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Wednesday, December 12, 2012
Sunday, December 9, 2012
Wednesday, November 28, 2012
Monday, November 26, 2012
Student Problem 7
Unit K Concept 10
- What is it about?
- This problem is from Unit K Concept 10. Concept 10 is about writing a repeating decimal as a rational number using geometric series.
- What does the viewer need to pay special attention to in order to understand the concept?
- To solve these problems we need to come up with our general rule. We do this by looking at our repeating decimal and listing the repeating parts. Then we simply put the appropriate amount of decimals in front of each repeating factor. Next we find a sub one and r. once we have found these values, we need to plug them into the infinite geometric series formula. We solve our formula and we get our answer. The viewer needs to be careful about always leaving the answer in a fraction. Also the viewer needs to be careful about how many decimal places they need to use when writing the repeating decimal. They need to be careful because sometimes the repeating decimal is in the ten, hundreds or the thousands.
Wednesday, November 14, 2012
Fibonacci Haiku
Disneyland
Happiness
Mickey Mouse
Sleeping Beauty's Castle
Meet your favorite
characters
Watch
the fireworks and ride your favorite ride
Sunday, November 4, 2012
Student Problem # 6
What is it about?
This problem is from Unit J Concept 6: partial Fraction
Decomposition with repeated factors. This is the same as Concept 5 except one
of the factors is repeated.
What does the viewer need to pay special attention to in
order to understand the concept?
This concept is similar to concept 5 except that in this
concept is repeated. This means that we have to count up the powers. For
example if we have x^3 we would need to write A/X + B/X^2 + C/X^3
Student Problem #5
What is it about?
This problem is from Unit J Concept 5: Partial Fraction
Decomposition with distinct factors. First we had to start by having a factored
equation and then we had to add it. After we added it, we needed to do partial
fraction decomposition to get our answer. In this concept each of our factors
is different.
What does the viewer need to pay special attention to in
order to understand the concept?
In this concept we need to start by separating our fraction
into three different fractions. Then we need to multiply each fraction by the
common factor. Next we had to combine like terms. From those like terms we had
to create equations and use elimination to find the value of each variable.
After that, we plug in the values form the variables back into the original equation.
The tricky part about this concept is making sure you factored right and also
that you didn't forget any negatives.
Thursday, October 18, 2012
Tuesday, October 16, 2012
Thursday, October 11, 2012
Unit H Concept 7 Student Video 3
- What is this video about?
- This video goes over Unit H Concept 7. Concept 7 is about finding logs when we are given approximations. Before we start solving these problems, we need to add our secret clues. The first one is log base b of b equals one. The second clue is log base b of one equals zero. Once we have added the clues, we need to break down the number we are trying to solve. We break it down into numbers that fir in with the clues. Next we need to take those numbers put a log in front of them and expand them. We replace the logs with the value of the clues, we simplify it and we have our answer.
- What does the viewer need to pay special attention to in order to understand the concept?
- These problems are easy to do and don't take that much time. In these types of problems we are given three clues plus our two secret clues, but we will not use every clue that we are given. We also need to be aware of the fact that we will sometimes use a clue more than once. Those are some things to keep in mind will solving problems from Unit F Concept 7.
Wednesday, October 3, 2012
Unit G Student Video 2
- What is this video about?
- This video works out a problem that is from Unit G Concept 1-7a. Concept 1-7a is about graphing rational functions. In this video, we go over finding the asymptotes of the function, finding holes if there are any, finding the domain, finding the x and y intercepts, and we also graph the rational function.
- What does the viewer need to pay special attention to in order to understand the concept?
- This graph has a horizontal asymptote and two vertical asymptote. The viewer needs to make sure that they find the horizontal asymptote and that they also find both vertical asymptotes. The viewer also has to make sure that their answer doesn't have any holes because this problem doesn't have holes. Another thing to pay attention to is graphing the asymptotes at the right value.
