Once you come up with your final equation, you are one step closer to finishing your graph. Now, all you need to do is figure out how to graph the equation. Well, if you look at the line equation mx+b=y, then you have to plug the numbers in. The b in your equation would be your y intercept, and you can put that on the line first since that is where the y axis and your point meet. Then you have your m, or your slope. If you think about what the slope is, its basically your change in your y over your change in your x, or can be portrayed as the rise over the run. From the y intercept, you would count the rise, or your y, however many spaces up or down depending if its negative or not. Then you take your the run which is the change in your x, and move it up left and right however many spaces.
    From there on out, you can continue to go however many spaces up, down, left, or right in whichever direction the slope directs it to go in. The denotation of a line is; any two or more points that go in one direction forever. There is no end to it, and that's will correspond to the equation of your line. So first, plot the y intercept, then use your rise over run method, or the equation, y2-y1, over x2-x1. 

A line is merely plots on a graph that have been connected, but the equation is what sets it up. For example, if the equation is similar to y=-3/4-1, then you use the normal line equation. The normal line equation would be y=mx+b. The m in that case would be the slope, and the b would be the y intercept. You can always plot the y intercept first, because you can tell that is where the y and line meet. Some questions that you could ask, knowing the equation, could be, What is the slope, or the y intercept? Well that depends on the equation your working with.
      Another would be, which way is the line going? The line can go diagonally from one corner to the opposite one. Well, looking by your y intercept, you could tell. First, graph the y intercept, and look at m, or the slope. If the slope is negative, then the line is going to go down diagonally to the right. If the slope is positive, however, then the line is going to go up diagonally to the right. The slope is the main factor is deciding if the line will go in a negative, or positive direction.

Most individuals have the option of buying a liter of soda, or a twelve pack. They begin to ponder over the thought of which is the better buy. Simply looking at the price may or may not help. Whichever is the least expensive buy, they are sure to go after that one. Sometimes checking the quantity helps also. If the twelve pack has a greater quantity, that would most definitely be better, simply because the liter bottle only has a liter worth of soda. Things should be taken into great consideration when buying items, or you could be cheated out of your money. Even when comparing two brands together, the quality, and price should be looked upon. If you had two brands to pick from, one has a better quality and is slightly more expensive, or the cheap quality one, that is not as expensive.
         Back to the soda problem, which is the better buy? Well I believe that the liter is the better buy, because if you had a twelve pack it would be similar to that of buying each one individually. Sales is also an issue in these items. One might be one sale, and have a slightly cheaper price then the other, but if they were both on regular sale, I would definitely got with the liter. You can even experiment with using empty glasses to pour the liter in. Take in consideration, that you have twelve glasses, each about the size of the soda cans in the twelve pack. Then you can pour the liter of soda into each of the cups, making sure you get the measurement precisely. If the liter fills all twelve cups, then you can infer that the liter is most likely the better buy.

Great and hard challenges were identified throughout the course of this past semester. Memories also came with this past semester. Exponents, word problems, and factoring were main key points. I found that these lessons were the most important in regards to this past semester. For review, exponents were known as "the power to." Factoring could be easily done with a factor tree, and finally word problems. Word problems were the hardest for me to comprehend, simply because of the complicated procedures needed. I can say that I definitely remember the word problems the most. Some were easily solved, and the rest seemed to be harder. I never fancied the word problems much, but the majority helped me with the lesson.
          In all regards, this past semester had its difficulties, and easy points. Challenges weren't as they seemed, and most had to be taken into caution. Each memory had a significant reason behind it. The Pythagorean theorem should be remembered by all students, simply because it can be used with more sophisticated maths in the future. Finding one length on a right trianle is one

Throughout pre- algebra, I figured that the math isn't as simple as it seems. Some topics were more challenging then others, yet they all connected somehow.  I believe that the hardest topic was combing like terms. The confusing part was mainly what side of the equation to work on first, and the fact that there was only one answer. For the answer of these questions, I would finalize huge numbers, yet it was hard to explain how I got it. Combing like terms wasn't as hard when you only had one number, or variable on the other side of the equal sign, but when the have several numbers, and exponents, and variables, I would always become utterly confused.
           The solution was not coming to me. I would have to constantly go into my math teachers room, and she would explain the same things over and over. Combing like terms was not settling inside of my head. I looked inside the math book, and the examples made a bit more sense, then I went and asked my family, and I finally got a solution. It was simple, and I was amazed a how I had learned it in such a short period of time. Combing like terms is know one of my favorite concepts, and I'm thankful to those who had helped me overcome this challenge. 

