BQ#7: Unit V: Derivatives

1. Explain in detail where the formula for the different quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc).

The formula of the different quotient comes from the plots of a graph. Since the formula helps us find the slope of a straight line, we use the slope formula; but the different quotient helps you find the slope of any curve or line at any single point. Both the different quotient and the slope formula is simply the change in "y" divided by the change in "x"; but the difference is that the slope formula's "y" is used as the y-axis, while in the different quotient the change of "y" is called f(x). The formula for the different quotient helps us find the derivative. The derivative is the slope of the tangent line to a graph f(x), and is usually denoted f '(x). A tangent line is just a line that touches the graph once. A function can have different derivatives at different values of "x". To solve the different quotient, you first have to find f(x+h), then simplify f(x+h)-f(x), you then divide the result from step 2 by "h" (or multiply the result by 1/h). 

Slope Formula

Difference Quotient

Tangent Line

Secant Line

References:
Mrs. Kirch's Unit V SSS Packet
http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg
http://www.coastal.edu/mathcenter/HelpPages/Difference%20Quotient/img002.GIF
http://clas.sa.ucsb.edu/staff/lee/Tangent%20and%20Derivative.gif
http://clas.sa.ucsb.edu/staff/lee/Secant%20and%20Tangent%20lines.gif

BQ#6: Unit U: Continuities and Limits

1. What is continuity? What is discontinuity?
A continuous function is predictable (goes where you think it should go).  It also has no holes, no jumps, and it can be drawn without lifting your pencil/no breaks in the graph. A discontinuity is made up of two families, which consist of removable discontinuities and non-removable discontinuities. A removable discontinuity has a point of discontinuity, which is also known as having a hole. A non-removable discontinuity has jumps, is unpredictable, you lift your pencil when you draw it, has vertical asymptotes, has different limits from values. A jump discontinuity has the right and left different. An oscillating behavior is wiggly. An infinite discontinuity, which is also known as unbounded behavior, occurs where there is vertical asymptote, which increases or decreases without bound of infinity or negative infinity.

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 exists at a point of discontinuity, which is also known as a removable discontinuity or hole. A limit exists as long as you reach the same height from both the left and right. In order for a limit to exist, both the right hand limit and left hand limit must be the same, so the limit does not exist at a jump discontinuity (left&right different), oscillating (doesn't have a certain point/wiggly), and infinite discontinuity (has a vertical asymptote). A limit is the intended height, but a value is the actual height. 

3. How do we evaluate limits numerically, graphically, and algebraically?
Numerically: You set up a table to help evaluate limits. You first draw a table and you add 1/10 on one side, and subtract 1/10 to the other side; with this, you slower get closer to the middle number.

Graphically: To help evaluate limits, you can draw a graph to help you pain a picture to where your limits are and it helps you explains when why you get undefined for your values. Or, you can use your fingers to help you. You start on one side and the other finger on the other side, and you try to make them come together to see if it's a removable or non-removable discontinuity.
Algebraically: To help evaluate limits, you do problem solving by using three methods: direct substitution method, factoring method, and conjugate method. When you use the direct substitution method, all you do is plug in the number that x approaches and you're done. But if you get 0/0 or indeterminate form, you keep on going to the other two methods. When you use the factoring method, you factor the denominator and numerator, and you then cancel what you can, but remember to not combine when you can cancel something from it, but remember, to always try direct substitution first! When using the conjugate method, you multiply to where the radical is and multiply by "one" and after you cancel, you get your answer.

