BQ#7- Unit V

                             Unit V Big Questions

Where does the formula for the difference quotient come from? Well know i now this question because i learned it from the wonderful Mrs. Kirch :D

       In this graph, we can see our understanding of where the difference quotient is derived from. The first point is (x, f(x)). There is delta x  and delta equals change in distance between the first and second point meaning the second point is (x + delta x, f(x + delta x)) that is what they used delta x but, for this we will reference to delta x as 'h' so our second point can also be written as (x+h, f(x+h) so that we can get a better understanding of a variable that we know so it wont be so confusing. The line that touches the graph at two points is called the secant line, very different from a  tangent line that only touches the graph once. To find the slope of the secant line we are going to use the slope formula. The slope formula is m = y2-y1 all divided by x2-x1. When we plug in our two points from the first graph we have f(x + delta x) - f(x) all over x + delta x - x. Then we substituted h for delta x and we get f(x+h) - f(x) all over x + h -x. Then we notice in the denominator we have a x and -x and what happens when you get a positive and negative number or variable is this case variable you CANCEL THEM, leaving only h. Then you get the Difference quotient f(x+h) - f(x) divided by the letter h, that's the difference quotient :D

There is a video that verbally shows you how to derive the difference quotient by clicking here

                                                     Works Cited 

Image

Image2

BQ#6- Unit U Concept 1-8

              Unit U Big Questions

1) What is continuity?
 Continuity is when something does not stop or is not interrupted by anything. A continuous function is predictable. It has no breaks in the graph, no holes, and no jumps. A function that is continuous can be drawn with a single, unbroken pencil stroke. In terms of limits, a function is continuous if the limit as x approaches a number of f(x) is equal to the height f(x) that reaches at that point.

This is a picture of a function that goes on forever in both directions and is not interrupted by anything, meaning there are no jumps, holes, or breaks in the graph.

What is discontinuity?

Discontinuity is when something doesn't go on forever in both directions and is not continuous. There are two families of discontinuous functions. They are removable and non-removable discontinuities. The removable discontinuity is just called point discontinuity, also known as a hole. The non-removable discontinuities are jump discontinuity, oscillating behavior, and infinite discontinuity, which is known as unbounded behavior and occurs where there is a vertical asymptote. The removable also means that there is going to be a limit that does exist while at non-removable discontinuities there all going to be the limit does not exist.

Removable: 

Non-removable:

2) What is a limit?  When does a limit exist? When does a limit not exist?  What is the difference between a limit and a value?
A limit is the intended height of a function. A limit exist when you reach the same height from both the left  and the right of the graph which only happens in a point discontinuity. A limit does not exist when its in the non-

removable family. Jump discontinuity DNE b/c of different L/R. Oscillating Behavior b/c of oscillating behavior, theres no name for it, and Infinite discontinuity b/c of unbounded behavior. The difference between a limit and a value is that, a limit is the intended height and the value is the actual height on a graph.

3) How do we evaluate limits numerically, graphically, and algebraically?
We evaluate limits numerically by using a table. With the table, we find the intended heights of the whole function. 

To evaluate limits graphically, we plug in our function into the y= screen on our graphing calculator, hit trace, and trace to the value we are looking for. If you hit trace and you don't get a value, the limit does not exist and the reasons for that situation are those three reasons.

Lastly, we evaluate limits algebraically by using three different methods: 

Direct substitution method
In which you just plug in the numbers that is approaching the limit of f(x) and solve it. You can get either a numerical answer, 0/# which equals to ZERO, #/0 which is UNDEFINED so the limit DNE,and if you get 0/0 it is a indeterminate form so then you have to use the next method which is divide and factor method.
Dividing out/Factoring method
We use this method when we get indeterminate form. We then factor both the numerator and denominator and cancel the common term to remove the zero in the denominator. Then we use direct substitution with the simplified expression. BUT always be sure to use direct substitution first!!
Rationalizing/Conjugate method
This method is also helpful with a fraction, especially if it has radicals. You multiply by the conjugate of either numerator or denominator depending where the radicals are at, then FOIL the conjugates and DO NOT multiply out the non-conjugate part, leave it factored because you then get something canceled out which is a good thing and then you simplify and once again use direct substitution with the simplified expression.

