WPP#10 Unit L CONCEPTS 9-14

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Unit L Concept 4-8: WPP #9

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WPP #7: Unit K Concept 11

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Student Problem #7 : Unit K Concept 10

• What is this problem about?

This picture covers a problem from Unit K Concept 10. This concept deals with writing repeating decimals as a rational number using geometric series. In this type of problem what you are basically trying to do is break down the decimal to discover the rational number or the sum of the decimal. After breaking down the decimal into repeating units you then are able to figure out the first term of the series and then the common ratio of the series which allows us to then plug it into the equation that has the sigma with the infinity sign. Then, you plug in a sub 1 and r into the summation notation formula in order to simplify and figure out the rational number. 

• What does the viewer need to pay special attention to in order to understand the concept?

In order to understand the concept the viewer needs to pay special attention to how the decimal is broken down. What we do is basically take the repeating decimal and set it as its own. Then we out an addition sign which then allows us to break down the decimal by putting zeroes in front of the repeating decimal. The point of this is to make sure that you get the correct value for a sub 1. Also, when you have wrote out the series it is important that you write the broken down decimal correctly so that when we divide one term to the other, we and up with the correct common ratio (r).

Fibonacci Haiku

Fibonacci Haiku

Family

Love

My Foundation

My Support System 

I will always love them 

They are my motivation to do better everyday. 

Fibonacci Haiku 

Basketball

Perseverance

Hard Work 

Passion to play 

It is apart of me 

Helps maintain the balance I need in life.  

Student Problem #6 Unit J Concept 6

Student Problem #6 Unit J Concept 6

• What is this problem about?

This problem covers an example from Unit J Concept 6. This concept is very similar to Concept 5 except now we are dealing with Partial Fraction Decomposition with repeated factors. In this type of problem when you try in factor the denominator out you will find that one of the factors is repeated. Meaning that now when you separate the factor into separate equations for the repeated factor you must include the factor as a denominator as many times as the exponent. For example if the denominator as shown is (x-1)^3 you must have one fraction be A/(x-1), the nest one be B/(x-1)^2, and the last one be C/(x-1)^3. This is the only difference, then you would just solve as you did in concept 5. 

• What does the viewer need to pay special attention to in order to understand the concept?

In this concept the viewer must pay special attention to the common denominator so that when multiplying the numerator by what a part of the fraction is missing then your system comes out to being correct. Also, you must pay attention to how to right the denominators when separating the fractions. You must also pay close attention making sure your answers are plugged in correctly in the end. 

Student Problem #5 Unit J Concept 5

Unit J Concept 5

• What is this problem about?

This problem goes over a problem from Unit J Concept 5. Unit J Concept 5 covers a lesson on Partial Fraction decomposition with distinct factors. This concept is much easier than it seems. There are 7 steps in which you must take to complete this problem and they are as follows: Factor out the denominator, separate each of the factors into their own fractions and put the letters A,B,C, etc. on top, get a least common denominator for the factors by multiplying it by what its missing, combine like terms, set the coefficients of the numerator equal to like term letters on the right, set up the system and solve, and finally put A, B,C, ect. back to where there variable is at in the spit up fractions in the beginning. 

• What does the viewer need to pay special attention to in order to understand the concept?

The viewer needs to pay special attention not to forget to put zeroes in the numerator if there is just an x, or if something is missing. This is important because when setting up your system of equations you may forget and get confused of what number goes on the other side of the equal sign. Also, you must pay attention to how you take out the x's once writing your system. You need to be sure to take the x's out so that you can combine terms an solve for A or B, it is not necessary to keep those x's in the equations at this point. 

Student Problem #4: Unit I Concept 2

Student Problem #4 Unit I Concept 2

• What is this problem about?

This problem goes over a real life example of a problem from Unit I Concept 2. Concept 2 deals with an example that is a logarithmic graph and by identifying it as that, we know that x=k and there are no restrictions on the range. Because it is a logarithmic graph we have a new equation in which we use: log base b (x-h)+k. Also, for this type of graph it is only important that we understand that the domain is dependent upon the asymptote. 

