1. Arithmetic and Geometric Series

This is probably the most applicable math skill in life that I have learned this year. It will help me calculate when I deal with large numbers of people, products, etc.

Arithmetic Series: Series that have a common difference, which is being added to each number.

Formulas for finding the sum of an arithmetic series.

Example: Find the sum of this series. 8, 5, 2, ……,-64

Here we are not given n, but we know , thus we can solve for n using this formula:

Now let’s find the sum:

Geometric Series:

Geometric series is a series of numbers which have a common ration, meaning a number that is being added to the terms.

Formulas:

There are two types of geometric series: finite and infinite. Finite means that it will have a definite sum and infinite means that it will not have a sum, but there is a possibility to find a sum for an infinite series.

Infinite series can diverge, where the numbers get bigger, or converge, where the numbers get smaller.

2. Systems of Equations

The second thing I learned that I found pretty interesting was systems of equations.

Systems of equations means two equations that can substituted in each other in order to find where they intersect.

example:

y=x+5

This can be solved graphically or algebraically.

Graphing:

Algebra

x=-3,1

points of intersection: (-3,2)(1,6)

3. Graphing inequalities

Graphing inequalities was what I thought the most artistic part of math class, because we got to colour in graphs

Let’s take:

Graphed it will look like this:

The way we can know which part is coloured in is because when we factor the equation (y=(x+3)(x-1)) , we get the x intercepts, which will tell us that the solutions (the part that is coloured) will lie between them, or outside of them. Because the inequality sign also includes an equal sign, the line will be solid and it will include the points/ x-intercepts.

The way we know where to colour is by testing a point. I will test (0,0)

0>0+2(0)-3

0>-3

This is a true statement, so I know my solutions lie within the parabola which I will colour.

.

4. Graphing Absolute values

The first time we graphed absolute values was a “wow” moment for me. I just thought it was really cool how everything has to get reflected into the pozitive sone (quadrants)

Quadratic equation absolute value:

We also have to know about piece wise notation, which is the values of x (so that everything is positive) for the actual equation and the absolute value equation.

When solving an absolute value algebraically, it would be:

f(x)= Piecewise: x>1, x<

This is the same principle as the quadratic. If the linear equation has a positive slope, then the absolute value equation will have a negative one. \

f(x)=x+5, x>-5

f(x)=-1(x+5), x<-5

5.Thinking critically

The most important thing I learned in math was to think critically. I tend to do a lot of silly mistakes, and it can be very frustrating.

An example of a silly mistake is

lets take:

x=7, x=-2

This is just a mistake of not paying attention to the sign. The middle term (5x) had to be positive, but I accidentally made it negative, which changed my whole equation and now I’d have to restart.

This week in math class we started a new unit: trigonometry.

Last year we learned SOH CAH TOA, but this only works for right triangles.

S stands for Sine- so 

C stands for Cosine- so 

T stands Tangent -so 

This year we looked at other types of triangles, that were not right triangles. Miss Burton called them the “magic triangles”. These two triangles are formed from two other geometric shapes: the square, and an equilateral triangle.

First, however let’s look at some terms.

Rotation angle- angle formed between the initial arm and terminal arm. It is measured in a counter clock-wise direction.

Th initial arm is on the x axis, and the terminal arm is the side that rotates.

Reference angle-measures how far a rotation angle is from either 180° or 360°, depending on which one is closer.

If we imagine that a full rotation makes a circle, then it is easier to picture how a triangle would be made. A radius of that circle would become a hypotenuse.

Thus, our new version of SOH CAH TOA

adjacent= x

opposite =y

hypotenuse= r

The Cast Rule: shows in which quadrant sine, cosine, and tangent will be positive or negative.

In quadrant one, all are positive.

In quadrant two, only sin is positive.

In quadrant three, only tangent is positive.

In quadrant three, only cosine is positive.

The magic triangles:

In a square, all sides and angles are equal – it is equilateral and equiangular, therefore, if split in half, it creates two congruent triangles ( that are equal to each other). Let’s say each side of the square was 1. When split in half, those sides still remain as one, but the line thourgh the middle of the square now becomes a side of the triangle. That will be the hypotenuse of the triangle.

If we want to find the length of the hypotenuse, we will use the Pythagorean theorem:

This type of triangles will always follow this pattern

An equilateral triangle is also equiangular, therefore it will also create two congruent triangles when it is split in half. If all its angles are 60°,  then when it is split in half at the top it will create a 30° angle. Here the hypotenuse will always be the largest side. The opposite side (from the 30° angle) will always be half of the hypotenuse. The is also obtained using the pythagorean theorem like we did for the square. This type of triangles will have a pattern of

The Unit Circle:

This picture of the unit circle shows us the quadrantal angles, and it gives us the value for x,y, and r at those angles. (The r value is 1 here)

The Sine Law

The Sine law is helpful for finding sides or angles of a triangle when we are given at least three variables(two from the same ratio and a different one)

A triangle has sides measuring 38cm and 22cm . The angle opposite the 22cm side is 35º. Determine the angle opposite the 38cm side.

The sine law can be reciprocated thus:

we have sinA, and a, and b so we will use only the first two ratios.

=

Cosine Law:

When not all variables are given, for example when only sides are given, or when only a variable is given from each ratio of the sine law, then the cosine law will be really helpful.

Let’s take and find

This week in math class we learned how to apply the knowledge we’d acquired about rational expressions and apply it to world problems. Here are two examples that I found other than what we did in class.

Example 1:

Clayton, the team manager, can sweep the gym floor of Tate Christian School in 36 minutes. Morgan, his assistant, can sweep it in 24 minutes. How long would it take them to sweep the floor if they work together?

