shown in the picture. The steps 3,4, and 5 are shown above clearly. Step 6 is where we draw the triangle. In this example, the x and y swap off in order to get that 60 degree shape. 

4. This activity assisted me in deriving the unit circle by teaching me 3 of the most important degrees I need to know for this unit. The x and y values are identical to those on the unit circle and by recognizing these special right triangles I can label the unit circle accordingly and solve real problems with ease.

5. The quadrant the triangles in this activity lies on is the 1st quadrant. If I placed the triangles in quadrant two, then the x value will be negative. If I placed them in quadrant 3, then both values would be negative. If I placed them in quadrant 4, then the y value would be negative. In this picture we see the 3 triangles being placed in different quadrants. The 30 degree triangle has its x value as a negative just like we mentioned. The 45 degree triangle has both ordered pair values being negative. The 60 degree triangle has only its y value being negative.Another change that occurred was how the overall angle measurements of each angle has changed. the 30 degree triangle is a reference angle for sure, but is 150 degrees from the initial axis. The 45 degree triangle is 225 degrees, and the 60 degree triangle is 300 degrees.

Inquiry Activity Reflection:

1. The coolest thing I learned from this activity was how this related to the unit circle I did last year and how easier this activity made my life easier by teaching me the core basics of the unit circle. 

2. This activity will help me in this unit because I will be able to derive the unit circle with greater ease by knowing the most important degree properties: 30, 45, and 60. 

3. Something I never realized before about special right triangles and the unit circle is how easy it can be after you do this activity of how to solve the properties of the 30, 45, and 60 degree triangles.

RWA #1: Unit M Concept 4-6: Conic Sections In Real Life Scenarios

1. Parabola: the set of all points the same distance from a point, known as the focus, and a line, known as the directrix.
2. Regarding the standard form of a parabola, if the parabola has a vertical axis of symmetry, then the equation is 
(x-h)²=4p(y-k), with (h,k+p) being the ordered pair for the focus. A parabola with a horizontal axis of symmetry will have the standard equation of (y-k)^2=4p(x-h) and the focus ordered pair being (h+p,k). The difference between the two is whether x or y is in the front of the equation and where the p value goes in the ordered pair of the focus. Visually (graphically), a parabola looks like a curved line, with both arrows pointing in the same direction, but meeting in the middle along the aforementioned curved lines. What determines the direction of the parabola are two factors. If the value of p is positive, just like the graph, the parabola is bound to be going up or right.If p is negative, then of course, the parabola will go down or left.The second factor on determining the direction of the parabola is if the equation has an (x-h)^2 or (y-k)^2 as its standard form. If it is x^2, then the  graph goes up or down. If its y^2. then its left or right. The size of the parabola depends on the value of |a|. The smaller the value of |a|,then the wider the graph. A parabola with a |a| of 0.3 will be much wider than one with an |a| of 1. This applies if the standard form is expressed as y=ax^2+bx+c. In this case we place a heavier emphasis on p.
        An axis of symmetry shows the exact middle of a parabola where it touches the vertex. The vertex is the origin of the parabola, and its value depends on the h and k values of the standard form. Regarding its origin graphically, it is in between the focus and the directrix. The focus's value,as mentioned before, depends on if the graph's axis of symmetry is horizontal or vertical. If its vertical, then the ordered pair is (h,k+p).If its horizontal, (h+p.k) will be the ordered pair. The focus is situated WITHIN the parabola. What this means is that the focus will always be in the same direction as where the arrows of the parabolas point to. The directrix's value is the p value added or subtracted conversely to what it was added or subtracted for the focus. However, we do not write it as an ordered pair. We write it either equal to x or y, depending on if the value goes up or down, or in other words, on the the direction of the axis of symmetry. The value of p,as mentioned previously, takes the value of the coefficient of the non-squared coefficient of the variable and sets it to 4p. This value determines how far the focus and directrix will be from the vertex. A key note is that the directrix is always behind the parabola, or vertex, and the focus is always in front of the parabola,or vertex. The focus affects the shape of the parabola tremendously due to its connection with the eccentricity of the parabola. The closer the focus is to the parabola, the skinnier the parabola will be due to its relation with the difference between the focus and vertex being closer to 0. The closer a conic section is to 0, then the more circular it will be. Therefore, if the focus is farther away from the vertex, than the curves of the parabola will stretch and become less circular.
                               
                                https://people.richland.edu/james/lecture/m116/conics/conics.html

      This picture depicts the various parts of the parabola. The vertex is now clearly shown as the origin of the parabola, since that is where all the other factors revolve around. The directrix is shown below the parabola, as the parabola protrudes upwards. The focus is between the two curves, and the axis of symmetry shows the middle of the parabola. The d1 and d2 lines showcase that the distance from the focus to a point on the parabola to the directrix is the same for both lines. Also, in this graph, we see that the focus is considerably farther away from the vertex than common examples with a p value such as 1/16, so the graph becomes wider.

                           
                         http://xahlee.info/SpecialPlaneCurves_dir/Parabola_dir/parabola.html

3.      A real life example of a parabola would be a car's headlights. In this example, to describe why it is a parabolic example, it uses a parabolic reflector. What this means is that a light source is placed where the focus is. This causes the rays to bounce off the parabola line and ricochet parallel to the axis of symmetry. These particles of light being concentrated to emit an array of light beams represent the function of this parabolic example. However, regarding the real life application of this example, it is sometimes necessary to lower the amount of rays being parallel to the axis of symmetry. Therefore, a filament should be used to control the angle of the light rays. If the filament is placed behind the focus, then the light rays will converge. If placed in front of the focus, the ray will diverge instead. If placed above, then the ray will be directed downwards, with it being placed below causing the headlights to show light upwards.
        The vertex is behind the more important part in the headlights- behind the focus point. The directrix is not of huge importance as well since the main point of this example is to highlight the usage of the focus, axis of symmetry, and to an extent the vertex (because the vertex provides the curve which is needed to alter the direction of light reflected off of it). The axis of symmetry is of course in the middle of the parabola and like we said earlier, the more light rays that are reflected parallel to the axis of symmetry, the greater the concentration of light that will be emitted. For the value of p, the greater it is, the greater the distance the focus will be from the vertex, resulting in a wider headlight, which means the light rays covering more area. Overall, this real life application is a paramount example of how important the focus and axis of symmetry to particular aspects of life.

4.
In this video, the equation of a parabola is emphasized. Life I mentioned before, the vertical axis and horizontal axis dictate much of a parabola's information, as it influences the axis of symmetry, focus point and directrix. The equation (x-h)^2=4p(y-k) and (y-k)^2=4p(x-h) plays a huge part in this video as the lady goes on the explain the effects each one has on the shape of the graph as well as the focus point and directrix. An example is even shown as it portrays the steps to solving a parabola correctly.

4. References:

• http://www.wyzant.com/resources/lessons/math/algebra/conic_sections
• http://www.mathwarehouse.com/quadratic/parabola/focus-and-directrix-of-parabola.php
• https://people.richland.edu/james/lecture/m116/conics/conics.html
• http://www.pleacher.com/mp/mlessons/calculus/appparab.html
• http://xahlee.info/SpecialPlaneCurves_dir/Parabola_dir/parabola.html
• http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php