Wednesday, September 26, 2012
Unit G Summary Question # 10; DIVAH
10. While the domain of a rational depends on DIVAH, what do you think the range of a rational function depends on? Give an example.
DIVAH stands for Domain Is Vertical Asymptotes and Holes. The vertical asymptotes and holes are the things that restrict our graph. Our domain will be everything except this bad values. The range is the y-values. We can get a bad y-value from a hole, but we can't get one from a vertical asymptote. We get bad y-values from horizontal asymptotes, but they don't always produce range restrictions because a graph can cross through the middle of a horizontal asymptote. Our range will depend on horizontal asymptotes and holes.
DIVAH stands for Domain Is Vertical Asymptotes and Holes. The vertical asymptotes and holes are the things that restrict our graph. Our domain will be everything except this bad values. The range is the y-values. We can get a bad y-value from a hole, but we can't get one from a vertical asymptote. We get bad y-values from horizontal asymptotes, but they don't always produce range restrictions because a graph can cross through the middle of a horizontal asymptote. Our range will depend on horizontal asymptotes and holes.
Unit G Summary Question # 9; 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.
There are two ways to find the x-intercepts for a rational function. There is a long way and a shortcut way. the long way is illustrated below. First we set our function equal to zero. Then we multiply and the denominator will get canceled. Next, we factor the polynomial. Once we have the factors we set them equal to zero and we put them into ordered pairs to get our x-intercepts. The shortcut way is similar to the long way but it has less steps. In the short cut we simply set the numerator equal to zero, factor it, solve and we have our x-intercepts. The shortcut way makes sense because when we cancel the denominator in the long way, we only concentrate on the numerator. In the shortcut we immediately start working with the numerator so that's why the shortcut makes sense.
There are two ways to find the x-intercepts for a rational function. There is a long way and a shortcut way. the long way is illustrated below. First we set our function equal to zero. Then we multiply and the denominator will get canceled. Next, we factor the polynomial. Once we have the factors we set them equal to zero and we put them into ordered pairs to get our x-intercepts. The shortcut way is similar to the long way but it has less steps. In the short cut we simply set the numerator equal to zero, factor it, solve and we have our x-intercepts. The shortcut way makes sense because when we cancel the denominator in the long way, we only concentrate on the numerator. In the shortcut we immediately start working with the numerator so that's why the shortcut makes sense.
Unit G Summary Question # 6; Holes
6. How do we find the appropriate place to put a hole if the y-value is undefined when plugged into the original equation?
First of all, we know that to find out holes we need to factor our equation. We say "factor the top and if something cancels then your graph will have a hole". We factor our equation and we see that we can cancel , let's say (x -1). We cancel those to factors and our hole will be at (1,_). When we plug in 1 in our calculator, it is undefined so we need to plug it into our factored equation. The answer we get when we solve is the y-value for our hole. Our hole will be at for example, (1,3). That is the actual value for our hole but on our calculator, it will show up as undefined.
First of all, we know that to find out holes we need to factor our equation. We say "factor the top and if something cancels then your graph will have a hole". We factor our equation and we see that we can cancel , let's say (x -1). We cancel those to factors and our hole will be at (1,_). When we plug in 1 in our calculator, it is undefined so we need to plug it into our factored equation. The answer we get when we solve is the y-value for our hole. Our hole will be at for example, (1,3). That is the actual value for our hole but on our calculator, it will show up as undefined.
Tuesday, September 25, 2012
Student Video # 1: Unit F Concept 10
- What is this video about?
- This video
covers a problem from unit F Concept 10, which is finding all zeroes of a
polynomial when the zeroes could be real and or irrational/
imaginary.We chose problem number two from page 13 of the SSS. In this
video we go over the steps to finding zeroes. Which is finding the p's
and q's, doing the sign changes, synthetic division,
and doing the quadratic formula to find
the irrational zeroes.
- What does the
viewer need to pay special attention to in order to understand the
concept?
