WPP 10

For concept 9, it is basic probability so Ted counts as 1 person and what are the chances of him getting pick out of 4 other people is 1/5. For concept 10, we are trying to figure out the chances of having the 34th law land on the 15th page, so to land on 1 of the 100 laws and 1 of the 50 pages, we would multiply them together and you get your probability of 0.02%. For concept 11, we want to figure what the chances of the first two pick will be without replacement, so there are a total of 2/10 ladies taken and 8/9 single ladies left. Multiply them together and you will get 8/45. For concept 12, Barney is a babe magnet and doesn't what care of women he takes home, so he plans to take home all the married ladies (1/5) or half of the single ladies (4/10) back to his place. The chances of Barney being able to do that is if we add up the two fractions together and figure out that Barney has a 6/10 or 3/5 chance of having his wishes granted. For concept 13, Barney needs to know the probability of one of the ladies choosing an ace or a diamond. We find out there are 4 aces in a 52 pack of cards and 13 diamonds in that same pack. But we know there is an Ace of Diamond already so we subtract that fraction to the sum of the other two and receive 16/52 or 4/13 chance. For concept 14, we would have to use nCr again but instead with fractions. We have to make sure that the values of the n in the numerator add up to the value of the n in the denominator and the same goes for the r value. The first nCr in the numerator is for how many of something you know and the # of something you want. The second nCr in the numerator is how many of something is left or you don't know and how many of something you might get wrong. The nCr in the denominator is just the total of the nCr's in the numerator. So we would solve it all out and get two percentages for each mini problem.

WPP 9

For concept 4, to find the number of possible outfits, we would have to multiply all the numbers together and by doing this we will get our answer, so 10 times 5 times 20 is equal to 1000 totally different outfits. Concept 5 shows you how to use combination and the equation nCr. Usually the biggest number(1000) is equal to the n and the smallest number of groupings(4) is equal to r. If you plug it into the calculator, you will get 3,921,225 different ways to arrange each tie in his closet. For concept 6, it is similar to concept 5 but since two things are happening at the same time, we would multiply both together. But you have to make sure that each number goes with its corresponding pair such as women would be group up with a different nCr than the men. So we would take 20 C 12 times 52 C 2 and get 167, 036,220 different groups that Barney can form and hire. Concept 7 uses a new formula called nPr and this formula tells us that the order in which things happen matters. If Barney called one number from his list of 12, he would have 11 numbers left to call the second time, and 10 for the third. So we would write  it out as 12 times 11 times 10 or 12 P 3 and get that Barney can call 3/12 numbers in 1320 different orders. Concept 8 is the same as concept 7 but wants to get rid of repitions, so we count up the total number of letters in LEGENDARY and put it as a factorial for the numerator and we add factorials for each repeating letter in the word such as the letter E. It appeared twice in LEGENDARY so it would have a 2! in the denominator.

Student Problem 7: Repeating Decimals into Rational Fractions

1) What is this problem about?
This problem shows us how to convert repeating decimals into rational fractions. In this problem, our repeating decimal was 0.11111. So we can rewrite it out as 0.1+0.01+0.001 because they will add up together to equal the repeating decimal again. To find the value of r, we take the second term which was 0.01 and divide it by a sub 1(first term) which was 0.1. We find out that r is equal it 0.1 or 1/10. Now we take our infinite geometric series formula and plug in what we know. We already know that a sub 1 is 0.1 and the r-value is 1/10 or 0.1. So now the equation looks like S sub ∞=(1/10)/1-(1/10). We would solve it out and find out that our answer is 1/9.
2)What must the reader pay close attention to in order to not make a mistake?
Readers must remember to split up the repeating decimals into multiple terms. By doing that, we are able to form our a sub 1 and be able to find our r-value with those terms. Also remember that a # divided by a fraction is like a # multiplied to the fraction reciprocal. In this problem, we saw that the tens were diagonal from each other and were able to cancel them out to equal 1. 

Haiku(s)

Diva

Fierceness

Bad Girl

Remember my name

All of the boys stare at my body

They want to lick me from my head to my toes with cream

The boys will be wanting me and the men will be thinking that they went to heaven when they see me

Strutting my stuff, walking down the streets in my heels, the girls will be staring me down while the women will be shaking their heads, saying oh that girl, what will she do?

Student Problem 6: Partial Fraction Decomp. w/repeated Factors

1) What is this problem about?

This problem is an example of how to solve partial fraction decomposition problems but with repeated factors. We are given one equation in which we have to split it up into multiple fractions with (+) and (-) signs between them.

2) What must the reader pay close attention to in order to not make a mistake?

