During the last week of math 10 I spent most of the time studying for the last exam but one thing that I had to refresh my memory about was factoring. I hadn’t done it in a while so I didn’t remeber much but luckiy I was able to go back in the workbook and relearn it. When you are factoring it’s really important that you remember the 1 2 3 rule because that will save you a lot of time. The 1 2 3 rule is to check how to factor it, 1 is there anything in common, 2 is it a binomial and a difference of squares and 3 is there three terms written as , x and then a number. You don’t have to do all 3 for one question though, you just need to use the one that fits. Here’s an example:

Let’s say the question is + 10x + 24 and you need to factor. using the 1 2 3 methode we can see that hey don’t all have something in common so we can’t use 1. Going to 2 we can see that it isn’t binomial nor is it a difference of squares so in this case we need to use the third. 3 works because it is a trinomial written in the correct format. So because of that essentially all we need to do is find 2 numbers that add up to 10 and when multiplyed it gives us 24. First is finding all the equations that when multiplyed it equals 24. So that’s 1 and 24, 2 and 12, 3 and 8, 4 and 6. next step is finding which one off that list gives us 10 when added together. We can obviously see that it’s 4 and 6 so luckily it isn’t going to be that difficult. And finally to write it we would do ( x + 4 ) ( x + 6 ) and that’s our final answer.

This week in math 10 I learned how to solve a system using the elimination method. It’s an easy way to find the x and y coordinates and even though there are quite a bit of steps, it’s actually really easy. There is another way to do it, it’s called the substituion method and they both work well but I prefer the elimination one so that’s the one i’ll be explaining. Here’s an example:

Let’s say you have the equations, 3x + 9y = 12 and 6x + 15y =3 and we need to find the x and y coordinates of them. So when using the elimination method, the first step is to find a zero pair. It’s not exactly necessary but it’s very helpful and it makes solving a lot easier. So in order to do that we have to multiply one of the equations so that we end up with a zero pair. So in this case we can multiply the whole first equation by -2 for example and then that would make the equations, -6x – 18y = -24 and 6x +15y = 3 but it doesn’t matter what we multiply as long as we get a zero pair. Now we can start finding the coordinates and to do that we are going to add all the like terms in the two equations together and we’re going to get rid of the zero pair. So after that we’ll be left with -3y  = -21. Next step is to isolate the y and to do that we just need to divide both sides by -3 which will result in us having y = 7, and just like that we’ve found the y coordinate. Now to find the x, all we need to do is insert that 7 into the spot of y in one of the equations. It doesn’t matter which one we pick because we’ll get the same answer either way so we’ll just use the first one. So we insert the 7 into the y’s spot which will look like, 3x + 9(7) = 12. Now we multiply the 9 and 7: 3x + 63 = 12, and now we want to isolate the x and to do that we need to take 63 away from both sides and we’ll be left with, 3x = -51. Then to isolate the x even more, we divide both sides by 3 and finally we’ll get x = -17. So those are our coordinates, x = -17 and y = 7.

This week  in math 10 I learned how to find the y and x intercept when given an equation in general form. It’s very helpful to know this so that you can easily graph the slope without having to do a lot of rearranging. Here’s an example:

Let’s say you have the equation 2x – 3y + 12 = 0. We can see that it’s written in general form so obviously we can’t tell much about the actual slope but by doing this trick we can quickly find an easy way to graph it. So the first step to finding the y intercept is to rewrite the equation without the x, which will leave us with: -3y + 12 = 0. Now we want to isolate the y and in order to do that we need to move it to the other side of the equation resulting in -3y = 12. Now to isolate the y even more we need to divide both sides of the equation by 3, and that will give us y = 12/3. and to simplify that even more we can turn the 12/3 into a 4. Finally we will be left with y = 4, which is our y intercept. Now to find the x intercept it’s basically the same thing except with the x. We start by getting rid of the y, giving us 2x + 12 = o. Then we isolate the x: 2x = -12. And then we divide it all by 2: x = -12/2. And once again we simplify the -12/2 which would give us -6. So in this case x = -6 is our x intercept. Now to graph it we would just put those two intercepts on the graph and connect them to get our slope. That’s all 🙂

This week in math 1o i learned how to change an equation in general form into slope intercept form. It’s good to know how to change general form because general form is pretty useless and it doesn’t really give you an information. So turning it into slope intercept form is best because it’s the easiest and it doesn’t have anything like brackets which would make it harder.

So let’s say you have the equation 2x + 5y – 15 = 0 and you want to change it into slope intercept. The first step is isolating the y, so we’re just going to move it to the other side of the equation and by doing that we need to take 2 and 15 off of both sides. So we’ll need to subtract 2 from o and 2 and then add 15 to 0 and -15. Which will leave us with 5y = -2x + 15. Now we need to get the y by itself so to do that we need to divide everything by 5. So 5y/5 = -2x/5 + 15/5. We can see that they all divide nicely: 5y/5 = y and 15/5 = 3, except for -2x/5 so we’ll just have to leave that one as a fraction in the final answer. So finally we are left with y = -2/5x + 3, where -2/5x is the slope and 3 is the y intercept. And that’s how you turn an equation in general form into slope intercept form.