- In
this problem, we have a list of six p's and q's so the viewer needs to
make sure they find all of them.
- Also,
it was difficult to do the synthetic division because there are
many numbers to choose from, so you just need to make an educated
guess and try
to pick the numbers that will give you zero heroes.
Unit G Summary Question # 4; Vertical Asymptotes and Holes
4. What is the difference between a graph having a vertical asymptote and a graph having a hole?
In the example below, the graph has a vertical asymptote and a hole. When a graph has a vertical asymptote, it means that it has a vertical line going though it at some number value. In the example below, the vertical asymptote is at x= -1/2. It is important to remember that graphs can never cross through vertical asymptotes.When a graph has a hole, it means that the graph does not exist at that point. In the example below, the graph has a hole at (0,1). When you plug this function into your graphing calculator, and trace to x=0, there will be no x value. That is because we have a hole and there is no value there. Holes are written as ordered pairs such as (1,3) and vertical asymptotes are written as an equation like x = 3.
In the example below, the graph has a vertical asymptote and a hole. When a graph has a vertical asymptote, it means that it has a vertical line going though it at some number value. In the example below, the vertical asymptote is at x= -1/2. It is important to remember that graphs can never cross through vertical asymptotes.When a graph has a hole, it means that the graph does not exist at that point. In the example below, the graph has a hole at (0,1). When you plug this function into your graphing calculator, and trace to x=0, there will be no x value. That is because we have a hole and there is no value there. Holes are written as ordered pairs such as (1,3) and vertical asymptotes are written as an equation like x = 3.
Unit G Summary Question # 8; Y- Intercepts
8. How do you find the y-intercept of a rational function? Does this need to be done in the original or simplified equation?
There are two ways in which we can find the y-intercept of a rational function. The first way is by plugging in zero into our original equation. Or as shown below, we can use the factored equation and plug zero in there. We will get the same answer from the original equation and the factored equation. There is one exception to finding the y- intercept and that is when we have a hole at the y-intercept. If we have a hole at the y-intercept, then our answer will be zero over zero and were going to have to use the factored form in order to find the real value for the y-intercept.
There are two ways in which we can find the y-intercept of a rational function. The first way is by plugging in zero into our original equation. Or as shown below, we can use the factored equation and plug zero in there. We will get the same answer from the original equation and the factored equation. There is one exception to finding the y- intercept and that is when we have a hole at the y-intercept. If we have a hole at the y-intercept, then our answer will be zero over zero and were going to have to use the factored form in order to find the real value for the y-intercept.
Unit G Summary Question # 7; Limit Notation for Vertical Asymptotes
7. Describe how to write the limit notation for vertical asymptotes and what the notation means.
To write the limit notation for vertical asymptotes, we use the value that we got got for the vertical asymptote. When we do limit notation for vertical asymptotes, we use little pus and minus signs. The little plus sign means "from the right side". The little minus sign means "from the left side". The notation means "as x approaches a number from the right, the graph goes up" or "as x approaches a number form the left the graph goes down". Here is an example of limit notation for vertical asymptotes:
To write the limit notation for vertical asymptotes, we use the value that we got got for the vertical asymptote. When we do limit notation for vertical asymptotes, we use little pus and minus signs. The little plus sign means "from the right side". The little minus sign means "from the left side". The notation means "as x approaches a number from the right, the graph goes up" or "as x approaches a number form the left the graph goes down". Here is an example of limit notation for vertical asymptotes:
As
x → 2 +, f(x) → ∞
As
x→ 2 -, f(x) → -∞
Monday, September 24, 2012
Unit G Summary Question #5; Asymptotes
5. Describe the conditions in which a graph can cross through an asymptote.
There are some conditions in which a graph can cross through an asymptote. First, we have horizontal asymptotes. The graphs for horizontal asymptotes can cross through the horizontal asymptotes but only if it crosses towards the middle of the graph. It can NEVER cross towards the far left or far right of the graph. Next, we have slant asymptotes. Like horizontal asymptotes, the graph for slant asymptotes can sometimes cross towards the middle of a graph. Finally we have vertical asymptotes. The graphs for vertical asymptotes can NEVER EVER cross through a vertical asymptote.