                           WORK CITED

Continuous Graph
Point Discontinuity
Oscillating Bahavior
Jump Discontinuity
INFINITE DISCONTINUITY
Evaluate Numarically

BQ#4- Unit T- Concept 3

Why is a "normal" Tangent uphill, but a "normal" Cotangent graph downhill? Use unit circle ratios to explain.
According to the unit circle, tangent is positive in the first and third quadrants which is why on the graph, tangent is above the x axis which is positive from 0 to pi/2 and pi to 3pi/2. Also because the asymptotes for tangent are pi/2 and 3pi/2, that explains why tangent is going towards the asymptotes meaning that because tangent stops being positive at the end of the first quadrant it goes toward the asymptote to finish the cycle and then starts going negative to start the second quadrant and so on. That is why a normal tangent graph is uphill at one period.

http://www.mathsisfun.com/algebra/images/tangent-graph.gif

According to the unit circle, cotangent is positive in the first and third quadrants just like tangent that is why on the graph, cotangent is also above the x axis (positive) from 0 to pi/2 and pi to 3pi/2. Also, because the asymptotes for cotangent are 0 and pi, that explains why cotangent is going towards those asymptotes, which means that cotangent stops being positive at the end of the first quadrant going towards the asymptote to finish the cycle and then starts going negative to start the second quadrant and so on. Another reason why it is downhill is also because it is the inverse of tangent so the direction is almost like a mirror image going in a different direction since they are reciprocals of each other since they have different asymptotes it changes the pattern of the graph which makes it go downhill this time instead of uphill. That is why Cotangent is going downhill.


https://algebra2c.wikispaces.com/file/view/Cotangent(X).png/34612489/Cotangent(X).png


                                    WORKS CITED PAGE

TANGENT
COTANGENT

BQ#3 Unit T Concepts 1-3

2) How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.Well, sine and cosine don't have asymptotes but the rest of the trig functions do have asymptotes because at certain marks on the graph, sine can be 0 and cosine can be 0. These two functions relate to the others because all of the others have sine or cosine within them in one way or another.  

a) Tangent 
tangent is sine/cosine so when cosine is 0 that means that the trig function is undefined so there is an asymptote. The Tangent's asymptotes are at pi/2 and 3pi/2.

http://mythi-trig.fateback.com/images/basicall.gif

b) Cotangent 
cotangent is the inverse of tangent which means because tangent is sine/cosine, cotangent is cosine/sine. When sine is 0 there is an asymptote because the whole thing is going to be undefined at that moment. Tangent's asymptotes are at pi/2 and 3pi/2 so, to contrast, this means that cotangent's asymptotes are at 0 and pi. Using the graphs you notice how the asymptotes of the graph of y = tan(x) are the x-intercepts of the graph of y = cot(x). There are vertical asymptotes at each end of the cycle.  The asymptote that occurs at pi repeats every pi units.

http://image.tutorvista.com/content/feed/u1593/cotangent.GIF

c) Secant 
secant is the inverse of cosine which means that it is 1/cosine and when cosine is 0, secant becomes undefined as a whole since it is the reciprocal of cosine. Anytime we have an undefined as our answer, we automatically know that there is going to be an asymptote, there are vertical asymptotes.  The asymptote that occurs at pi/2 repeats every pi units.

http://www.analyzemath.com/trigonometry/graph_cosecant.gif

d) Cosecant 
cosecant is the reciprocal of sine making cosecant 1/sine so when sine is zero, cosecant becomes undefined anytime sine is zero resulting in an asymptote. The x-intercepts of y = sin x are the asymptotes for y = csc x. There are vertical asymptotes.  The asymptote that occurs at pi repeats every pi units.

http://www.scs.sk.ca/cyber/elem/learningcommunity/math/math30c/curr_content/mathc30/GRAPHICS/RECIPROCAL_GRAPHS/MOVIE8.GIF                                                                                                  WORKS CITED PAGE
TANGENT
COTANGENT
SECANT
COSECANT

BQ#5:Unit T- Concepts 1-3

1.) Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain. 