• What does the viewer need to pay special attention to in order to understand the concept?

The viewer needs to make sure to pay attention on how to find the asymptote, instead of it being y=k it is now x=h for the asymptote so you need to make sure to know what h equals by looking back at the formula given. Yo must also play close attention to how you solve for both the x and y intercept because since we are now dealing with logs and natural log then we need to be aware that we know how to get rid of them, so that we can simplify further. Another important fact that we must remember and pay attention to is how to write the domain and range. For this type of logarithmic graph we should remmeber that there are no restrictions on range and for the domain will include the asymptote and an infinity sign. 

WPP # 6 Unit I Concept 3-5

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Student Problem #3 Unit I Concept 1:Graphing exponential functions

STUDENT PROBLEM#3 NOT ONE 

• What is the problem about?

This photo covers a real life problem of Unit I Concept  1 which deals with the graphing of exponential graphs. In this concept we deal with the equation y=a(b)^x-h+k and this equation is the basis of solving to find the rest of the elements in the problem. Once again, this concept deals with previous concepts that have come back to actually play another role in solving this type of problem such as knowing how to find the x and y intercepts involving logs and ln, how to find the domain and range of the equation, and how to label the numbers correctly with the right letter by remembering the equation used for this section. In this problem we also deal with figuring out the horizontal asymptote since it is an exponential graph and it has specific rules for the domain and range. 

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

In order for the reader not to make a mistake when trying to graph an exponential graph is that you need to make sure that you memorize the equation y=a(b)^x-h +k. This is the basis of this problem because you need to know which numbers plug into the variables or else you may get the incorrect answer. You also need to make sure you know how to solve for the x intercept by setting y=0 and how to find the y intercept by setting x=0. Since this problem is an exponential graph you also must remember that these graphs have no restriction on domain but the range is restricted. The range depends on the asymptote and also on the number that a ends up being in your equation. The range is also depended on the variable a because if it is positive you have to make sure the range is (the #, inf.), and if (A) is negative it will be (-inf., #). 

Student Video #3: Unit H Concept 7

• What is this video about?

This video covers an example from Unit H Concept 7. Concept 7 is all about using what we have learned from previous concepts and applying them to another more complex problem, such as applying the knowledge we have of bringing one log together and then expanding it. We also use substitution in this concept when figuring out the expanded out log and plugging in the correct variable or number into the equation. In this lesson we also review how multiplication while solving logs, turns into addition. The main focus of this concept is to use the clues given to us to figure out the log given so that we can expand the log and substitute the values of the logs we expanded to form a new simplified equation. 

    

• What does the viewer need to pay special attention to in order to understand the concept?

The viewer needs to pay attention to each of the steps taken to ultimately figuring out the correct solution to the problem of finding logs with clues. Also, to be successful in solving this type of problem we need to remember to use two of the log properties to add onto our list even though they are not written on the paper. We need to be responsible enough to remember to the two properties that are log base b of b= 1 and log base b of 1=0. These two properties are two extra clues that can be used to helps us figure out what to substitute once expanding the logs. In order to also understand the concept it is critical that you remember that the logs with fractions may need to be multiplied by a number on top or bottom so that one of the clues given can show up. In your ultimate answer there should not be a log it should be the values given when substituted. 

Unit G Summary Question 10: DIVAH range of a rational function

10. While the domain of a rational function depends on DIVAH, what do you thin k the range of a rational function depends on? Give an example.
While the domain of a rational function depends on DIVAH, the range of a function now depends  on the horizontal asymptote. It depends on the opposite of what the domain does, the horizontal asymptote is what now determines the range of a rational function because it can go through the middle of the asymptote but never on the far left or right. This is the difference of domain and range because a vertical asymptote cannot pass through an asymptote while the horizontal can. For example, if there was a rational function and we were to graph it and it ended up going through the horizontal asymptote in the middle and contained a hole then that would give us a restriction for the range but sometimes it doesn't give us restrictions. 