References:
Mrs. Kirch's Unit U SSS Packet

BQ#4: Unit T Concept 3: Difference Between a Tangent and Cotangent Graph

4. Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill? Use unit circle ratios to explain.
Asymptotes are based on the Unit Circle, so since tangent is sin/cos and cotangent is cos/sin, their denominators differs and so do their asymptotes. For tangent, cosine is its denominator, so we have to see where cosine is zero (0,1) (0,-1) and that leads to undefined which is an asymptote. For cotangent, sine is its denominator, so we have to see where sine is zero (1,0) (-1,0) and that will lead for your answer to be undefines which is an asymptote. Both tangent and cotangent are positive in the first and third quadrant, and negative in the second and fourth quadrant. Because we know this, we understand that in a "normal" tangent graph it goes uphill because of its asymptotes, at pi/2 or 90 degrees, and 3pi/2 or 270 degrees. On the other hand, a "normal" cotangent graph goes downhill because of its asymptotes at zero and pi or 180 degrees.

BQ#3: Unit T Concepts 1-3: Relations to Sine and Cosine Graphs

3. How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.

Since Sine (y/r) and Cosine (x/r) have “r” as their denominator which equals one; they do not have to follow the same rules as Cot, Tan, Sec, and Csc because they will not be able to get zero as their denominator, so they will never get undefined. Cot, Tan, Sec, and Csc have “x” or “y” as their denominator; and because of this, they are able to get undefined as their answer because they are able to get zero as their denominator. Cotangent and Cosecant both have “y” as their denominator, so when their denominator equals zero, they will become undefined (1,0) (-1,0). Tangent and Secant both have “x” as their denominator, so when their denominator equals zero, they will become undefined (0,1) (0,-1). Because those trig functions can get undefined, they will have asymptotes, which are the boundaries (line) that can never be touched even they get really close to them. On the other hand, Sine and Cosine don’t play by these rules, because their denominators will always be one, so they can never be undefined.

Also, since the graph goes on forever (cyclical), we show a period so it can fit in the window so we can see how the graph develops in a rotation. We also don’t have to use radians as our x-axis, but we like to use it because it helps us see the whole graph, while if we use degrees, we can’t see the whole graph. When the graph starts to go up the x-axis, it shows that it is positive in that designated part of the Unit Circle, and if the graph goes below the negative x-axis, then it demonstrates that it is negative in the designated part of the Unit Circle.

BQ#5: Unit T Concepts 1-3: Graphing Sine, Cosine, Secant, Cosecant, Tangent, and Cotangent

5. Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use the unit circle ratios to explain.
Sine (y/r) and Cosine (x/r) do not have asymptotes, because their denominators will never equal to zero, since their denominator is "r", which equals to one. Since Cosecant (r/y), Secant (r/x), Tangent (y/x), and Cotangent (x/y) don't have "r" as their denominator, they are able to get zero as their denominator which will be undefined which turns into an asymptote. Since Secant and Tangent both have the asymptote in the same location; the points they share is (0,1) and (0,-1). Cosecant and Cotangent both have the asymptote in the same location; the points they share is (1,0) and (-1,0).

BQ#2: Unit T Concept Intro: Trigonometric Graphs

2. How do the trig graphs relate to the Unit Circle?
The trig graphs relate to the Unit Circle in that it helps define where and why the graph is negative or positive. For example, since Sine is positive in the first quadrant and the second quadrant, the graph will be up and positive when the degrees is between those two quadrants. But since Sine is negative in the third and fourth quadrant, the graph will do down and become negative. In Cosine's case, it is positive in the first and fourth quadrant, and it is negative in the second and third quadrant; so Cosine's graph will go up and be positive because it's in the first quadrant, then go down to negative because it is in the second and third quadrant, and then go up to be positive when it goes to the fourth quadrant. For Tangent/Cotangent, it is positive in the first and third quadrant, and negative in the second and fourth quadrant, so the graph will go up to be positive in the first quadrant, then go down to negative in the second quadrant, then go up again and be positive in the third quadrant, and then go down to be negative in the fourth quadrant.

A.) Period?- Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
A period is when the graph cycles one time through their cycle. The period for Sine and Cosine is 2pi, whereas the period for Tangent and Cotangent is pi, because that's how long it takes them to repeat the pattern.

