WPP #10: Unit L Concept 9-14

                                     

SV #4: Unit I Concept 2: Solving Logarithmic Equations

**to watch my video click here

For this concept when you're finding the h and the k, the h will be coming from the number that is inside the parenthesis in the equation and it will be opposite sign of what is inside. The k will be coming from the last number in the equation. The range in this case will always be from negative infinity to positive infinity. The domain will be from whatever you h is to infinity. For the x-intercepts you have to put y=0, for the y-intercept you have to put x=0.

SV #3: Unit H Concept 7: Finding Logs with Given Approximations

**to watch my video click Here

For this concept it is important that you realize that since you are putting the simplified numbers in a fraction you need to put the subtraction sign I between separating the tops from the bottom. Also try to remember that if you have log(base4)4 the your answer will be 1 because 4 to the power of 1 is 1 . Also if it is log(base4)1 the that will equal 0 because anything to the power of 0 is 1.. If you use one of those two don't forget to add it to the end of your answer.

BQ #4: Unit T Concept 3

Why is a “normal” tangent graph uphill, but a “normal” Cotangent graph downhill? Use unit circle ratios to explain.

In tangent's ratio in the unit circle it's y/x. When you're trying to find the asymptotes for it you have to look where x would be undefined which means that the x has to equal 0. You would have to find the ordered pairs in which 0 is in the x spot, that would be in the 90 & 270 degrees. When you're going to put the asymptotes it would be on pi/2 & 3pi/2. Therefore the asymptotes would be at the 2 & 3 "quadrant", that will make the graph start from the bottom and make its way to the top without touching the asymptotes, being an uphill graph.
                                 
For cotangent the signs on the graph would be the same but the only thing that will make the difference would be the ratio, you now have the ratio of x/y. That means that you have to find the degrees in he unit circle where y will be 0. That could be found in 180 & 360 degrees. The asymptotes will shift from what they were in tangent. The asymptotes would be from the 3rd and 4th quadrants which will start of with the positive and then go down to negative. The graph will now be a downhill graph.

BQ #3 Unit T Concepts 1-3

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

Tangent

                                

When you look at the unit circle you realize that tangent in positive in 1st and 3rd quadrant and negative in the 2nd and 4th quadrants, so when you unfold the unit circle in the graph you divide it by quadrants. To mark the quadrants you label them with the radians of the quadrants so it's pi/2, pi, 3pi/2, and 2pi, so since tangent is positive in the 1st & 3rd quadrant they would have asymptotes at pi/2 and 3pi/2.. In the graph the second "quadrant" is negative while the third quadrant is positive, the graph has to stay in between the two asymptotes so the tangent graph will start in the negative and go up to the positive without touching the asymptotes. 

Cotangent

                                 

For cotangent everything is mostly the same but the ration is the inverse of tangent,which makes a huge difference in the graph. The ratio is now x/y which we now have to find where x=0, x is 0 at pi and 2 pi. The asymptotes for cotangent will now lie at pi and two pie which means that they just shifted, when they shifted the signs now start from positive to negative which will make the graph downhill.

Secant 

Secant, just like cosine, is positive in the 1st and 4th quadrants and negative in the 2nd and 3rd quadrants. The asymptotes for this graph will lie on pi/2 and 3pi/2. Therefore that means the in quadrant 1&4 the parabolas will be above the positive cosine but with the inverse direction. Now in quadrants 2&3 will be below the positive cosine and will also have the inverse direction. 

Cosecant 

For cosecant, just like sine, it will be positive in the 1st and 2nd quadrant and negative in the 3rd and 4th quadrant. The asymptotes for cosecant would be on pi and 2pi since you have to find where y=0. That means that in the 1&2 quadrant it will have a parabola at the peak of the amplitudes with an inverse direction. Then in the 3&4 quadrant it will be negative  and heading at an inverse direction of  the amplitude. It would be very similar to the sine graph. 

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. 