Unit G Summary Question 9: X-intercepts of a rational function

9. Describe how to find the x-intercept of a rational function. Include both the long way and the shortcut way, explaining when the shortcut makes mathematical sense. 
Finding the x intercept for the rational function are very easy, you need to remember that for this equation you use the  factored out function. We need to make sure we use the factored out equation when trying to find the x intercept because the original function is no longer really necessary. The long way of figuring out the x intercept you have to set the factored out function to 0 and then multiply it on both sides by the denominator. you then are able to set your numerator equal to zero to find the point. The shortcut method is also quite simple because all you need to do is set the numerator, of the factored out function, to zero. Either method should give you the same answer because it is the same thing just a new shorter way of figuring it out. 

Unit G Summary Question 6: Y- Value when its undefined

6. How do we find the appropriate place to plot a hole if the y-value is undefined when plugged into the original equation?
To find the appropriate place to plot a hole when the y value is undefined you have to plug it into the factored equation. When trying to find this exact spot we have to use the factored equation because if we use the original one it can result in a totally different answer. Since we have already factored out our function beforehand, we no longer bring into account the original one. All we need is the factored out rational function to find the rest of the stuff we need to find such as domain, x intercepts, and y intercepts. Its imperative that we don't pay attention to the original function as much because it can effect our answer. 

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Unit G Summary Question 5: Crossing through Assymptotes

5. Describe the conditions in which a graph can cross through an assymptote. 
For each of the asymptotes there are conditions in which you know that they can or can't cross through a graph. In Horizontal asymptotes, graphs can sometimes cross the graph but only in the middle of the graph. The graphs can never cross at the far left or far right of the horizontal asymptote. Slant asymptotes also have the same conditions for the graph, they can cross but only towards the middle of the asymptote. As for vertical asymptotes, vertical asyptotes, the graph are never to cross the asymptote. To sum it up, Horizontal and slant asymptotes are able to have graphs that cross through there asymptote but vertical asymptotes should never cross ways!

Vertical Asymptotes Never Cross!!!!

Student Video #1: Unit F Concept 10

By Kassie and Cecilia

• What is this video about?

This video covers an example from Unit F Concept 10. Concept 10 in this unit deals with us putting all the concepts together of this unit to ultimately figuring out the zeroes and the factorization of the polynomials. Such concepts included in this once concept are using the pq's, figuring out how many positive and negative real zeroes there are, and using synthetic division to figure out the zeroes. Also we are to now put the zeroes into there factored form so that we find the complete factorization of the polynomial. This video covers the complete process of concept 10 but also includes other practicing techniques of the same unit in this single problem.

• What does the viewer need to pay special attention to in order to understand the concept?

The viewer need to pay attention to each step we do so that they could ultimately figure out the correct answer. For example, you should not just blow off the pq's because when using synthetic division to find zeroes it is important that you limit your options so that you don't waste so much time trying to guess which numbers fit. In order to all understand the concept you must make sure that you pay attention to the detail and steps taken to figure out the right answer.

• What is this video about?

This video covers an example from Unit H Concept 7. Concept 7 is all about using what we have learned from previous concepts and applying them to another more complex problem, such as applying the knowledge we have of bringing one log together and then expanding it. We also use substitution in this concept when figuring out the expanded out log and plugging in the correct variable or number into the equation. In this lesson we also review how multiplication while solving logs, turns into addition. The main focus of this concept is to use the clues given to us to figure out the log given so that we can expand the log and substitute the values of the logs we expanded to form a new simplified equation. 

    

• What does the viewer need to pay special attention to in order to understand the concept?