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 graphs have an amplitude. Amplitudes are half the distance between the highest and lowest points on the graph. They can be found by looking at the equation at the values of the absolute value of 'a'." The fact that Sine and Cosine have amplitudes of one relate to what we know about the Unit Circle is that Sine and Cosine can't be greater than 1 or less than -1. We can explain this since Sine is "y/r" and Cosine is "x/r", and "r=1", so it creates a restriction since that is what an amplitude is.

References:
Unit T SSS Packet

Reflection #1: Unit Q: Verifying Trig Identities

1. The meaning of verifying a trig identity is to prove that the equation is true and we do this by showing that both sides are equal to each other.

2. Tips and tricks that I found helpful are to try to always change the identities to sin(x) and cos(x) when possible. Another tip I would use is to try to convert the equation into a Pythagorean identity when it applies. Another important tip is to NEVER TOUCH THE RIGHT SIDE OF THE EQUATION! Always start with the left because you are verifying if it does equal to the right side. We love having our equations to have squares, so if your answers can not go further, you then can square both sides to get a Pythagorean identity out of it. A very important tip to always do when squaring both sides is to check your answer in the end to make sure it works, because the answer can be extraneous. A trick I would give you is to use the "u-substitution" because it is helpful when factoring trigonometric expressions. And it makes it easier during the factoring process because after you factor, you just plug everything back in! So easy!

3. My thought process through the steps when verifying trig identities is to look at the equation in general. I first check to see if we want the left and right side of the equation to equal to each other or if we are looking for a specific answer (degrees/radians). If we are looking for an answer, than the first thing I would do is to try to change the equation into sin(x) and cos(x). I would then look for a Pythagorean identity in order to have the same identity throughout the problem. I would also check to see what quadrants it lands on because it helps us determine where our answer will lay in. I would also see if I can use the "u-substitution" or "m-substitution" to make my life easier. If we were looking to see if the equations on both sides equal to each other, I would start with the left side and leave the right side alone and then bring it back down when in the end. I will also see if I can separate the equation, get a common denominator, FOIL, combine, cancel, replace, convert, or change identities. I need to have my options open to how I want to approach the problem. I also use my right side equation as guidance to where I want to be. My right side equation can tell me if a should combine and cancel or change and convert to a different Pythagorean identity. What I recommend to do is to look at the equation first and analyze the starting point and then see where we want to end. In my opinion, that is the key. Analyze the equation before making a move.

SP #7: Unit Q Concept 2: Finding all trig functions values when given one trig function and quadrant (using identities)

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

I/D #3: Unit Q Concept 1: Using Fundamental Identities to Simplify or Verify Expressions

INQUIRY SUMMARY ACTIVITY

1. Where does sin^2x + cos^2x=1 come from to begin with?

     An "identity" is proven facts and formulas that are always true. The Pythagorean Theorem is an identity because it has been proven to be always true. Since the Pythagorean Theorem consists of x, y, and r, it is also the same as a, b, and c.

    Since a^2 + b^2=1, we divide r^2 on both sides. After MEMORIZING the Unit Circle, we knew cosine is (x/r)^2 and (y/r)^2 is sine. The ratio for cosine on the unit circle is x/r and the ratio for sine is y/r.

Sin^2x + cos^2x=1 is referred to as a Pythagorean Identity because we used the Pythagorean Theorem to find it.

  I will choose one of the “Magic 3” ordered pairs (30, 45, or 60 degrees) from the Unit Circle to show that this identity is true; in this case, I chose 60 degrees.

2). Show and explain how to derive the two remaining Pythagorean Identities from sin^2x + cos^2x=1.

The picture below shows and explains how to get tan^2x + 1 = sec^2x. The identity with Secant and Tagent is tan^2x + 1 = sec^2x.

The picture below shows and explains how to get 1 + cot^2x = csc^2x. The identity for Cosecant and Cotangent is 1 + cot^2x = csc^2x.

INQUIRY ACTIVITY REFLECTION

1). The connections that I see between Units N, O, P, and Q so far are that the Unit Circle and its ratios play a big part in all these units so “MEMORIZE THEM” and they also involve triangles to use the Pythagorean Theorem to get the answer or the missing side/angle.