Sine and cosine do NOT have asymptotes because that is how they are, they are in a wavy line when they are fully done in the graph and they run infinite on the x-axis. It refers to the unit circle because the ratio for cosine is x/r and since r is always 1 cosine can never be undefined since r can never be zero which is what makes a function undefined when the denominator is zero . The same goes for sine, in the unit circle the ratio for sine is y/r and since r is always 1 sine can never be undefined since r can never be zero, which is what is needed if you want to result in an asymptote to get that you need the denominator to be zero.

http://www.kirupa.com/developer/animation/images/cosinegraph.jpg

Cosecant, Secant, Tangent, and Cotangent all have asymptotes because of how x and y are zero in this case. Going back to the identities we know that secant is 1/cos which is, using the ratio of secant which is r/x. X can be any number and when it is zero secant automatically becomes undefined and when it is undefined we know that its going to have a asymptote. The same goes for cosecant, which is r/y, except that when y is cosecant it becomes  undefined and has an asymptote. Tangent is sin/cos so when cosine is 0 it becomes undefined and makes an asymptote. The same goes for cotangent except that since cotangent is cos/sin, when sine is 0  cotangent becomes undefined and makes an asymptote as well. 


http://www.mathipedia.com/GraphingSecant,Cosecant,andCotangent_files/image014.jpg

http://www.mathipedia.com/GraphingSecant,Cosecant,andCotangent_files/image008.jpg


http://www.pindling.org/Math/Learning/Ti_Calculator/Inverse_Trigs/3ac34ecf.jpg

http://www.calculatorsoup.com/images/trig_plots/graph_tan_pi.gif

                             WORKS CITED PAGE
IMAGE
Secant Asymptote
Cosecant Amplitude

Cotangent
Tangent

BQ#2-Unit T Concept Intro: Trig Graphs and the Unit Circle

1.)How do the trig graphs relate to the Unit Circle?
well the trig graphs relate to the unit circle because we are still using the unit circle but it is now in a line this time and you still have the quadrants there but its just different it practically everything the same in the unit circle just that we don't use the degrees as an x-axis because when you put it in your graphing calculator you wont be able to see the whole thing of what you want only a little bit of it so that's why we use radians instead but everything else relates to the unit circle.
a)Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
The period for sine and cosine is 2pi because that is how long it takes to repeat it, their pattern  doesn't repeat after four marks. On the unit circle the quadrant angles are (0,90,180,270, 360)  but we like to do (0, pi/2, pi, 3pi/2, and 2pi) because as i told you its better that way when it comes to plugging it in the calculator you can actually see a snap shot of it that is good enough. Sine is positive in quadrant 1 and quadrant 2, and negative in quadrant 3 and 4 which makes the pattern (+ + - -). Cosine is positive in quadrant 1 and quadrant 4, and negative in quadrant 2 and 3 which makes the pattern (+ - - +). When you see it on the unit circle the pattern for these two trig functions repeats only after one full revolution and since one revolution around the unit circle is 360 degrees it has a period of 2pi. If it was one period on a graph it's like saying one revolution on the unit circle. 

http://www.mathwarehouse.com/trigonometry/systems-sine-cosine-functions/images/picture-system-sinex-cosx.gif

Tangent and Cotangent have a period of pi because their pattern of being positive or negative only repeats after two marks or half of a revolution. Tangent is positive in quadrant 1 and 3 and negative in quadrant 2 and 4. Cotangent is the same which means that both of the patterns are (+ - + -). When it comes to putting it on the graph you can notice after every two marks that tangent repeats. 