The viewer needs to pay attention to each of the steps taken to ultimately figuring out the correct solution to the problem of finding logs with clues. Also, to be successful in solving this type of problem we need to remember to use two of the log properties to add onto our list even though they are not written on the paper. We need to be responsible enough to remember to the two properties that are log base b of b= 1 and log base b of 1=0. These two properties are two extra clues that can be used to helps us figure out what to substitute once expanding the logs. In order to also understand the concept it is critical that you remember that the logs with fractions may need to be multiplied by a number on top or bottom so that one of the clues given can show up. In your ultimate answer there should not be a log it should be the values given when substituted. 

Unit G Summary Question 8: Finding the y-intercept

8. How do you fin the y intercept of a rational function? Does this need to be done in the original or simplified equation?
Finding the y-intercept in a rational function is really simple. All you need to do is plug in zero into anywhere in which the equation has an x and simplify both on top and on bottom to get your y intercept.Finding the y intercept needs to be done using the factored out equation because sometimes the original equation contains x's on both top and bottom which means that it will be zero. But that is not correct, you need to use the factored out equation because yo get a complete different answer than when using the original function. Remember that there can only be one y intercept!

Unit G Summary Question 7: Limit Notation for Vertical Asymptotes

7. Describe how to write limit notation for vertical asymptotes and what the notation means. 
Vertical asymptotes deal with factoring and simplifying the rational function on both the numerator and the denominator. Once you have factored out the function you then set the denominator equal to zero so that you are able to figure out which numbers you are going to use to find the limit notation. For example, say you have factored out the bottom and set the factor of x+1=0, this will leave you with x= -1. With this number you will write out the notation like this: as x -> -1(to the right), f(x)-> either positive or negative infinity and you would also use as x-> -1 (to the left), f(x)-> either pos. or neg. The positive or negative depends on how your graph is to the right of the boundary line of negative 1 and how it looks to the left side of it. The notation lets us know whether that graph is in a positive or negative direction. 

Unit G Summary Question 4: Vertical Asymptotes and holes

4. What s the difference between graphing a vertical asymptote and a graph having a hole?
When graphing rational functions, Vertical asymptotes and holes have a different function. A vertical function serves as a boundary line and is a guide for you to put your graphs in the specified spot. Holes are different, instead of being a boundary line, they are plotted as points on the graph to make your graphs a little more accurate. When dealing with a rational function that has a vertical asymptote you need to remember that you factor the function out as much as possible and then you are able to solve for your boundary lines. Another way of differentiating between the two is that the holes are a plotted point where the vertical asymptote is not. 

Unit G Summary Question 2: Limit Notation for Horizontal Asymptotes

2. Describe what limit notation for horizontal asymptotes actually means.
Limit notation is written so that we can identify the end behavior of a graph. The limit notation allows us to have a visual or guide to how the graphs would will look. In a horizontal asymptote the limit notation serves as a type of guide for the graphs. It tells you where the horizontal y axis is just as a fact of how the behaviors are when approaching positive and negative infinity. More specifically, it lets you know if the graph is heading towards the number given.

Unit G Summary Question 3: Slant Asymtote

3. When does a graph have a slant asymptote? How do you find the equation of a slant asymptote?
A slant asymptote is a line that is slanted and either goes uphill or downhill. A slant asymptote only exists if the degree of the top is ONE bigger than the degree on the bottom. To locate aslant asymptote you must perform long division. After performing long division everything but your remainder is now the equation of the slant asymptote  of the line. To find the equation you also use y=mx+b. Something to remember about slant asymptotes is that the graphs can sometimes cross through towards the middle of the graph. 

Unit G Summary Question 1: horizontal asymptotes

1. How do we know if a graph has a horizontal asymptote? What are the three options?
A horizontal asymptote would jut be a horizontal dotted line. To figure out the horizontal asymptotes you have to compare the degrees of the numerator and the denominator. The first option will be if the bigger degree is on the bottom the asymptote will be y=0. Second, if both the numerator and denominator have the same degree then the asymptote is the ratio of the coefficients. But, if there is a bigger degree on top, there will be no asymptote. One thing we got to remember about horizontal asymptotes is that graphs can sometimes cross through them but only towards the middle not to the far left or right.