2). If I had to describe trigonometry in THREE words, they would be burdensome, exhausting, and painful.

BQ#1: Unit P Concepts 1 and 4: Law of Sines and Area of an Oblique Triangle

   1. Law of Sines:
                                                                           
                                         

   Why do we need it?

       We need the law of sines because not every triangle is a right triangle, so we use the law of sines to solve for non-right triangles. We cannot use the Pythagorean Theorem or the normal trig function to solve for non-right triangles like we do for right triangles, but the law of sines luckily works for any triangle.

     How is it derived from what we already know?

·         The first thing we do is draw and label our non-right triangle.

·         We then drop down a perpendicular from “angle B,” which makes two triangles, and we call it “h”.

·         Since we now have two triangles, we can use the normal trig functions like sin, cos, tan, csc, sec, and cot; so “we can drop perpendiculars from the other two vertices and get the other relationships.”

·         After getting the two sin ratios, we see that they have “h” in common, so after we simplify both of the equations by getting rid of the denominators, we equal both to each other since both of them equal to “h”. We then get cSinA=aSinC.

·         We then want to get Sin and its angle by itself so we divide by the coefficients, which are “a” and “c” in this case, and you then get “sinA/a = sinC/c”.

·         It all depends where you make your perpendicular line “h”. So we can find angle B if only the perpendicular line is from a different angle like A or C and you do the same process.

 4. Area of an Oblique Triangle:

1     Area formulas:

·   “The area of a triangle is A=1/2bh, where ‘b’ is the base and ‘h’ is the perpendicular height of the triangle”.

·    “The area of an oblique (all sides’ different lengths) triangle is one-half of the product of two sides and the sine of their included angle (the angle in between them)”.

·    Formula: It depends on what angles they give you, but the options you have are A=1/2bcsinA, A= 1/2acsinB, or A= 1/2absinC.

How is the “area of an oblique” triangle derived?

·    The area of an oblique triangle is derived by cutting the triangle into two, which makes the triangle into a right triangle; this is done by using “h” as a line. Because we now have two right triangles, we can use the normal trig functions. Depending where “h” is dropped down from a different angle, we can see which one we are using from the options of sinC=h/a, sinA=h/c, sinB=h/(a or c). Angle B has two options because “h” is dropped down so it can apply to both triangles.

·    Since we want to get “h” by itself, we multiply by the denominator to both sides to get rid of it.

·    We then plug each of the equations into the formula for the area of a triangle, which is “A=1/2bh”.

·    We then substitute in our regular area equation for “h,” and we get A=1/2b(aSinC) or A=1/2b(cSinA). This also applies to angle B when you do it in the other triangle.

How does it relate to the area formula that you are familiar with?

It relates to the area formula that we are familiar with by plugging in the values into “h,” which make new formulas, and it also uses the normal trig formula for the area of a triangle. The area of an oblique is conditional; it needs to obtain two side lengths, which cannot be the same letter, and their angles must be included. Or else it will not work.

References:

Unit P SSS Packet

WPP #13 and 14: Unit P Concept 6 and 7: Law of Sines and Law of Cosines

This WPP13-14 was made in collaboration with Ashley V. and Tina N.  Please visit the other awesome posts on their blog by going here and here.

Create your own Playlist on LessonPaths!

WPP #12: Unit O Concept 10: Angle Of Elevation And Depression

Create your own Playlist on LessonPaths!

I/D #2: Unit O Concept 7-8: Deriving Patterns for Special Right Triangles

Inquiry Activity Summary

The purpose of the activity was to help us apprehend the special right triangle and how its patterns gets what it has. With the special right triangle, it helps us understand where the patterns come from, and instead of memorizing the patterns for both the 45-45-90 degree triangle and 30-60-90 degree triangle, you see how we get each of their side lengths by deriving them.

