WPP #9: Unit L Concept 4-8 - FCP, Combinations, and Permutations

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SP #6 - Unit K Concept 10: Writing Repeating Decimals as Rational Numbers

Be sure to add the whole number to your fraction at the end. Convert the whole number to an improper fraction with the same denominator as your other fraction and add! It is wise to keep most of your numbers as fractions. As you can see, I changed .144 to 144/1000. Remember when you are multiplying your denominator by its reciprocal, also multiply that reciprocal by the numerator! What you do to the bottom, you must do to the top! Finally, it is easy to make mistakes by simply writing the decimals that you put into the series. Just make sure you put the correct amount of zeroes everytime and you should be fine. 

Fibonacci Haiku: Rudolph is my Spirit Animal

Rudolph

Reindeer

Love me

That red nose

You're always my number one 

I think I make a good tenth reindeer

http://popculturehasaids.files.wordpress.com/2010/12/rudolph-red-nosed-reindeer-001.jpg

SP #5: Unit J Concept 6 - Partial Fraction Decomposition with repeated factors

Be careful in this problem because you are dealing with FOUR variables rather than 3. There are some 2 variable equations in this problem so combine them together to make a 3 variable equation. Then, combine 4 variable equations to create a 3 variable equation (cancel out a variable). Combine those two 3 variable equations, and from there you can solve this system! Remember that when you are writing out the denominators for the partial fractions in the beginning, remember to count up because you are dealing with repeated factors.

SP #4 - Partial Fraction Decomposition with distinct factors

PART 1: This problem is about COMPOSING when given individual fractions to create a larger, combined fraction. We are using Algebra I/II skills here by adding fractions and making sure that our fractions have common denominators.

               Save yourself some work by multiplying the factors you are multiplying to a specific fraction. As you can see, for the first fraction, I multiplied x(x+3) to create x^2+3x, so it is a lot easier for me when I distribute that 6. Don't worry too much about the denominator, we only care about the numerator at this point.

 





PART 2: This problem is about DECOMPOSING. We are given a fraction (11x^2+40x+24/x(x+2)(x+3)) that we need to find the factors that created it. We know the factors that create the denominator and place them in individual fractions and through decomposition, we find the numerators of each fraction.

               Make sure to multiply out the factors that you are multiplying each fraction by, as stated before. It saves work in the long run. When combining like terms, remember you cross out anything "x" related because only care about the coefficients. When you create your system, make sure to see if you can take a GCF out of any of them (I divided 6 from my last equation) because we always like smaller numbers in the long run.

 

PART 3: This is simply what you type into your calculator. This is a matrix, as we have a 3 variable equation on our hands.

PART 4: This is your reduced matrix, which tells us our answers. Because the first terms represent a (IN THIS CASE, USUALLY IT IS x)and there is a 1--then we have 6 in the "answers" column/term, a=6. Our 2nd term in the 2nd row equals 1, so b=1. We already knew this plugging it in and from our system but the last row tells us that c=4.

SV#5 - Unit J Concepts 3-4: Matrices

Be careful when first approaching the system. Look for any equations that might need to be simplified. Also, when looking for your 0 for Row 3 Term 2, remember that most likely, you will need to use Row 2, because if you use Row 1, your work (your 0's in row 3) will be erased. Also, take note that because you ended up with an ordered pair as an answer, your answer is a CONSISTENT, INDEPENDENT system. If you at any point have all 0's in a row and another number that is NOT 0 in your answer column, then you have an INCONSISTENT system. If you have all 0's as every term in a row, you have a CONSISTENT, DEPENDENT system.

WPP #6: Unit I Concept 3-5 - Compound Interest, Continuously Compounding Interest, and Investment Application

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SV #4: Unit I Concept 2 - Logarthmic equations

It is important to be careful that your h, when negative will look positive in your equation. Say your h is -3... it will look like (x+3). Be sure to remember that your equation originally looks like this (x-h). Another thing to be extremely careful is that when plugging in your equation, you cannot just plug it in how it looks. You must use your knowledge from UNIT H of Change of Base formula. Plug in ln(x-h)/ln b +k. Also, keep in mind, when you graph the equation (in your calculator), the graph will seem as if it does not continue on forever (in concerns for range). However, your function does indeed go on forever, the calculator just does not show it. Make sure when you draw your own, draw arrows indicating that it continues on.

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

It is important to remember that by knowing whether "a" is positive or negative, we know if the graph is above or below the graph (positive, above; negative, below). Remember that if your asymptote is a positive, but your graph is above it, there will be no x-intercept. Also, if your asymptote is negative, but your graph is below it, there will also be no x-intercept. Another way to remember that there is no x-intercept is if you, at any point of solving, have to log a negative--it is impossible. Helpful sayings also make your life easier like this one: the YaK died. This means that our asymptote will be Y=K (remember our skeleton: y=a*b^(x-h)+k)and our domain is unrestricted resulting in (-inf, inf).