When you look back at the unit circle for SINE, COSINE, and TANGENT we had ratios to which we derived the SOH-CAH-TOA where we could see the where the ratios came from. When you see sine and cosines ratio they are both over "r". For sine it's ratio is y/r and then for cosine it is x/r. In the unit circle, it's whole radius is equal to one, therefore that means that sine and cosine will not have asymptote because hey are just over 1 unlike all the other functions. Cosecant, secant, tangent, and cotangent are all over different numbers which makes them have an asymptote, the asymptote being the number the equals either the x or the y. In cosecant and secant their ratio has the r on the top of the fraction leaving the bottom with other variables. 

WPP #9- Unit L Concepts 4-8

SP #6- Unit K Concept 10: Writing a repeated decimal as a rational number using geometric series

For this concept you have to pay most attention to where the numbers come from and why he have the decimal where they do. Like the .18 comes from 4.181818 and it is put over 100 because it's in the hundredths place. The 100 goes down to 99 because you had to subtract 1 from it like its given to us in the equation.

SP #3- Unit I Concept 1: Graphing Exponential Functions

When graphing exponential functions you have to make sure that when you are trying to find the x intercept and you have to use the natural log you have to remember that you can't take the natural log of a negative number or fraction. To find the x intercept you have to make y=0 and to find the y intercept you have to make x=0. The domain of the graphs will always be from negative infinity to positive infinity.

BQ#2- Unit T Concept Intro

How do the trig graphs relate to the unit circle?

The trig graphs are just the unit circle unraveled. If you were to take the circle and stretch it into a line then you get your four quadrants but just in a straight line. In the unit circle each quadrant angle could be turned into radians, the first quadrant would be pi/2, the second quadrant would be pi, the third quadrant would be 3pi/2, and the fourth quadrant would be 2pi. For sine the pattern would go POSITIVE for the first two quadrants and NEGATIVE for the last two quadrants. In cosine the pattern is POSITIVE, NEGATIVE, NEGATIVE, POSITIVE. Finally for tangent it will start of with POSITIVE,NEGATIVE, POSITIVE, and NEGATIVE. as you see the "All Students Take Calculus" also applies to the graphs. 

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 unlike tangent and cotangent the pattern for sine and cosine does not repeat until the second time around the "unit circle". The patterns aren't repeated which means that for a full period it has to reach 2pi. For tangent and cotangent it's period is only pi because it's pattern is repeated (positive,negative,positive,negative) which means that there is no need to reach 2pi because it would be exactly the same. 

Amplitude?- how does the fact that sine and cosine have amplitudes of one (and ogre trig functions don't have amplitudes) relates to what we know about the unit circle?

Sine and cosine both have the ratio of having "r" as the denominator. Cosine's ratio being x/r and sine being y/r, in the unit circle "r" always equals one since it is the radius of the unit circle.  With that you realize that the boundaries for sine and cosine is between 1 and -1. Now for tangent it's different because the ratio for tangent is y/x and the x can be multiple values not only 1 and -1 which means it has no restrictions. 

Reflection #1: Unit Q: Verifying Trig Functions

1. What does it mean to actually verify trig functions?

In order to verify a trig function you have to make sure that the final product is the same on both sides. They start of by giving you a complex problem on one side of the equal sign and on the other side is the simplified version of it. You have to continue by trying to find identities or ways to simplify the equation on the left to equal the one on the right. If you want to use an identity you have to make sure that it is actually and identity. At the end both side need to equal the same and if they don't then you will have to try to solve for it again.

2. What tips and tricks have you found helpful?

For concept 5 I found out that the tricks is that you first have to see if you can find the least common denominator or try to separate the equations if there is one identity at the bottom and if all odds fail then you will have to turn whatever you have into sine and cosine. There are times in which you can't tell but you are able to cancel things out or you are able to substitute it for an identity. You can try to simplify. When you have to do the conjugate you have to make sure that you distribute and factor it out correctly.

3. Explain your thought process and steps you take in verifying a trig function?

For me verifying trig functions is extremely hard for me but I just try to find any identities but then if I can't then I have to see if I can either separate the functions or try to use the conjugate. Once I find anything then I try to use my identities and canceling things out or trying to simplify it.