There are some conditions in which a graph can cross through an asymptote. First, we have horizontal asymptotes. The graphs for horizontal asymptotes can cross through the horizontal asymptotes but only if it crosses towards the middle of the graph. It can NEVER cross towards the far left or far right of the graph. Next, we have slant asymptotes. Like horizontal asymptotes, the graph for slant asymptotes can sometimes cross towards the middle of a graph. Finally we have vertical asymptotes. The graphs for vertical asymptotes can NEVER EVER cross through a vertical asymptote.
Unit G Summary Question # 3; Slant Asymptotes
3. When does a graph have a slant asymptote? How do you find the equation of the slant asymptote?
Slant asymptotes can only exist if the degree from the numerator is ONE BIGGER than the degree on the bottom. This means that if we have the function (x^2-1)/(x-2), we will have a slant asymptote because we have x^2 on the top and x on the bottom. The equation for the slant asymptote is y=mx+b. To find the equation for a slant asymptote, first we need to do long division. Once we have our answer, we will use everything but the remainder and that will be the equation for the slant asymptote.
Slant asymptotes can only exist if the degree from the numerator is ONE BIGGER than the degree on the bottom. This means that if we have the function (x^2-1)/(x-2), we will have a slant asymptote because we have x^2 on the top and x on the bottom. The equation for the slant asymptote is y=mx+b. To find the equation for a slant asymptote, first we need to do long division. Once we have our answer, we will use everything but the remainder and that will be the equation for the slant asymptote.
Sunday, September 23, 2012
Unit G Summary Question # 2; Horizontal Asymptotes Limit Notation
2. Describe what the limit notation for horizontal asymptotes actually means.
The limit notation for horizontal asymptotes has plus and minus signs included. Instead of saying "as x approaches infinity, the graph goes up" , we are now saying as we go right (or left) we approach as number. Next to infinity there is a positive sign next to it to indicate that it is positive. For asymptotes, the limit notation is different because we do not say if the graph is going up or down. We state which number it is getting close to.
I could't find a picture for limit notation so I'm going to put an example below:
As x → + ∞,
f(x) → 2
As
x → - ∞, f(x) → 2
The limit notation for horizontal asymptotes has plus and minus signs included. Instead of saying "as x approaches infinity, the graph goes up" , we are now saying as we go right (or left) we approach as number. Next to infinity there is a positive sign next to it to indicate that it is positive. For asymptotes, the limit notation is different because we do not say if the graph is going up or down. We state which number it is getting close to.
I could't find a picture for limit notation so I'm going to put an example below:
As x → + ∞,
f(x) → 2
As
x → - ∞, f(x) → 2Unit G Summary Question # 1: Horizontal Asymptotes
1. How do we know if a graph has a horizontal asymptote? What are the three options?
To determine whether a graph has a horizontal asymptote, we need to compare the degrees of the numerator and the denominator. If the bigger degree is on the bottom then the asymptote is y = 0. If the coefficients have the same degree, the asymptote is equal to the ratio of the coefficients. Lastly if the bigger degree is on the bottom, then there is no asymptote. Sometimes the graphs cross through a horizontal asymptote. It is okay if they cross towards the middle of the graph. It is never okay if they cross towards the far left or far right.
To determine whether a graph has a horizontal asymptote, we need to compare the degrees of the numerator and the denominator. If the bigger degree is on the bottom then the asymptote is y = 0. If the coefficients have the same degree, the asymptote is equal to the ratio of the coefficients. Lastly if the bigger degree is on the bottom, then there is no asymptote. Sometimes the graphs cross through a horizontal asymptote. It is okay if they cross towards the middle of the graph. It is never okay if they cross towards the far left or far right.
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