SV #3: Unit H Concept 7 - Finding logs when given approximations

It is important to remember that when you have a root of any kind, that root turns into an exponent as well. From that exponent, you can create a coefficient in front of your log, furthering the expansion. Also, it is good to remember that you have two hints of your own (which are not given) from our basic knowledge of logs. If you have log(base b)b, the answer, or it equates to 1. If you have log(base b)1, your answer would be 0 because anything to the 0 power is 1. It is always good to refer back to your knowledge of the exponential formula. Also, despite not being shown in this particular video, it is good to remember that at times, when our log's quantity is a fraction, we may have to multiply the numerator and denominator by a common number, because the given log may not have any factors that correspond with our hints.

SV#2: Unit G Concepts 1-7 - Finding all parts and graphing a rational function

The student video is about finding asymptotes (slant, horizontal, vertical) and holes when given a ration function. After finding this information, you should be able to graph the function. Using basic knowledge from previous concepts like long division, interval notation, setting denominators equal to zero, and factoring, you will be able to graph the function. You should take note that, because there is a hole in this function, we will be plugging in many values into our SIMPLIFIED EQUATION, meaning the equation you are left with after crossing out common factors found both in the numerator and denominator. When finding the y-value of your hole, you plug your x-value into your simplified equation. This is the same when we plug in numbers when finding x-intercepts and y-intercepts. Another thing you should pay close attention to is that when you write the notation for your vertical asymptotes, this will help you get an idea how your graph will look, thus making it easier for you to draw it when the time comes.

SV#1: Unit F Concept 10 - Finding all real and imaginary zeroes of a polynomial

*warning: There's a little noise throughout the video but it shouldn't be too distracting. Sorry for the inconvenience.* Our problem is about finding imaginary and irrational zeroes on top of finding real zeroes, which are accounted for in Decarte's Rule of Signs and our P/q's. It is important to remember to not include certain fractions in our p/q's because we may have already written the reduced form of it already. It is also good to remember that in Decarte's Rule of Signs, that we must account for irrational or imaginary zeroes by writing "3 or 1" real zeroes. A helpful step is also to take out a GCF from the quadratic (after finding 2 zeroes) because it makes smaller, nicer numbers for when we plug them into our quadratic formula. Concerning writing factors, remember to multiply x by the number of the denominator of the irrational zero so we have the entirety of it over one number. It is also good to remember to distribute any GCF we took out previously, to a factor that contains a fraction.

SP#2: Unit E Concept 7 - Graphing a polynomial and identifying all key parts

This problem is about being able to sketch a relatively accurate graph when given a polynomial. We are able to get an idea of how the graph should look because after factoring, we are able to find and plot the x-intercepts/zeroes, the y-intercept, as well as identify the end behavior (if the graph goes up when we move left, if it goes down, etc. Also, looking back at the zeroes, we should be able to identify the multiplicity of each point, and determine whether the graph will go through (multiplicity of 1), bounce (multiplicity of 2), or curve (multiplicity of 3). To go a step further and create an even more accurate graph, we would find the extremas (min and max) and the intervals of increase and decrease to determine the behavior in between the end behaviors.

A few things that would make graphing these polynomials easier would be to pay attention to end behavior as well as remember that you are only able to go through the x-axis through a point. If you know how the end behavior looks for each kind of graph (even positive, even negative, odd positive, odd negative), you will know where to start and where to end. Also, knowing that you are only able to go through the x-axis through the points, helps us create an accurate graph with little to no mistakes. A trick to remember to go through the points is to think that the x-axis is a wall. Those points are doors. We cannot walk through walls. See my point? (PUN NOT INTENDED... UNTIL NOW.)

WPP#4: Unit E Concept 3 - Maximizing Area

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WPP#3: Unit E Concept 2 - Path of Football (or other object)

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SP#1: Unit E Concept 1 - Graphing a quadratic and identifying all key parts

        This problem is allowing us to understand how to change a standard equation into a parent function. This will make our lives that much easier because all needed points in a graph are visible in this equation. In this problem, you are to find the vertex, y-intercept, axis, and x-intercepts. Many steps are taken in order to find these answers. However, the steps will make graphing easier and accurate for us.

        Special things to pay attention to include the (h,k) within the parent function. The formula for a parent function is y = a(x-h)^2 +k. Remember that the "h" will be the opposite of its sign. Also, we should remember that in x-intercepts, imaginary numbers occur (as seen in this problem), in which case, you are unable to graph the function. 

WPP#2: Unit A Concept 7 - Profit, Revenue, Cost

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WPP#1: Unit A Concept 6 - Linear Models

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