1. 30-60-90 Triangle

An equilateral triangle makes up of 60 degrees for each of its 3 sides; and making each of its sides equal to one since all its sides and degrees are equal. But when cut in half, its degrees change, and so does its sides. Its degrees become 30-60-90 degrees; the number across 90 degrees stays the same since it side was not cut off, the number across 30 degrees is now “1/2” because 1 was cut in half, so it value was also cut in half, so know the only thing we know need to find is the number that is across 60 degrees.

We use the Pythagorean Theorem to solve for the number side across 60 degrees, which we then get √3/2 as the answer. We then see how each side effects and correlates to the other sides. To get rid of the fractions from each sides, we multiply each side by 2, which gets rid of the fractions; we then get n, n√3, and 2n. After doing that, we multiply each side by “n”. Since “n” is a ratio, it helps the sides get bigger but its value will always stay constant. 

2. 45-45-90 Triangle

When we cut a square that has its four sides equal to 90 degrees and making each of its sides equal to 1 will change when cutting it across diagonally. It cuts the 90 degree angles in half, making it into a 45-45-90 degree triangles. Since two sides of the now triangle stood the same, it is still equal to one, but we now need to find the other side that was cut in half, so we will now use the Pythagorean Theorem to find the hypotenuse, which is the missing side. After finding the hypotenuse side, √2, we put everything together. We then see how each sides correlate and effect each other.

Since there are two 45 degree angles, both of their sides now equal to 1; and the 90 degree angle equals to √2. We then multiply “n” to all three sides and we end up with n, n, and n√2. Since “n” is a ratio, it always stay constant, so even if the numbers you plug in to “n” make the sides bigger, its value will stay constant.

Inquiry Activity Reflection

1. Something I never noticed before about special right triangles is how even though you cut them down, its value will always stay consistent. 

2. Being able to derive these patterns myself aids in my learning because it helps me understand where everything comes from and how one side effects the other sides rather than just itself.

I/D#1: Unit N Concept 7: The Unit Circle

     The Special Right Triangle (SRT) and the Unit Circle relate to each other in that the SRT helps give you the 30, 45, and 60 degrees points in the UC, which will then help you find the rest of the points in the other three quadrants. But you have to remember the order the points have to be in for each quadrants since they do not go in order.

INQUIRY ACTIVITY SUMMARY

1. 30 Degree Triangle

The first thing I did was plug in the degrees, so I can be able to set it up correctly. I labeled the hypotenuse, vertical value, and horizontal value and it's value for each side so it would be easier for me to know where to put my answers in either "r", "x", and "y". 

Since this is a special right triangle, the hypotenuse or "r", will always be one. In order to turn 2x to 1, you divide 2x to it to get one. Since you did it to "2x", you must also do it to "x radical 3" and "x"; so "x radical 3" turns into "radical 3 over 2" and "x" turns into "1/2". So r=1, x=radical 3 over 2, and y=1/2. After doing all this, you get your 30 degree point, which is (radical 3 over 2, 1/2).

2. 45 Degree Triangle

The first thing I did was plug in the degrees, so I can be able to set it up correctly. I labeled the hypotenuse, vertical value, and horizontal value and it's value for each side so it would be easier for me to know where to put my answers in either "r", "x", and "y". Since this is a SRT, I equaled "r" to 1, and so far, left the rest blank.

In order to get "x radical 2" to equal one, I will have to divide it by "x radical 2". Since I did that to one side, I also do it to the other sides, so both the x's will then equal to "radical 2 over 2". r=1, x=radical 2 over 2, and y=radical 2 over 2. In the end, you get the 45 degree point which is (radical 2 over 2, radical 2 over 2).

3. 60 Degree Triangle

The first thing I did was plug in the degrees, so I can be able to set it up correctly. I labeled the hypotenuse (2x), vertical value (x radical 3), and horizontal value (x); and it's value for each side so it would be easier for me to know where to put my answers in either "r", "x", and "y". Since this is a SRT, the hypotenuse (r) will equal to one.

