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simplifying and identifying rational expressions

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simplifying and identifying rational expressions

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understanding and finding the angle of rotation

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understanding and finding the angle of rotation

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Simplify the following and state any restrictions.

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Find the angle of rotation, principal angle, reference angle, co-terminal angle (+ and -) and reference for the angle above.

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Sine Equation:
y=3sin(x+45)-0.5

Cosine Equation:
y=3cos(x-45)-0.5

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(x+7) <- Most simplified version

Restriction(s):
x cannot equal -5

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Angle of Rotataion: 367°
Principal Angle: 7°
Reference Angle: 7°
Co-terminal (+): 727°
Co-terminal (-): -353°
Related Angles:
Q1: 7°
Q2: 173°
Q3: 187°
Q4: 353°

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y = ((x-3)/2)^1/2 - 1

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1) I first identified the maximum and minimum points on the graph

Finding those two points are crucial because they can help you find all the other variables.

2) I found c by using the equation (max + min)/ 2. By doing that with my y coordinates from my maximum and minimum I got -0.5. The reason is that 2.5+(-3.5)= -1 and that divided by 2 is -0.5.

3) I used the amplitude equation to find a. The equation is (max-min)/2, and if I put my y maximum and minimum coordinates in it it'll be (2.5 + (-3.5))/2. Which gives us 6/2, then 3.

4) Then I found k by first finding the period by using the x coordinates of both maximum and minimum. I then found the distance between both the max and the min then multiplied by two. The reason is that their distance is half of the period, so by finding that and multiplying it by 2 it'll help us find the whole period. Then to find k you need to put it into the equation k = 360/P.

For this question the period I got was 360 because the distance between the max and min was 180, and 180 x 2 is 360. So then to find k I just put it into the equation, k = 360/P = 360/360 = 1. So that's how I knew k was 1.

5) First I decided to find the sin equation. I had everything I needed, I was just missing the variable d. To find that I just looked at the graph and counted where the mid-point was on the graph and how far it was from x = 0. For this graph the mid-point was 45 units to the left of x=0. So, d was -45.

The reason why I did that was because in the sin function the starting point is at the mid-point. D is the variable where it tells the function how many units to shift veritically. So, d depends on where the mid-point moves.

6) I then placed all of the variables in their spots in the sin equation. So, that gave me y = 3 sin(x + 45) - 0.5.

7) I then solved for the cos equation. The cos equation is basically the same as the sin equation, however, it's d variable is different. For cos, in its parent function it starts at the max. So, d will be depending on where the max is. For this equation the max shifted 45 units to the right, meaning d = 45.

8) Then I substituted the variables into the cos equation. Leading to this final equation: y = 3 cos (x - 45) - 0.5.

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1) I factored the numerator from x^2+12x+35 to (x+7)(x+5).

The reason I factored was because the question asked me to simplify and to do that I needed to factor as many things as I can, so if it's possible, I can cancel stuff out. The reason why I didn't factor the denominator is because it couldn't be factored.

2) I cancelled (x+5).

The numerator and denominator both had (x+5), so I could cancel it out.

3) Stated the restrictions of x cannot equal to -5.

The reason why the restriction was -5 is because during the the simplifying process (x+5) was in the denominator. So, if x equalled to -5 then the equation couldn't be solved. Since (x+5) -> ((-5)+5) that'll equal 0, and if 0 was ever in the denominator then the equation isn't solvable.

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1) The first thing I did was to jot down the important information that the question was giving me. For example some key information, the population was currently at 3000, doubles every decade, etc.

2) I copied down the table the question supplied for me and added the y points accordingly. To find the ones in the negative, first I divided the current population by 2 and just kept going. The reason why I divided it was because division is the opposite of multiplication, so to reverse something that was doubled was to divide. For the positive one I multiplied by two because the question said the population multiplied by 2.

3) I found the equation by first identifying what was the starting number, which was 3. Then I identified what was changing the 3, which was 2 because the numbers were doubling. Then I had to figure out where to put the x. I then noticed that it was increasing exponentially, so I put the x where the exponents were. That's how I got the equation, 3(2)^x.