http://etc.usf.edu/clipart/36700/36731/tancotan_36731_lg.gif

B) Amplitude?-How does the fact that sine and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the Unit Circle?
sine and cosine have amplitudes of one because sine and cosine have restrictions that can't be greater than 1 or less than -1 so that is where the lowest point (-1) and the highest point (1) should be in the graph, relating to using the unit circle when sine=0 it should equal to one by using the ratio of sine to find out that it is one and when you do cos=0 that should equal to -1. Tangent and the rest of the trig functions do have an assymptote so when the functions are graphed in between the asymptotes the vertex indicates where the function begins to go up to infinity or down to negative infinity.

http://www.teacherschoice.com.au/Maths_Library/Functions/sine_function.gif

source: http://www.education.com/study-help/article/pre-calculus-help-other-trig-functions/


                                                      Works Cited Page
sine and cosine
Tangent and Cotangent
Image
IMAGE

Reflection#1- Unit Q: Verifying Trig Functions

 1.) What does it actually mean to verify a trig identity?
          To verify a trig function really means to like simplify an expression and make it equal to what they want you to verify it to. Verifying a trig function involves simplifying it to make it easier for you because then its either using reciprocal, ratio, and pythagorean identity which you should memorize and it can equal 1 or many other variations.
2.) What tips and tricks have you found helpful?

  Tips and tricks to give you ummm.. good luck buddy your on your own JK :D. Tips that i can give you is that you should really memorize the Identities because most of the strategies and rules come from memorization. Also, is that make smart choices when you are using pythagorean,ratio, and reciprocal and the other ones as well. Another is that don't think so hard on the problem because you probably will get it wrong. Some tricks are that if you get stuck with one of them you should convert everything to cosine and sine because that's the easiest to work with and usually something cancels out as well.

3.) Explain your thought process and steps you take in verifying a trig identity.  Do not use a specific example, but speak in general terms of what you would do no matter what they give you.

  From what i have learn from doing all my PQ problems is that if you do the first step right then you might have the answer and if you dont that if fine you will know that you cant use that and you can go on doing different other options that might work. When i get a fraction i know right away that we can use conjugate to make the expression easier because you will use one of the identities and that would either go to another identity or it will cancel. The one thing that is really important is that dont think very hard about the problem because if you do it is most likely that you will get the answer wrong. so just relax take your time and you will be fine you will figure it out.

Well hope this helped for you, future Math Analysis Students :)

Unit Q Concept 2: Find all trig functions when given on trig function and quadrant using identities

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I/D#3- Unit Q- Pythagorean Identities

INQUIRY ACTIVITY SUMMARY:
1) sin^2x+cos^2x=1 is a Pythagorean Identity which means that it can be a proven fact and formula that is always true. So to start deriving it we use the Pythagorean Theorem. Pythagorean Theorem is always a^2+b^2=c^2 but we have learned different when dealing with unit circle. We know that the leg are the x-axis and the one going up and down is the y-axis and we know that the hypotenuse is r when it comes to the unit circle. So know we just switch it really instead of having a^2+b^2=c^2 we can a new one that is x^2+y^2=r^2 and we want to get to sin^2x+cos^2x=1. To do that we divide everything by r^2 and we should get x^2/r^2+y^2/r^2=1 it then turns to (x/r)^2+(y/r)^2=1. Right here we should notice something about the variables and the ratios. We know that cosine is x/r and we know that sin is y/r so, then we plug it into our formula and (y/r)^2 is sin^2x and (x/r)^2 is cos^2x so know we put it together and we should get that identity which should be sin^2x+cos^2x=1

I choose the 60º from the "Magic 3" in the Unit Circle to show that this identity is true. The picture is below to see that it can be proven.