This week in math 10 I learned about the vocabulary for slopes. I learned the four main slopes and how to tell which one they are. The four main slopes are a positive slope, a negative slope, a zero slope and an undefined slope. I also learned about parrallel and perpendicular lines and what they mean. And finally what it means when a slope is collinear.

Basically a positive slope is a slope that is always going on a upwards angle. A negative slope is a slope that is always going on a  downwards angle. A zero slope is a slope that is perfectly horizontal and an undefined slope is a slope that is perfectly vertical. They’re called this because when writing the fraction these are the symbols you would use. The undefined and zero are called that because they are not at an angle and therefor you can’t find numbers to put into a fraction. A parrallel slope is two lines that never touch because they both always have the exact same angle. A perpendicular slope is two lines that are one negative and one positive and when put together they create a 90 degree angle. And to tell if a slope is collinear you just have to look to see if all the points on the line are perfectly lined up on the angle. That’s basically all the key points for the vocabulary and if you know those you’ll good for the most of it.

This week in math 10 I learned about function notation and how to solve it when given different inputs and outputs . It looks difficult but it’s actually really simple. All you have to do is put your input everywhere it’s mentioned in the equation and then solve like normal or do the steps backwards to find the input if you don’t already have it. Here’s an example,

Let’s say you’re given f(x) = 2x + 5 and then you’re asked to solve it if f(17). So all we need to do is insert the 17 into all the places in the orginal equation where there is an x. Now we have f(17) = 2(17) + 5 and we can basically solve as normal. So first 2 x 17 is 34 and now we add the 5 and we’re left with 39. And that’s essentially it, our answer would be f(17) = 39 and written in coordinates it would be (17,39).  Now if we were given the same equation of f(x) = 2x + 5 except we were told to solve using F(x) = 23, we would need to take similare steps. So for thr first one we were being asked for the y coordinate but in this case we are being asked for the x. So what we need to do is find what number can go into the equation as the x and then equal 23. I prefer doing the equation backwards so that’s how I will explain it. We can start by taking 23 and subtracting 5 instead of adding and then we’d be left with 18. Next step is taking the 18 and dividing it by 2 instead of multiplying which would result in the answer being 9. So then we’d have the final answer as f(9) = 23 or as coordinates, (9,23).

This week in math 10 I learned how to tell if something is a function or just a relation. A function is when the input value only has ONE output value. Everything with more than one output value is considered a relation. Seeing that on an arrow mapping diagram would look like just one arrow pointing to another, not more than one pointing to the same number. Here’s an example on a graph.

Say we’re graphing principle square roots. So we can have 4, 9, 16 and 25. We already know that their is only one answer when trying to find those principle square roots, so we automatically know they are functions, because again, there’s only one output number. It would be: 4= 2, 9 = 3, 16 = 4, 25 = 5. Now on a graph, the way we would be able to tell if it’s a function, is by seeing if an of the points overlap. There should only be one dot on each vertical line or else in isn’t a function. As shown in the picture above, none of the dots are on the same vertical line so that’s another given that those are functions. So whe using an arrow graph, none of the inputs should have more than one output and on a graph, none of the vertical lines should have more than 1 dot.

This week in math 10 I learned how to find a linear relation and it’s equation by using the table of value. When you have a pattern of numbers and you need to find the equation, a table of values is really helpful because it’s an easy way to organize your numbers and find your answer. Basically all you need to do is split your table into two columns, one is labeled x and the other is y. The x column is where you write the order of the numbers so like wether it’s first, second, third etc. The y colum is where you write the number itself. So once you have it all written out you can do some multipilcation and addition or subtraction, whichever one works and that’s it.

Here’s an example. Let’s say you have the numbers 5, 9, 13, 17… and you need to find the pattern and equation. The first step is putting the numbers into the column, so we’re gonna start by putting 1, 2, 3, and 4 into the x column and then 5, 9, 13 and 17 into the y column. Now looking at the numbers in the y column, we can notice that it increases by 4 each time so that will be our first clue. Now to find our equation we’re going to do 4 mutiplied by the numbers in the x column and then add or subtract what we need to get our numbers in the y column. So starting with the first one, 4 x 1 is 4 but we need to get to 5 so we can add 1. 4 x 1 + 1 = 5. So that works for the first one but now we need to check the others. 4 x 2 is 8, plus 1 is 9. So that one works too. 4 x 3 is 12, plus 1 is 13, that one also works. We don’t have to do the last one to see that that equation works for them all. And that’s it, our final equation will be (4)(x) + 1.