## Week 17 – Math 10

Arithmetic Series are a way to find the sum of a whole group of numbers quickly as long as you have the first and the last number ($t_1$ & $t_x$, x being the final term). When you add the first and last number together, you get the sum, and the sum of the rest of the numbers. If you were to add the second number and the second to last number together, you would get the same sum as the first and last! This continues throughout the entire series of numbers.

If you are looking for the sum of all numbers together, you need to do an extra step. You divide the total number of terms by two, and multiply it with the sum of the first and last numbers. When you do this, you get the sum of all the numbers in the entire sequence (or at least the amount of numbers you want from the sequence). This works even if there is an odd number of terms because for example, if it has 19 terms (18 pairs), and 19 divided by 2 is 9.5, the .5 has the missing number inside of it.

ex.

This only works with Arithmetic Sequences.

## Week 16 – Math 10

This week we started our unit on Sequences. Sequences are how terms change consecutively, whether they go up and down by the same amount (Arithmetic). Multiplying or dividing (Geometric). Or another way (Other Sequences).

This will focus on arithmetic sequences, sequences that go up or down by the same amount.

In a sequence where you are only given one number, you will likely be given $d$, which stands for difference as in the difference between the two numbers. If it is positive, then it goes up as you move to the left and down as you go to the right, and if it’s negative, it’s vice versa.

ex.

_,_,_,0,_,_ $d=2$ to  -6, -4, -2, 0, 2, 4

If you are looking for a specific term or number in an arithmetic sequence, you can find it rather easily without having to use the rule over and over again until you get to that term.

If you use a specific formula, you can find the number you need without finding the difference over and over again.

Here is the formula and what it all means:

## Week 15 – Math 10

We learned more about graphing and linear equation. W learned about different ways to find solutions to equations. They are Inspection, Substitution, and Elimination.

Inspection is only used when a system is very easy and understood. It is basically just eyeballing the question and guessing what the solution is. You can test your solution by plugging the numbers in the appropriate spots (x and y).

Substitution is another method to find solutions, but it uses algebra. You can choose one of the equations, and then choose on of the variables to isolate. Then you plug it into the correct spot on the other equation. You then use algebra to solve the equation, and then use the answer to plug in to find the other variable.

ex.

Elimination is used when there are no coefficients of 1. You start by adding the two equations together (you can subtract but it doesn’t work as well). You want to make a zero pair by making either the x or y cancel the other out. If they don’t do this from the start, you can multiply one of the equations or both of them to get one, and then do your adding. Once you have the answer to one variable, you plug it into the equations and find the other.

ex.

## Week 13 – Math 10

This week we further continued unit 7/8 by learning about point-slope form and general form. Both of which are put into a different form than slope formula. Point slope form is useful for doing quick algebra, and general form has no fractions and other “imperfections”.

Point-slope form looks like the following: $m(x - x_2) = y - y_2$

$m$ is the slope. $x$ is not connected to a point, but $x_2$ is. The same goes for $y$ and $y_2$. You can take an equation or “hints” to make a point-slope formula.

ex.

To further turn this into slope formula and make it easier to understand, you use algebra and steps.

continuing with ex.

General form is the “pretty useless” form, pretty, but useless. It contains no fractions or decimals, but doesn’t tell you about the graph itself. The equation usually looks something like ax ± ny ± b = 0. the x is always first, y is always second, x is always positive, and everything is always on one side equalling zero.

You can change all forms/formulas into general form.

ex.

1. slope formula.
2. Point-slope form.
3. slope formula with fractions.

General form uses no fractions and follows several rules as listed in the paragraph above. It is used to take away fractions and “imperfect” parts of a formula, but it doesn’t tell you anything until it is changed into another form.

## Week 12 – Math 10

This week we continued unit 7 and learned about slopes. A slope is like moving from one “nice point” to another. A nice point being a point on an exact measurement, basically on a vertex of a graph’s square. It gives you a fraction, for example 4/5. The 4 being the vertical distance you need to reach, and the 5 being the distance you need to cross horizontally. A slope is shown with the variable $m$, example: $m$ = 3/2. The slope is basically telling you how steep a line is. You know the slope is correct when it always hits a nice point whenever it’s used. It should work on the same line forever.

The steepness is shown in the fraction in the form of rise/run (rise over run). The rise is y, run is x. When the rise is 0/n (0), it is a horizontal line, and n/0 is vertical, and is an “undefined” slope.

Lines can be considered negative or positive. This can be determined just by looking at them. A line facing one way is positive, and if it’s the other, its negative.

ex.

Sometimes you are given two coordinates/points, and asked to find the slope between the two. To find this, you must subtract its x’s and y’s. $x_1 - x_2$ and $y_1 - y_2$.

ex.

## Week 11 – Math 10

This week we spent most of the time readying for our unit test and taking the test. Plus, this week was only three days long, one of which with an early dismissal. In this short amount of time however, we did learn something new. How to find the distances between pints and the midpoint of a line.

Sometimes you can count the space between two points if they are close together, but sometimes they are so far apart, that it would be ineffective to count. To find the distance between two points this far apart, you need their coordinates. If this line is completely horizontal or completely vertical, you can find the distance with just the x or y coordinate. If a line is horizontal, you just need the x coordinate, and if it’s vertical, just the y.

ex.

Let’s say we have a horizontal line with the coordinates (10,12) and (24,12). To find the length, you subtract one x point from the other. But what’s important to remember is that the distance between the points will always be positive, so we can do 24-10=14, which is positive. But we can also do 10-24=-14, all you have to do is take away the negative.