2) OTHER PYTHAGOREAN IDENTITIES

    The first pythagorean identity contains Secant and Tangent. The only way we could get that is by dividing by either sine or cosine. In this case we divided by cosine so that we can get what we want which is tangent and secant. So we divide by cos^2x and for sin^2x/cos^2x you should know well you should memorize that it is always going to be tan^2x. for the second part IT SHOULD NOT BE ZERO IT SHOULD BE ONE.The third part we use the reciprocal identity and it should be sec^2x. So when you put it all together you get tan^2x+1=sec^2x

The other pythagorean identity contains Cosecant and Cotangent. Since we divided the last one by cosine lets divide this one by sine now. we notice for part 1 that IT IS NOT GOING TO BE ZERO IT IS GOING TO BE ONE DONT FORGET. The second part you should know by memory that it should be cot^2x. The last part is the reciprocal so the reciprocal of sin is csc so it is csc^2x so your final answer should be 1+cot^2x=csc^2x

INQUIRY ACTIVITY REFLECTION:

“The connections that I see between Units N, O, P, and Q so far are…” The connections are that we are still using the trig functions cosine sine tangent .etc. and the references of the unit circles.

“If I had to describe trigonometry in THREE words, they would be…” Stress, overwhelming RATIOS

WPP#13 & 14: Unit P Concept 6-7: Applications with Law of Sines & Law of Cosines

"This WPP13-14 was made in collaboration with Peter Nguyen.  Please visit the other awesome posts on their blog by going here"

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BQ#1: Unit P Concept 1 and 4: Law of Sines and Area Formula

1.) Law of Sines- Why do we need it? How is it derived from what we already know?
       The Law of Sines is needed to solve triangles that are not right triangles. The normal trig functions are defined for a right triangle and are not directly useful in solving non right triangles but we can use the trig function to determine the law of sines. So first you have a non right triangles labeled as ABC. Then you make a imaginary perpendicular line from angle B and we call that h. Now we have formed two triangles. Now we use the trig function of sin which is hypotenuse over adjacent and then on the other triangle we have the same as well but we don't know the numbers so for the triangle on the left side it is SinA=h/c and for the right it is SinC=h/a. We then want h by itself so we multiply by c and on the other by a and we get cSinA=h and the other aSinC=h then we equal them together cSinA=aSinC and you divide by ac and you get SinA/a=SinC/c and that is exactly what the law of sines is.

To get a picture in your head of how to do it instead of me telling you there is a video here.

4.) Area Formula- How is the "area of an oblique" triangle derived? How does it relate to the area formula that you are familiar with? 
       The area formula is derived from the area equation and the trig function. The area equation you should know form Geometry is A=1/2bh where b is the base and h is the height. We then draw a perpendicular line on triangle ABC in half from angle B to create two triangles with a sharing side of h. We then know that SinC=h/a and when you get h by itself you get h=aSinC. Then you substitute it in the area equation for h and we get A=1/2b(aSinC). The area of an oblique triangle is one half of the product of two sides and the Sine of their included angle.

To get a visual in your head of how it should look like and example problems as well watch this video here.
      It relates to the area formula by using 1/2bh as a start to find the area formula for an oblique triangle. We still use 1/2 and it still consists of two sides and the base for b but we have to find our own h by using the trig function of sine.

Works Cited
http://www.youtube.com/watch?v=zJsw_ltZxFo
http://www.youtube.com/watch?v=gAX_IleqeJQ

WPP#12: Unit O Concept 10: Solving angle of Elevation and Depression word problems

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

INQUIRY ACTIVITY SUMMARY:
1) 30º 60º 90º TRIANGLE

  The 30-60-90 triangle comes from an equilateral triangle, which has equilateral angles of 60 degrees. The equilateral triangle is cut in half, making: 2 triangles, (2) 30º degree angles, and (2) 90º degree angles.  Then we cut it in half and (b) will equal to 1/2 because its half of 1, while the other untouched side (c) the hypotenuse would remain as 1. However, one side remains missing with its value (a), and in order to find it we use the  Pythagorean Theorem (a^2 + b^2 = c^2).

As you can see from the pic above to the left we use it and it should 1/2 squared should be 1/4 and now we subtract it from 1 and it should be radical 3/4 but 4 is a perfect square so we just have a equaling radical 3/2 so now we know that (a) is radical 3/2 (b) is 1/2 and (c) is 1. So now were not done because we dont want a fraction so all we have to do is multiply by 2 on each side and by looking at the pic above on the right side you see that when we multiply by 2 on (a) it should now be radical 3 and (b) should be 1 lastly (c) should be 2. Then we use use "n" to represent any value for the triangle creating n radical 3 on (a) 1n or just n on(b) and 2n on (c). 