4) I then used the chart and plotted the points down on a graph and that's how I achieved the exponential function that reflects this situation.

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Angle of rotation:
I got that answer because the angle of rotation is just the given angle. To graph the angle on the circle you'll have to find its coterminal if its smaller then 0 or larger then 360. For example, for the question I got it is larger than 360. So I found its coterminal by subtracting 360, and I got 7. Therefore, I graphed the 7 and I know that's where my angle of rotation is on the circle.

Principal angle:
The principal angle is the angle that is postive and between 0-360 that is equivelent to to the given angle. So, for 367 the principal angle for it is 7. The reason is that 367 -360 = 7, and the reason I subtracted the given angle by 360 is because 360 is a whole cycle. So, by subtracting a whole cycle we will end at the same place but our angle will be between 0-360, which is waht we want.

Reference Angle:
The reference angle is how far our angle point is away from the x axis. So, with each quadrant there is a different equation to help us find the reference angle. However, we need to put the principal angle in the equation and to figure out which equation to use you need to see which quadrant the principal angle is in and use accordingly. So, for quadrant 1 the equation is just the principal angle, quadrant 2, 180-principal angle, quadrant 3, 180+principal angle, and quadrant 4, 360-principal angle.

For this question, the reference angle I got was 7. The reason is that the principal angle is in quadrant 1, so I know the reference angle is just the principal angle.

Co-terminal:
Co-terminals are angles that has the same positions on the circle, and since a circle is 360, the distance between co-terminal angles is 360. So, that's why I either subtracted or added 360 to my given angle.

This question asks me to find both the positive and negative co-terminal, so for the negative I subtraced the given angle until it turned negative and for the positive one I just added 360.

Related Angles:
Related angles are the angles that have the same reference angle. So you usually find the reference angle and use the the equations to find the final answer in each quadrant. That'll give you all of the related angles.

So, that is what I did for my given angle. I knew my reference angle is 7, so I put them in all of the equations.
Q1: 7
Q2: 180 - 7 = 173
Q3: 180 + 7 = 187
Q4: 360 - 7 = 353

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1) The first step I did to find the inverse was to swap x and y.

The reason why I did this was because finding the inverse
of one equation is just finding the reflected version of the given function. So if we swap x and y and solve for y we would be able to get the opposite. Therefore, the inverse.

2) I started to solve for y

3) I subtracted x by 3

To cancel out +3 you need to -3.

4) I divided everything by 2

I did that because (y+1)^2 was multiplied by 2, so to get rid of the 2 I had to divide. So with that action on the x side it would of been x-3/2

5) Now without the 2 I had to square root the whole equation

To isolate the y, I had to get rid of the squared (^2) that was around y and 1. To do that I had to square root everything.

6) The last step is to subtract 1 on both sides

The reason I did that is because to get why by itself I had to cancel the +1, and the opposite of +1 is -1, so I had to subtract 1 on both sides.

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The reason why I chose to cover this topic on my portfolio is because this was one of the topics I wasn't very confident on before but with more practice I became better and more comfortable with it. So, I chose this one to show how much more I understand and the progress I have made.

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The reason why I chose this topic from unit 3 to be on my portfolio is because I believed I didn't demostrate my learning as well as I could on both my final exam and on the unit task. So, I believe now on my portfolio I can better show my understanding and knowledge.

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I chose to include this topic in my portfolio because during my exponential skills test I did something wrong during the graphing question. I also believe that I didn't do very well on creating an equation from a situation on my final exam, so I included a bit of that in there. Therefore, I can prove that I know and understand this topic.

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Here is the link to my planning document c: 

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Here is the link to my planning document c: 

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Create a new document and edit with others at the same time -- from your computer, phone or tablet. Get stuff done with or without an internet connection. Use Docs to edit Word files. Free from Google.

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I chose to do the angles because during my unit 2 test I did some stuff incorrectly in the angles question. So, I'm using this opportunity to show that I do know and understand how to find angles and get other information from my given angles.

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I chose to do inverse on the portfolio because I got an inverse question wrong on the unit test. I also struggled with inverse when I was reviewing for the final exam. So, I used this opportunity to solidify my skills and to prove that I can do it.