The reader must at first know how to factor out the bottom polynomial into factors that have an x-variable to the power of 1 such as x or (x-2). Readers also must know from how many factors there are, there will be the same number of letters as numerators for the factors. Readers also must know how to find the LeastCommonDenominator but with this problem, he/she must pay really close attention to what to multiply to each equation. Because some fractions already have a variable such as (x+2) and (x+2)^2, so all you need to do is multiply with (x+2) to get the second power, you don't need to multiply (x+2)^2 to (x+2) cause that will get you a fraction to the third power. Readers also need to know how to solve a system of equations through elimination, substitution, or matrix, but with this equation we were lucky enough to only use substitution.

Student Problem 5: Partial Fraction Decomp.

1) What is this problem about?
This problem is an example of how to solve partial fraction decomposition problems. We are given one equation in which we have to split it up into multiple fractions with (+) and (-) signs between them.
2) What must the reader pay close attention to in order to not make a mistake?
The reader must at first know how to factor out the bottom polynomial into factors that have an x-variable to the power of 1 such as x or (x-2). Readers also must know from how many factors there are, there will be the same number of letters as numerators for the factors. Readers also must know how to find the LeastCommonDenominator and know how to solve a system of equations through elimination or substitution.

Student Problem 4- Graphing Logarithmic Functions

1)What is this problem about?
This problem is an example of how to solve logarithmic equations but in this case, natural logs. We are given the equation and we have to determine what is the asymptote and also if the graph has a y-int and where the x-int is.
2) What must the reader pay close attention to in order not to make a mistake?
Readers must understand what the equation log sub (x-h) +k. The only part reader must know from this equation is the (x-h). Switch the operation sign that is in front of the h-value and that new value will be your x asymptote. If your x asymptote is 0 or higher, then you will know for sure that there will be no y-int. Readers also need to know how to exponentiate the (ln) with the letter (e)

Student Problem 3-Exponential Graphs

1)What is this problem about?
This problem is an example of how to solve exponential graphs. We are given the equation and we have to determine what is the asymptote and also if the graph goes left/right or stays above/below the asymptote.
2) What must the reader pay close attention to in order not to make a mistake?
Readers must understand what the equation a•b^(x-h)+k means. A determines if graph goes above/below asymptote and to determine, readers must remember positivism makes a person's mood go up so the graph will be above the asymptote and vice versa. B determines if graph will get close to the asymptote on the right or left side. If the absolute value of b is <1, then the graph will be closer on the right side and if absolute value of b is >1, then it will be closer on left side. (x-h) will shift the graph left or right, but we will be using a calculator to get the points that will shift the graph for us. Last is the k which represents the asymptote. Whatever k is, then y is the same(y=k)

October 15th Extra Credit

The answer is I math because the square root of -1 is an imaginary number which equals the letter i and the square root and the power of two hearts cancels out so that leaves the . Then the y/x represents slope which also equals the letter m. As for the radical c^2 + b^2 is a because you use the Pythagorean theorem which is the two shorter sides squared and added together equals the hypotenuse and in this cause it would be c and b according to the triangle below. So you would solve it and get the letter a. As for the ln e ^t natural log and e cross cancels each other out leaving just t. If we drew a perpendicular line to side (a) starting from the 90 degree angle that would create the height of the triangle. By multiplying that height with side(a) would give us the area of a square/rectangle. The other way to find the area of a square/rectangle is by multipling side b with side c. By using substitution property, the areas will create BC=AH and we just divide a to both sides and we are left with cb/a=h. Therefore making the last equation equal to h.

Student Video #3: Unit H Concept 7

1) What is this video about?

This video shows viewers how to solve for logs through a process known as approximation. We are given a key to determine what the value of the log is with a given value. This video shows an example and describes the process on how to find the factors of the log and with substitution, find the answer/value of the log.

2) What does the viewer need to pay special attention to in order to understand the concept?
Viewers need to pay special attention that there is a value that isn’t listed in the key. If the log base # is the same as the # after the log, then their values will cancel out and equal 1. For example: Log 10. Since the log is log base 10, the base will cancel out with the # after the log in log 10 which is 10 to make 1. Viewers also need to learn how to find factors by dividing the log within the given values in the key. By doing that, we can find the factors out much faster, but if no factors can be found, then we will have to find a multiple of the log that can provide the factors we need.

10) while the domain of a rational function depends on DIVAH, what do you think the range of a rational function depends on? Give an example.

Since the domain depends on the vertical asymptotes and holes, then the range depends on the horizontal/slant asymptote. Usually the range is equal to everything except for the value that the horizontal asymptote represents. For a slant asymptote, the range can't equal any value that is on the slope or function.-

Unit G Concept 1-7 X-intercepts

9) Describe how to find the x-intercepts of a rational function. Include both the long way and the shortcut way, explaining why the shortcut makes mathematical sense.

