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. 

SP#7: Unit Q Concept 2 Finding trig functions with one trig function and quadrant

Please see my SP7, made in collaboration with Jose Leal, by visiting their blog here.  Also be sure to check out the other awesome posts on their blog

WPP 13-14: Unit P Concets 6-7: Applications Of Law Of Sines & Cosines

Please see my WPP13-14, made in collaboration with Jose Leal, by visiting their blog here.  Also be sure to check out the other awesome posts on their blog here

BQ#1:Unit P Concepts 1 & 4: The Law of Sines AAS/ASA & Area of an Oblique Triangle

Law Of Sines AAS/ASA

1. We need the law of sines because sometimes we have to solve for triangles that are non right triangles. We can cut the triangle in half to form two triangles as shown in the picture below.

http://www.ck12.org/book/CK-12-PreCalculus-Concepts/r508/section/4.6/

and from the perpendicular line drawn from the two verticies we can find the other relationships as shown in the picture below. 

http://www.lhs.loganschools.org/~rweeks/trig/law_of_sines.jpg

Area Of An Oblique Triangle

The are of an oblique triangle is derived from A=1/2bh which is 1/2xBasexHeight. This relates to the area formula i am familiar with because we use sin of some angle for example sinB=h/a we multiply by "a" on both sides to get "h" alone and get h=1/2b(sinB) we substitute "h" for in the equation for "sinB" 

http://hotmath.com/hotmath_help/topics/law-of-sines.html

Resources

http://www.ck12.org/book/CK-12-PreCalculus-Concepts/r508/section/4.6/

http://hotmath.com/hotmath_help/topics/law-of-sines.html

http://www.lhs.loganschools.org/~rweeks/trig/law_of_sines.jpg

WPP# 12 Unit O Concept 10: Solving Angle Of Elevation & Depression Word Problems

Vanessa's Tootin' Train Express

A) The very cool express train is on the very coolest train track towards a super cool tunnel. The angle of elevation to the top of the super cool tunnel is 22°8'. If the base of the super cool tunnel is 848 feet from the very cool express train how high is the super cool tunnel?

Pickel Extravaganza

B) There is a man standing at the top of the Tunnel his name is Swanson Giggleton OH NO! He see's a pickle on the train track and doesn't want it to get squished. He measures the angle of elevation to the pickle to be 44°. He also knows that he is 345 feet from the ground. If there is a zip line that takes him in a diagonal direction straight to the pickle how long will it take him to reach the pickle & save the pickles life? 

I/D 2: Unit O-How can we derive the patterns for our special right triangles?

30-60-90

We label all the sides of the triangle 1. We then cut the triangle in half and notice that since all the angles are 60 the angle at the top will be cut in half and made into 30 degrees after this we can label the triangle sides "a", "b" and "c". The horizontal line was one and was cut in half so we know that the length of the new side is 1/2. We see that we are missing side "b" so we use the pythagorean theorem to solve for the missing angle. when we do a^2+b^2=c^2 we get "b" to equal radical (3/2) we know that we can't have any fractions so we multiply every side by two and get a=1 b=radical 3 c=2. we then can add the variable "n" to show that this pattern can be applied to another function.

45-45-90

i started by cutting the square diagonally. We know that each angle in the square is 90 degrees so by cutting it in half we know that half of 90 is 45 degrees. We also know that the two angles that are not the 90 degree angle are the same measurement. we are given 1 as the side length to all the sides and we label the sides "a", "b" and "c" we are missing "c" so we use the pythagorean theorem to find "c" we end up getting "c" to equal the square root of 2. We know that we can add the variable "n" to show that this also is a pattern that can be applied to another function. 

I/D#1:Unit N - How do Special Right Triangles and The Unit Circle Relate?

Inquiry Activity Summary

This activity helps me derive the Unit Circle because the angles that are given and plotted if plotted all the way around make the unit circle with all the same radians as well. As shown in this picture below.

We can also see that as the triangles are changing quadrants the values of the radians change also shown in the picture below

Inquiry Activity Reflection

1. The coolest thing I learned from this activity was the pattern i see as i move around the circle with the different triangles
2. This activity will help me in this unit because it shows me the difference between all three triangles
3. Something I never realized before about special right triangles and the unit circle is that as we move around the circle the signs of the radians change