To find the midpoint, it is pretty simple. If you have the distance between the points, you can just take it and divide it in two. The midpoint is the exact point in the middle of a line. Similarly to eyeballing distance, you can sometimes find the midpoint without trying too hard, but sometimes, the midpoint is not on a line, and is somewhere in the middle, where eyeballing is more difficult. The best way to find it is to use the math.

ex.

With the same example, the midpoint would be somewhere between 10 and 24. The distance was 14, so the midpoint would be half of that. 14÷2=7. The midpoint is 7.

If the line we are using is an oblique line (meaning its on an angle), we must make a triangle. Wherever the lines two points make touch, marks the right angle point of the triangle.

From here, you need to use pythagorean theorem ( $a^2 + b^2 = c^2$). Since the adjacent and opposite sides are both straight, you can find their distance. Just use the coordinates of where the two lines meet. Use pythagorean theorem to find the distance of the original line. Sometimes the square root of $c^2$ in the equation is a number followed by multiple points. To find the exact answer, you can just leave it as the square root of ___.

To find the midpoint, you do something similar to what we did with straight parallel lines, but because an oblique line is not parallel to any axis, you need to use both the x and y axis to find coordinates, which in turn give you the midpoint.

ex.

Let’s say we have a line with the points (2,1) and (5,4). The distance of this line is the square root of 18. To find the midpoint, we use 2 (from (2,1)), and 5 (from (5,4)), 5+2=7, 7÷2=3.5. Now we do the same with the other two: 1+4=5, 5÷2=2.5, therefor the coordinates of the midpoint of the line is (3.5,2.5), which is in-between lines on both the x and y axis.

## Week 10 – Math 10

Although I was sick for half of this week, and wasn’t able to learn fully what the rest of that class did, or at least learned less “hands on”, I was still there for the first two days, and so what we learned then is what I understand most. We learned how mapping notation can be put into the form of function notation.

Mapping notation is where you use a math “sentence” to find an output with the use of an input. (went over semi-briefly on last blog post).

ex.

ƒ    :    x              →               3x – 2

name   input       changes into         output

Function notation is generally the same thing, but like how functions are relations but relations aren’t always functions, functions notation is the same. Function notation is helpful when finding inputs and outputs of functions. They are written slightly differently as well.

ex.

name   ↓input            changes into      output

ƒ         (x)             =                 3x – 2

“ƒ of x”

Both are used generally the same way, to find the output using an input. It is “ƒ of x” because the ƒ is the functions name, and the relation is a function.

Functions & Graphs

Using the inputs and outputs from mapping and function notation, you can plot points on a graph. The input is x, and the output is y. To get the output, you put the input in the correct spot on the opposite side.

ex.

f(x) = 3x + 1  →   f(3) = 3(5) + 1

Using them, you can get coordinates. (x, y)

## Week 9 – Math 10

This week we had our midterm and spent most of the week studying for it. But on Friday, we learned about functions, a kind of relation.

A function is a relation that is special and each input has one output, no more. A function is a kind of a relation but a relation is not a kind of function.

On a graph, if any of the points are on the same x axis, then it is not a function. Each point has to be in a different x coordinate.

ex.

A function is unique, and is often named a single letter (f, g, h, etc.), and followed by x, changing into blank.

ex.

ƒ:x  → 7x + 6

ƒ is its name, x is the input, the arrow signifies “changing into”, and the final numbers are the output.

## Week 8 – Math 10

This week we started our graphing and linear relations unit. One of the main things we learned was domain and range. The domain is all of the $x$ coordinates that the graph covers, and the range is all of the $y$ coordinates that are covered.

Domain and range can be shown in “curly brackets” such as in the following example.

{x|-4 ≤ x ≤ 7, x ∈ R}

Sometimes if the graph just contains a bunch of points, the domain and range can be given in specific numbers,

ex. D = {-2,0,1,4,7} or R = {1,3,4,9,12}

here’s what one of those graphs could look like:

But they can also be lines meaning their points can be anywhere on those lines,

ex. D = {x|-2 ≤ x ≤ 7, x ∈ R} or {y|1 ≤ y ≤ 12, y ∈ R}

here’s what one of those graphs could look like:

They can also be a line, but have no beginning and/or end. This graph would have lines with arrows to represent that it continues on.

ex {x|x ∈ R} or {y| y ≤ 12, y ∈ R}

here’s what one of those graphs could look like:

When writing in these curly brackets, especially with line graphs, you need to form a “sentence”. You start with the axis you are talking about (x/y), then the possible points, and then finish with x ∈ R or y ∈ R, which means x/y is an element of a real number.

## Week 7 – Math 10

This week we completed the polynomial unit and had our unit test, but we still learned a couple of things, one of which was difference of squares.

Let’s say we have the binomial $x^2 - 16$, 16 is a perfect square (4×4). We can use this information to break it down or “factor” it. The x is squared, so this means it can be broken in half. So now we know that both parts of the binomial can be broken in half, but there is one problem, the 16 is negative. A negative and a negative equals a positive, and a positive and a positive make the same, so one of the 4’s in 16 will have to be negative.

The resulting factored polynomial is: $(x-4)(x+4)$ (or vice versa).

But what if the binomial has a variable that is not squared but with an exponent of 4, 6, 8? As long as the variable has an exponent that is even, and the other term is a perfect square, you can use a difference of squares.

ex.