As you can see in the picture above, to find for the x-intercepts, plug the number 0 into the y-variable and solve for x. The long way to solving is to factor the numerator and multiply the denominator to both side. But a number multiplied to a 0 is still 0 so the other side is still zero. Then you use ZPP or quadratic formula to find the x-intercepts. The shorcut is knowing that a number multiplied to the other side(that has the number 0) will still result in a zero, so you can skip multiplying the denominator and just solve for the numerator.

Unit G Concept 1-7 Y-intercept

8)How do you find the y-intercept of a rational function?Does this need to be done in the original or simplified equation?

To find the y-intercept, just plug in the number 0 for every x variable and solve it out. Does it matter what equation you put it in? No not really...Haha I'm just kidding! It does, you have to make sure that all the x variables in the ORIGINAL equation is 0 before solving for y.

Unit G Concept 1-7 Limit Notation for vertical asymptotes

7)Describe how to write limit notation for vertical asymptotes and what the notation means.
The picture above is how you write the limit notations for the vertical asymptotes. The x-value will vary depending on how many factors you crossed out from the original equation. The (+) on the x-value just tells us that on the right side of the vertical asymptote the graph or f(x) will either go up and down. The (-) on the x-value just means left side of the vertical asymptote.

Unit G Concept 1-7 Holes and y-value

6)How do we find the appropriate place to plot a hole if the y-value is undefined when plugged into the original equation?

There is no appropriate place to plot a hole if you find out that the y-value is undefined. To solve that problem, you grab the x-value after you factored original equation, crossed out similar factors, and ZPP those factors. With those x-values, you plug it back into the simplified equation(one with all similar factors crossed out)and solve for the y-value. Now you have an (x,y) coordinate and just plot that hole onto the graph.

Unit G Concept 1-7 Conditions when graph can cross

5)Describe the conditions in which a graph can cross an asymptote.

People believe that the graph can not touch the graph but why does it? The reason is because they are being misguided by there own definition. The relation between a graph and asymptote is that the line will "approach" and get closer to the asymptote, but there will be times that the graph will cross and having x-ints can allow the graph to cross. A graph can cross slant/horizontal asymptotes but never vertical asymptotes.

Unit G Concept 1-7 Vertical Asymptote and Hole

4)What is difference between a graph having a vertical asymptote and one w/ a hole?

The difference is seen in the picture above. The vertical asymptote acts like a boundary line that prevents any lines to pass through and it is different from a hole because a hole is located on the line itself. However the hole disjoins the line at a certain point and continues on afterwards. So the difference is vertical prohibits line from touching it and hole just disjoins the line for a bit and is on the line.

Unit G Concept 1-7 Slant Asymptotes

3)When does a graph have a slant asymptote? How do you find equation?
You know a graph has a slant asymptote when the degree value in the numerator is ONE bigger than the degree value in the denominator. Afterwards, your next step would to do long division with the numerator as the dividend and the denominator as the divisor. In the end, your slant equation will be the whole quotient- the remainder in the end if there is one.

Unit G Concept 1-7 Limit Notation for Horizontal Asymptotes

2)Describe what limit notation for horizontal asymptote actually means.

Limit notations shows you that if x→+∞ or -∞, then the f(x) will equal some number. The reason for limiting notation is to show that the line on the graph will be very near to the asymptote but will never touch it unless there is an x-int then the "never crossing the asymptote rule" will be broken. As you keep the x-values going up and up and up or down, down, down, both ∞ values for x will most of the time have the same f(x) value. With same f(x) value, it will make another horizontal line parallel to the asymptote, but that line will never cross it. The reason why is if you plug ∞ and -∞ for all x-values, your f(x) will be a decimal answer with numerous 0's in it as you see in this example: 4/inf(and let's make ∞ equal to 1 billion). So you see that a small number divided by a very large number such as ∞, will result in an answer very close to zero. This explains why the line is so close to the asymptote and this is what limit notation is trying to show.

Unit G Concept 1-7 Learning About Asymptotes and Holes

1)How do we know if a graph has a horizontal asymptote?What are the three options?

To see what kind of asymptote we will have, we must first compare the numerator with the denominator. The first option shows us that if we have a bigger degree for the denominator, then the asymptote will be y=0. The second option is if there is a same degree in the numerator and denominator. The asymptote will be the ratio of the coefficients in front of the largest degree variable. So it will look like y=(#/#). The last option to find the horizontal asymptote is if the degree is bigger in the numerator, then there is no horizontal asymptote.