To be able to get "2x" to equal to one, you divide it by "2x". Since you do it to one side, you do it to the other sides. "x" when divided by "2x" will equal to "1/2". "x radical 3" when divided by "2x" will equal to "radical 3 over 2". So r=1, x=1/2, and y=radical 3 over 2; and the 60 degree point will be (1/2, radical 3/2).

4. This activity helps me derive the Unit Circle in the way that it gives me the points around the Unit Circle and it helps me understand where the numbers come from. It also helps you visualize the SRT in the Unit Circle and how each degree gets its point. You also see the correlation between SRT and UC and how the SRT draws you the details for the UC, but the UC gives you the whole picture and has everything put together.

5. The triangle that was drawn in the activity lies in the first quadrants, since everything is positive, and it is the base of the Unit Circle. To memorize the whole Unit Circle, you just need to memorize the five steps in the first quadrants, which are the 0, 30, 45, 60, 90 degrees, radiants, and points to it. After having those down, you just need to remember some patterns, but you get the concept. The values change when I draw the triangles in Quadrants II, III, and IV, because depending on what the ratios for the trig functions are, you'll know which ones are positive and negatives, but it is still the same.

(http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/ebaa19ac-ff8b-43a6-a793-a00d9ac15e86.png)

This right here shows Quadrants II, III, and IV, which came from Quadrant I. As you can see, all three are the same in angles, but the only difference is their degree and the quadrants that they are in. All three have the same reference angles and all three share the same points but different connotations, depending if it's positive or negative.

(http://www.regentsprep.org/Regents/math/algtrig/ATT3/reftriex.gif)

This is a 45 degree triangle that is in Quadrant II, which has the same values as Quadrant I, but the only differences is its connotation, since the x-value is negative, while in Quadrant I it is all positive. 

(http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif)

The left picture shows a 60 degree triangle that is in Quadrant III, which also has the same values as Quadrant I, but in this case, because it is in the third quadrant, its x-value and y-value are both negative. On the right picture, it also shows a 60 degree triangle, but is in the fourth quadrant. Like Quadrant II, III, and IV, it has the same values as Quadrant I. In Quadrant IV, the y-value is negative and the x-value is positive.

HEADING FOR THIS SECTION: INQUIRY ACTIVITY REFLECTION

1. The coolest thing I learned from this activity was that everything connects with one another even if it is in different quadrants, its values are the same.

2. This activity will help me in this unit because I am able to memorize the Unit Circle because I am now able to visualize the SRT in the Unit Circle, so I am able to remember the degrees, radiants, and the points, so basically, now I know the whole Unit Circle.

3. Something I never realized before about special right triangles and the unit circle is that if you put them together, you can see the connection between them two and how it works and the details that make the Unit Circle up.

Citations:
(http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/ebaa19ac-ff8b-43a6-a793-a00d9ac15e86.png)
(http://www.regentsprep.org/Regents/math/algtrig/ATT3/reftriex.gif)
(http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif)

RWA#1: Unit M: Concept 5: Ellipses

http://www.education.com/study-help/article/pre-calculus-help-conics-ellipses/

1) Ellipse: "The set of all points such that the sum of the distance from two points is a constant." (Taken by Mrs. Kirch's SSS Packet)

2)
 

(http://imanawkwardturtle.files.wordpress.com/2013/03/hyperbolas3.png)

The formula for an Ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 =1 or (x-h)^2/b^2 + (y-k)^2/a^2 =1

The formula of an ellipse can be determined depending what number is bigger on the denominator. As you see in the image below, if the denominator is bigger under the (x-h)^2, then you will know that your graph will be "fat" or "wide" horizontally. If the denominator is bigger under the (y-k)^2, then you will know that your graph will be "skinny" or "tall" vertically.

(http://www.mathwarehouse.com/ellipse/images/translations/general_formula_major.gif)

As you can see from the picture above, the ellipse is skinny/tall so you will know that it is vertical.