2) 45º 45º 90º TRIANGLE

The 45-45-90 triangle comes from cutting a square directly in half diagonally. Cutting the square diagonally in half will create: 2 triangles, (4) 45º degree angles, and (2) 90º degree angles. We just have to focus on one since we know that its a 45º 45º 90º triangle we know that (a) and (b) have to be the same but by looking at the pic above we know that they are both 1. To look for (c) we need to use the Pythagorean Theorem (a^2 + b^2 = c^2).

To find (c) we do 1 squared plus 1 squared equals c squared. (c) equals radical 2. Then we add "n" to represent any value for the triangle to create n radical 2 that is (c) and n or 1n for (a) and n or in as well for (b) because remember its a 45º 45º 90º triangle and you know that (a) and (b) have to be the same.

INQUIRY ACTIVITY REFLECTION:

  “Something I never noticed before about special right triangles is…” how they got those numbers as in the sides like radical 2 and what the n had to do with all of these in the first place. How similar it is for deriving a special right triangle is.

“Being able to derive these patterns myself aids in my learning because…” it gives me a better understanding on hoe they got the numbers and what n really meant instead instead of just knowing that it just had to be there and why someone chose these numbers for a reason and if i forget i can always remember where these patterns were derived from.

I/D#1: Unit N Concept 7: How do SRT and UC relate?

INQUIRY ACTIVITY SUMMARY
1.) 30º TRIANGLE

Well to start we have to label all the sides. The hypotenuse has to equal to 1, and due to the rules of the 30º,60º,90º right triangle the side that is adjacent to 30º (x) has a value of x radical 3, the side opposite (y) has a value of x and the hypotenuse (r) as 2x. Then we divided by 2x in the hypotenuse,  since we did that we must do the same and then simplify the other sides by 2x . So x-radical 3/2x turns to radical 3/2 which is your x value. Then we do the same for our y value and x/2x turns to 1/2 and that is your y value. Then we draw the coordinate plane on the triangle.The origin is (0,0), while  going across the x axis is (radical 3/2,0) and then going directly up from the previous coordinate is (radical 3/2,1/2).

2.)45º TRIANGLE

For the 45º triangle you have to label it accordingly by the special rule for the 45º.  (x) is the side adjacent to the given degree with value of x, the (y) is opposite to the angle but similar to the adjacent side too so it has the same value of x and the hypotenuse, (r) is equaled to 1 still and is x radical 2. Then you are suppose to divide the other two sides by x radical 2 so x/x radical 2 is 1/radical 2 but we could NEVER leave a radical in the denominator so we multiply it by radical 2 and it should be radical 2/ 2. Since the x and y have the same value y would have to be also radical 2/2. Then draw the coordinate plane of the triangle. You start at the origin (0,0) then go to the right and that is (radical 2/2,0) then go up from the previous coordinate and it would be (radical 2/2,radical2/2).

3.) 60º TRIANGLE

The 60º triangle is exactly the same as the 30º triangle. (x) is adjacent but has a value of x, (y) would be the opposite angle with a value of  x radical 3 and the hypotenuse (r) still equal to 1 and is 2x. Then divide the other two sides by 2x like the 30º triangle and x/2x would be 1/2 the value of x, for the value of y it is x radical 3/2x and that would be radical 3/2. When drawing it in the coordinate plane you start at the origin (0,0) then move alongside to the x and it is (1/2,0) then go up with the previous coordinate plane and that is (1/2, radical 3/2).

4.) HOW DOES THIS ACTIVITY HELP YOU TO DERIVE THE UNIT CIRCLE?

This activity helps me derive the unit circle by figuring out the certain points and if you know these three you can know the rest of the other points but you need to know there signs if its going to be negative or positive.

