RWA Unit : M Concepts 4-6 : Conic Sections In Real Life

Parabola❤︎

1)

Mathematical Definition: A parabola is curved graph, it resembles an arched structure.Any point on a parabola equidistant  from a point called the focus, and a straight line called the directrix. 

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

2)
Algebraically: The equation we use is y=(x-h)^2+k to find the axis of symmetry 

Graphically: This is what a parabola looks like graphically 

     ↘️

http://www2.lv.psu.edu/ojj-rcm27/topics/parabolas.html

Key Parts: ↗️

1. Axis of Symmetry 

2. Focus 

3. Vertex 

4. Directrix

This Link will explain thoroughly and clearly the conic section of a parabola {http://www2.lv.psu.edu/ojj-rcm27/topics/parabolas.html}

Foci: The foci affect the graph because the closed the foci get to the vertex the skinner the graph gets as shown in the image below                          

                     ⬇️

http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-geogebra-dynamic-worksheet

If we move the foci farther away then the graph gets wider and opened up more like this image below
⬇️

http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-geogebra-dynamic-worksheet

3)

A real world application that i found of a parabola was a satellite dish. when the dish reflects information it bounces off and goes straight directly out. These rays are all bounced off to the same focal point. 

⬇️

http://www.geo-orbit.org/sizepgs/tuningp2.html

Video ☀︎

https://www.youtube.com/watch?v=r-KmkpxVtGg

4)

Works Cited

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

http://www2.lv.psu.edu/ojj-rcm27/topics/parabolas.html

http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-geogebra-dynamic-worksheet

http://www.geo-orbit.org/sizepgs/tuningp2.html

http://www2.lv.psu.edu/ojj-rcm27/topics/parabolas.html

https://www.youtube.com/watch?v=r-KmkpxVtGg

BQ 6: Unit U - Concepts: 1-6

Continuity & Discontinuity ❤︎

Continuity happens in a continuous function. a continuous function is a function that is predictable, which means it will go wherever we assume it will go, it has no breaks no jumps no holes and you may draw the graph without lifting your pencil from the paper. Graphs are continuous on intervals. Continuous functions make good bridges. A function is also continuous if the value and the limit are at the same point. 

Discontinuity happens when the graphs do contain breaks and or holes. Discontinuities are broken into removable and non-removable families. in removable continuities there are point discontinuities these graphs are also known as holes. In point discontinuity the value and the limit can sometimes be the same. In non-removable discontinuities there are jump discontinuities, oscillating behavior and infinite discontinuities. There is a break in Jump Discontinuities, In oscillating behavior the graph is wiggly so we cannot tell where the point are and in infinite discontinuities there is a vertical asymptote which leads to unbounded behavior.

Limits∞ 

A Limit is the intended height of a function and the limit exists when the value and the limit are the same, it also exists in continuous functions as well. A limit does not exist when the value and the limit are different. Limits do not exist in a jump discontinuity due to the fact that the points arrive at different locations from both the left and the right of the graph, it doesn't exist in oscillating behavior due to the fact that the graph is very wiggly and we cannot tell where the points lye on the graph & it doesn't exist at infinite discontinuity because of unbounded behavior due to the vertical asymptotes. The difference between a limit and a value is that the limit is the intended height and the value is the actual height.   

Evaluating Limits ☁︎

We evaluate graphs numerically by making a table and start off with the number we are heading toward and as we go to the left we subtract 1/10th and as we go to the right we add 1/10 and as we do this we can see that we are getting closer to our value as we move to the middle. We evaluate graphs algebraically by using either the dividing out/factoring method & the rationalizing conjugate method. Finally we evaluate a graph graphically by putting one finger on the right side of the graph and one finger on the left side and we slide our fingers along until they meet and we see where they meet and if they do in fact meet at a certain point on the graph to determine whether they are removable or non-removable. 

BQ#4 – Unit T Concept 3

"Normal" Tangent & "Normal" Cotangent Graph❤︎☁︎☀︎

Tangent☽

Each space between the asymptotes is a quadrant from the unit circle. tangent equals sine over cosine and tangent is positive in the first and third quadrant and also negative in the second and fourth quadrant. the graph has to be uphill down to up because the graphs cannot touch the asymptotes. there are asymptotes where "x" is equal to 0. 

Cotangent❀

For Cotangent its the reciprocal of the tangent graph. the asymptotes determine how the graph will look. There are boundaries in both graphs these graphs cannot touch the asymptotes. cotangent equals cosine over sine. asymptotes will be graphed where "y" is equal to 0. We start by graphing close to the asymptote and end below the graph.

BQ#3 – Unit T Concepts 1-3

   (All Asymptotes are based on Sine & Cosine☀︎)

Tangent, Cotangent, Secant & Cosecant ❤︎

All Asymptotes are based on sine and cosine. Asymptotes for these graphs occur when cosine is equal to 0. tangent has an asymptote where cosine is equal to 0 because of the ratio y/x this will end up being undefined. All these four trig graphs contain sine and cosine. These graphs consist of repeating units. Each period is repeated in a negative and positive manner. The tangent graph is positive and the cotangent graph is negative, where as the cosecant and secant graphs can start or end in either positive or negative direction.

BQ#5 Unit T Concepts 1-3

Sine & Cosine 
These two trig graphs will never have asymptotes because the ratio for sin is y/r and the ratio for cosine is x/r. "r" will always equal 1 therefore this function will never be undefined.

Cosecant, Secant, Tangent & Cotangent
These functions may have asymptotes because of their ratios. csc=r/y, sec=r/x, tan=y/x & cot=x/y. We know that the "x" or "y" value could be 0 which will make anything divided by it an undefined graph. secant and tangent will have the same asymptotes because of the "x" value on the bottom of their rations. The same goes for cosecant and cotangent because the "y" value is in the denominator of both of them.

BQ#2 Unit T Concept Intro

Trig Graphs

• The trig graphs relate to the unit circle because they are repeating units due to the different quadrant values. (ASTC)
• Sine and cosine have a 4 part repeating unit whereas tangent and cotangent do not have repeating units. 
• we have amplitudes  because we have limits for sine & cosine we can only stay between (-1,1) its the lowest and highest that we can go in the unit circle.

Reflection#1-Unit Q: Verifying Trig Identities

Reflection

1. What verifying a trig function means is that we use either the ratio identities, the reciprocal identities or the pythagorean identities to simplify a ratio as much as we possible can. We must also look for a way to simplify it but in a way that isn't going to make the problem a bigger mess than it already might be. 

2. Tips i have found helpful are to do a lot of practice, memorizing all of the identities and also watching videos from other students not only the teachers because some teach it in a way that i better understand or they clarify something in a different way. Like in Concept 5 i was completely lost it took me a while but with all the practice and the videos I've re-watched it really helps to stay on task and to ask questions when you need to. 

3. The steps i take to solving a trig ratio are simple. I analyze the problem and try to figure out what would be the best way to approach it. I avoid making the trig ratio a big mess, i do not want to make it harder to solve. I also see and picture which identity i can use whether its the ratio identity, reciprocal identity, or the pythagorean identity. Then i look to solve and simplify it to the simplest answer i can simplify it to.