One day, while April had been walking outside, she had wanting to do something constructive. The first thought that came to mind, was painting. She would need something to paint on, so instead of painting on an empty canvas, she decided to paint the roof. The task would be a bit risky, but she knew that her ladder would be stable. She asked her parents for permission, then a couple seconds later, she was setting up the ladder. Her house was 14 feet tall, and the distance from the house and ladder was about 5 feet. She didn't exactly know what the height of her ladder was, and that would seem to be too dangerous. What is the height of her ladder? Well, the first step in this problem would be to square a and b. A and b represent the height of her house, and the distance from the house to the ladder.
             The next thing would be to square everything, then you can find the length of c. C will always be the hypotenuse, or the line that is across from the 90 degree angle of the triangle. Since the formula is a squared+ b squared= c squared, you would take square a, then square b. The result would be c squared, so then you would need to find the square root of c, and that will be your final resolution. The height of the ladder is approximately 15 feet tall.

                The ultimate definition of a square root, is a number that is mad up of two of the same numbers. For example, if you have the number 49, the square roots of 49 would be 7, because 7 times 7 is 49. Then, you would have to worry about the two types of square roots. Perfect square roots, and non perfect square roots. The perfect square root would be exactly like 49, because 7 is a whole number. 35 would have a non perfect square root, because two of the same whole numbers multiplied together wouldn't equal 35. Another name for a square root would simply be the two numbers multiplied together. When you have a square root, you can either decide if it is irrational, or rational.
                 Irrational number would be exactly like pi. Pi is a decimal that goes on forever, but doesn't repeat the same number. The symbol for a rational number would be the first two number of the decimal with a line over it, indicating that the number does repeat itself over, and over. Once you figure if it is rational or irrational, you can now find if it is real or unreal. Anything divided by zero would be considered unreal, and the other number are considered real. If you ever had a number that wasn't rational, you could always find the two numbers around it that have a perfect square root. Say, you had the number 45, you would find the closest perfect square roots, which would be 36, and 49.

            The ultimate definition of a power, or exponent, is to raise that you multiply that number the number the number of exponents there are. For example, if you had 3 to the 2nd power, it wouldn't be 6, but instead 9, because you multiply 3 twice. The positive and negative powers are very different. If there is 3 to the negative 2nd power the answer would be much different. It would be than one, but not exactly a negative number. Another way of expressing a exponent would be using the sign known as the carrot. The carrot isn't just used for math, but used for grammar too, especially if you want to make a correction somewhere inside your essay. The carrot has the same effect in math, but instead of using it for grammar, it depicts where the exponent goes. 7^2 power would mean exactly 7 to the 2nd power. 
             The number wouldn't exactly be a negative if you multiplied it be a negative exponent, because your not multiplying that number negative times. Instead, the solution would be to go into the process of multiplying negative exponents. A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, "x–2" (x to the minus two) just means "x2, but underneath, as in 1/(x2)".

• Anything to the zero power is "1", so 00 = 1.
• Zero to any power is zero, so 00 = 0.

              Exponents are used in most maths, especially algebra. Exponents are known to put a number in "the power of" something. For example, if you had 4 to the 2 power, you would multiply 4 times 4 and come up with 16. You wouldn't multiply 4 times 2, because and exponent is the same number multiplied by 2. If you have any number to the 2nd power, it's called square, and if you have and number to the 3 power it would be known as cubed. Exponents can appear anywhere in real life, and is a simplified version of long multiplying. You can only use exponents if the number is the same.
               If you had 4x4x4x4x4x4x4x5 you wouldn't say 4 to the 8th power, but instead 4 to the seventh power times 5. If The reason for exponents is also to shorten the equation, or expression. Exponents can be seen or used anywhere, but especially in most complicated mathematical equations. Having the power of exponents can have a great impact on shortening an equation. If, however, you have a negative power, you take the who thing, the number and the negative power, and you put a 1 over it. Then your equation should look something like 1/6 to the negative seventh.

            Number diagrams may seem pretty simple, until you come to integers. In the diagram you have to find to difference of two numbers, fractions, or integers, and type them into the box. If you get the wrong answer, the box will not light up. Figuring out the whole numbers was pretty simple, and didn't require scratch paper. I really needed the scratch paper when I came to the fractions. Some fractions were simple and easy and others were a bit more difficult. I found that the ones with different denominators were more difficult then the ones that had the same denominator. The integers was better than the fractions, in my opinion. For example, you would have a positive and negative number and you would need to find the difference.
            The diagram was specifically shaped like a diamond with a square. You would first need to figure out the questions in the square, and inch your way into the diamond. The boxes would glow up if you got the right answer, and when your finished, the whole pattern glowed up. Completing the first one was very easy, but as you went into the integers and fractions, it became more difficult.