5.)

Depending on which quadrant the SRT is found, the value will change by its sign positive or negative. If the triangle is found on Quadrant II, then x from the coordinate pair will be negative. In Quadrant III, both x and y will be negative. Finally in Quadrant IV, only y will be negative. In Quadrant I, all trig functions are positive, while in Quadrant II only sine and cosecant are positive and the rest negative. Meanwhile in Quadrant III, tangent and cotangent are the only positive. Finally in the Quadrant IV, only cosine and secant are positive. you can also memorize it by All Students Take Calculus.

6.) INQUIRY ACTIVITY REFLECTION

"THE COOLEST THING I LEARNED FROM THIS ACTIVITY WAS..." well i took cp Algebra ll so when i saw this i didnt know what this was at all as in the Unit Circle but the activity was Geometry so i knew that and learned that this activity was actually the first three in the unit circle in quadrant l.

" THIS ACTIVITY WILL HELP ME IN THIS UNIT BECAUSE..." I noticed that if we know the first quadrant then we can fill out the whole unit circle that is amazing and to help you more you can sing the song to get the ordered pairs and other methods as well.

"SOMETHING I NEVER REALIZED BEFORE ABOUT SPECIAL RIGHT TRIANGLES AND THE UNIT CIRCLE IS..." That i never knew that they were so related to one another. i didnt know that the coordinates can be different shapes as well.

RWA#1: Unit M Concpet 4-6- Conic Sections in Real LIfe

1. Parabola- Set of all points equal distance from a focus and the directrix
2.Algebraically

Graphically










                                                                                                 (image)
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The parabola has FOUR 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 that is perpendicular to the axis of symmetry. The axis of symmetry is a line that breaks the graph into two equal sections. The focus is the point inside the parabola. Lastly the distance from the focus to the vertex and the vertex to the directrix is the same and known as "p". 
For graphing the Parabola you need to make sure the formula is in standard form. After you have done that you can find your parts you know what the vertex is it is (h,k). Then plot the focus which you should know is p by now and you plot it away from the vertex meaning it should be inside the parabola. To know if p is going up, down, left or right you need to know that when "p" is positive and the equation is x^2 it goes up, if "p" is positive and equation is y^2 it goes right, if "p" is negative and equation is x^2 it opens down, if "p" is negative and the equation is y^2 it opens left. the effect the focus does to the shape is the distance from the vertex to the Focus which determines how skinny or fat the parabola is going to be. The closer the focus is to the vertex the fatter the graph will be, the further away the focus is from the vertex the skinnier the graph will be.The eccentricity of the parabola is equal to 1 all the time.You then put in the directrix which is a line p units away from the vertex in the opposite direction of the focus. Then put in the axis of symmetry which is perpendicular to the directrix. Notice that the vertex and the focus are both going to be ordered pairs while the directrix and the axis of symmetry are going to be lines.
3. RWA

http://upload.wikimedia.org/wikipedia/commons/4/48/Headlight_lens_optics_schematic.png The cars headlights can be a great example of having a parabolic shape."The light is placed in the focus of a parabolic mirror, as the light travels and meets the mirror, it is reflected in lines parallel to the axis . This is why the light beam from the headlights of cars are so strong."( http://www3.ul.ie/~rynnet/swconics/UP.htm) source should be the vertex that bounces all of the dispersed lights from the reflector in a straight line straight forward.

  "Parabolic reflector has the property that if a light source is placed at the focus of the reflector then the light rays will reflect from the reflector as rays that is parallel to the axis. To have a safe night drive you don't want all the light rays to be parallel to the axis. Some of the light must be aimed far down the road to the side upward for signs or bridges."(http://www.pleacher.com/mp/mlessons/calculus/appparab.html)

Here it talks more about the headlight and how you have to find the right vertex so that when it bounces all the rays stay straight.

4. Works Cited Page
 http://upload.wikimedia.org/wikipedia/commons/4/48/Headlight_lens_optics_schematic.png
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Car Headlights

Headlights