D.E.V-H[anna]H

Hi! We'll be talking about a variety of topics today and we will talk you through each problem! Our pictures are color coded and you can follow along easily with the colored text next to the picture! Enjoy learning!
-Hannah and Anna!

Our first problem:

Solve and find the domain for the equation in black.
The equation in blue represents the "difference of squares , which is the equation we will use to factor what is under the the radical.
In order to find the "A" and "B" values, you need to think of what you need to square to get your original term.Hence a=4x^2 and b=15
Just as any other formula, plug the terms into the blue equation to achieve what we have in purple.
In green you see there is no radical sign. This is because the radical is accounted for by using the greater than or equal to inequality. Remember, a radicals domain cannot be negative, but can be equal                      to  zero! 
 From there we than split the factors apart to find the x-intercepts(where the factors are equal to zero).
Using algebra, solve each inequality. In orange, you see there is a negative under the radical. This means it in not a real solution, therefore, this solution will not show up for the domain or on the graph. 
The answer in gray represents the real solutions, and will account for the domain and be visible on the graph.



In order to visualize the domain better, we graphed it. The part of the graph highlighted in yellow is where inputs produce negative values. Which is not possible in a radical function.
The final domain is in purple. Notice the square brackets, this is because it can be equal to those numbers. Notice the parenthesis, this means it cannot be equal to those numbers, but as close as possible. Negative and positive infinity are not numbers and can never be reached! Hence the parenthesis.You also might notice a funky "u" in between, this  means union. A union is the combining of two sets of information, and that information exists for the entire problem, or in this case, the Domain! 


Our next problem:

Solve the equation in black and find the domain. The pink highlighted equations are what you must use.
Start by find the domain of each equation, which is highlighted in yellow(find by solving for x)
From there,  put the f(x) equation into the g(x) equation every time you see an X. You should end with what is in green.
Set what is under the big radical greater than or equal to zero. (we told you why in the last problem!)

Now its time to use our algebra skills! Subtract three from both sides , than divide by -1 to get rid of the negative radical. Note:when you divide by a negative number, your inequality sign changes directions. Than square both sides.
Add four!
You should have the highlighted green part at this point!

Set this equal to zero because you cannot find the square root of an inequality because you cannot be +/- of a inequality. (Think about it, it makes sense!)
You than get, what is in orange!
Final domain should look like the highlighted yellow portion.Remember why we have square brackets? We already told ya!;)

To help visualize the final domain we put f(x) and g(f(x)) domains on a number line. Where they overlap is highlighted in yellow, and that is where x values are possible for both equations. The domain is written in black! (There are those tricky brackets again...don't forget what they mean!)





Our next problem:

Solve for X by completing the square! What's completing the square? Well... it's where you take a quadratic equation in standard form(ax^2+bx+c=0), and turn it into vertex form[a(x-h)^2+K]! Let us show ya how!
We subtracted 39 from both sides to get rid of the "C" value. 
What you see around the 16 in the red equation is reference how we get our new "C" value. We found "64" by using (b/2)^2. Remember to add that number to both sides because what you do to one side of the equation, you must do to the other.

Whoa! How'd we get that purple equation?! Well, you must use  B/2 to find you "h" value. Don't forget to square the factor!
Now it's time to solve for X. Take the square root of both sides to eliminate the square. Remember  the square root of a number is +/-. 
Next add 8 to the 5 & -5.
Your final answers should be X=3 and X=13.





Our next problem:
*Note:Watch in HD, it gives way better picture quality so you can see what we're writing!

Here's another way of looking at it!

Our final problem:


Reflections!
Anna
I think the main reason we chose these problems was because we felt like we were experts at this topic and that we could teach them/ explain them well. I also liked these problems because there was a variety of topics in each problem. In all five problems they spanned through many units so I felt we showed our understanding of the entire trimester as a whole and tried to incorporate as much of it as possible. I feel from this project I understand the problems even more so than when I began. It  forced me to given a good reason for why I did the things in the problem, not just because Mr.Jackson told me to.I feel  in the future this assignment should be given at the very beginning of the trimester because maybe it would be easier to keep certain subjects in mind and to brainstorm problems. Overall, it was a decent project and the freedom of it was a lot  different than other school projects.

Hannah
I liked the idea of this project but it was not my favorite to work on. I felt like in order for my group to get a good grade, we would have to go above and beyond and use programs that I didn't even know how to use. I felt that the requirements for media tools was a little high for what this project was trying to accomplish.
Like I said, before, I like the idea of this project. It was good review for me and I was able to touch up on some ideas that weren't so clear before. However, I found that we spent more time trying to figure out our different media tools than we did with other things.
Overall, this project was alright. Anna and I chose topics that we thought would challenge us the most, and wanted to go above and try to explain a wide variety of subjects. We did a lot of double checking when working and it was nice being able to talk through the problems with someone. I enjoyed working with Anna and was glad to have some confusion cleared up.