Horizontal Major Axis

(http://image.tutorvista.com/content/feed/u1989/hoe%20ellipse.GIF)

As you can also see from the above picture, the ellipse is fat/short, so you will know that it is horizontal.

    The key features in an ellipse is the standard form, center, skinny or fat, "a=", "b=", "c=", 2 vertices, 2 co-vertices, 2 foci, major axis, minor axis, eccentricity, and the graph.
-To get the standard form of an ellipse you need to complete the square, after you do that, divide everything from the other side to the variables so it can equal to one. After getting your standard form, you look at the denominators of (x-h)^2 and (y-k)^2, and if a^2 is bigger under "x" your graph will be fat, and if the denominator is bigger under "y" then it will be skinny.
-To get your center, you look at (x-h)^2 and (y-k)^2, you equal them to zero, and you get your center. "a=" will always be bigger than "b=", so you look at (x-h)^2 and (y-k)^2, and you square root their denominators; the biggest number is "a=" and the smallest is "b=". To get your "c=", you can use the equation "a^2-b^2=c^2", you plug in the numbers and after you square root the end, you get "c=".
-Your major axis will be the numerator equal to zero, that has the smallest denominator. While the minor axis will be the numerator equal to zero, that has he biggest denominator.
-To get your 2 vertices, you add and subtract the "a=" to the center point of either "x" or "y", depending what the major axis is. To get your 2 co-vertices, you add and subtract the "b=" to the center point of either "x" or "y", depending on what your minor axis is. To get the 2 foci, you put your minor axis and add and subtract to what "c=" and you leave the major axis by itself.
-To get your eccentricity, you use the equation "e=c/a" and you plug in the numbers and you will get your eccentricity, but remember to have it in the thousands place.
-To get your graph, you plug everything that you found, but to remember to use a solid line to show that it is your major axis and to use a dotted line for the minor axis, your foci will always be inside the ellipse, your vertices and co-vertices will be its boundaries, and the shape of your ellipse will match up with either skinny or fat, and your center will be in the middle. The bigger the ellipse, the farther the foci will be, but the closer the ellipse is, the closer the foci is. The foci is the same distance to any point in the ellipse.The foci affects the eccentricity because it determines the shape of the ellipse and how close it is to the ellipse's eccentricity, which equals one. 

3) (https://www.youtube.com/watch?v=lvAYFUIEpFI)
Conics of ellipses can be seen in buildings or statues around the world, but it helps find the height and length of it. The foci that are plugged in can determine what kind of eccentricity you would want as the outcome. Ellipses can be seen in power plants, buildings, statues, basketballs, car logos, and many more things. Just like it says in the video, you can determine your ellipse by putting in what it gives you. You can use what they give you, and you can slowly get the other parts when you connect everything together and do it step by step.


(http://ww1.prweb.com/prfiles/2009/11/19/248313/EllipseofFire.jpg)

-A fun example I will use is the the picture of the eye above. (If you know about the difference about single and double eyelids, then you can comprehend this example easily). Referencing its shape, you can see it as an "single-eyelid eye" or a "double eyelid eye". If the eye has a single-eyelid, you can infer that it is small so its foci would be closer. On the other hand, if the eye has a double-eyelid look, then you can infer that the eye is bigger, so its foci will be farther apart.

4) Work Cited:

• Mrs. Kirch's Unit M SSS Packet
• http://ww1.prweb.com/prfiles/2009/11/19/248313/EllipseofFire.jpg
• https://www.youtube.com/watch?v=lvAYFUIEpFI
• http://www.education.com/study-help/article/pre-calculus-help-conics-ellipses/ 
• http://imanawkwardturtle.files.wordpress.com/2013/03/hyperbolas3.png
• http://www.mathwarehouse.com/ellipse/images/translations/general_formula_major.gif
• http://image.tutorvista.com/content/feed/u1989/hoe%20ellipse.GIF

WPP #10: Unit L Concepts 9-14: Probability Mixes

Create your own Playlist on LessonPaths!