Example 2:

The sum of a fraction and its reciprocal is . Find the fraction.

This week in math class we learned how to add and Subtract rational expressions. The first thing to do is to factor. Once the factoring is done, the non-permissible values are stated. The next step and the most important thing to remember when subtracting or adding is to get a common denominator first. Once a common denominator is reached, all of the numerators need to be multiplied by whatever the denominator was multiplied by in to get the common denominator.

Addition example:

+

The common denominator for these fractions will be so the first fraction will need to be multiplied by 1 and the second fraction will be multiplied by ab. 

The fraction can be written as a big fraction with everything on the top and the common denominator on the bottom:

Subtraction Example:

Subtraction uses the same principles as addition does, however one important thing to keep in mind when subtracting is that if there is a negative sign in front of a fraction, all the signs of the terms in that fraction’s numerator will change.

Example –

These fractions must first be brought to a common denominator of 12ab. When written as a big fraction over the common denominator, the second fraction’s signs will be changed.

To get the fractions to a common denominator the first one has to be multiplied by 4 and the second one by 3a

unfortunately the numerator does not factor any further, so I will the answer as it is.

Fractional Equations

Just as with numbers there are a few ways to solve a fractional equation.

for example:

This fraction can be solved using these different ways:

• both fractions could be multiplied by a reciprocal

1st reciprocal: 3/5

⋅ =⋅

2nd reciprocal: 4/3x

⋅=⋅

• both fractions could be multiplied by the common denominator\

⋅=⋅

• the fractions could simply be cross-multiplied where the numerator of the first is multiplied with the numerator of the second and the numerator of the second is multiplied with the denominator of the first

This week in math class we learned about rational expressions. A rational number is a quotient of two integers (eg. 3/5)

We looked at equivalent fractions, which means fractions that have the same value but are written differently.

We know that 2/3 is equivalent to 6/4 because 6/4 can be reduced to 2/3. It is the same with fractions that involve quadratic or linear equations.

Example:

is equivalent to

If we simplify the first fraction, it would eventually reduce to

Non-permissible values

When working with quadratic or linear equations, we have to state the restrictions for the denominator. If a quadratic equation is in the denominator, it must first be factored.

Example:

The quadratic in the denominator can be factored:

Therefore, in the denominator, x cannot be equal to 1 or 2.(x≠1,2)

The reason x cannot equal those values is because they would result in a zero in the denominator, which is not allowed.

Multiplication 

Let’s take:

}$ ⋅

The first step here would be to factor:

$latex  \frac{(x-3)(x+2)}{(x+4)}$ ⋅

Once the factoring is done, we have to state the non-permissible values for x: (x≠-4,0,-2) Those are the values in the denominator which x cannot equal.

Then, we can simplify:

Simplifying is just like in numbers, because like factors cancel each other out, or result into a 1, which makes the factors become invisible.

Division. 

If given, 1/2÷2/4, most of us would probably know that the second fraction can be reciprocated and then we would simply multiply: 1/2⋅4/2=2/2=1

The same concept is used when dividing fractions involving equations:

$latex

This last week in math class we learned how to solve absolute value equations graphically as well as algebraically.

Let’s take the equation: 10=|x+4|

From this equation we know that y=10, and y=|x+4|.When we solve this graphically, the solutions will be where the lines intersect. This is very straightforward by just looking at the graph.

The solutions (only x-values) are 6 and -14

x={6,-14}

We can also do this with a quadratic equation.

If we take , we know that y=4 and

We can see that our solutions for x are 1, 3, -1, -3

Solving Algebraically

These solutions can also be figured out algebraically. Let’s take the first example that we graphed:  10=|x+4|

Before solving however, let’s talk a little about piecewise notation. Piecewise notation is a way of writing the absolute value so that it can be solved two different ways. One will be positive, one will be negative.

10=|x+4| will be written as {10=x+4, and 10=-(x+4)}

Now we can solve the equation two different ways.

10=x+4

10-4=x

6=x

AND

10=-(x+4)

10=-x-4

x=-10-4

x=-14

These solutions are in agreement with the solutions that I got when I solved graphically, so I know I am correct.

Let’s now solve the quadratic equation: 

This can be written as and as

AND

According to the earlier graph, the solutions are correct.

Reciprocal Functions

The word reciprocal usually refers to fractions. A reciprocal function happens when a graph is reciprocated. Thinking about this in a fractional style, I kind of imagine the numbers that are really big suddenly turn into very small numbers which are indicated by fractions, and the small numbers/fractions turn into big numbers.

Any number reciprocated will be a different value, except for 0,1,-1. The 1 and -1 one are always going to be the same value when reciprocated, therefore we call these the invariant points-the points on a graph that will never change. Zero cannot be reciprocated, and thus we call this the undefined, where the values on the graph will be undefined or unknown.

Let’s take f(x)=2x and reciprocate it: f(x)=1/2x. It will look like this

The green bent lines are known as hyperboles. The way they are graphed is that they first have to pass through the invariant points and then get smaller and smeller or bigger and bigger, but never touch the asymptotes, which are kind of like the boundary lines of how far the hyperboles can go. The equations of the asymptotes are: y=0 and x=0. For the x equation of the asymptote, the value will be the same as the x-intercept.

The restrictions for these functions is as follows

x∈ R, x≠0

y∈ R, y≠0

The reason they cannot be equal to zero is because that is where the asymptote is and we know that the hyperbole is not allowed to touch the asymptote.

Quadratic equations can also be graphed this way, according to the same concepts and rules.

The reason the hyperbole is only along the y axis is because there are no negative values for the parabola, and no x intercepts. Therefore, it only stays in the positive zone. This function has two solutions.

Here the hyperboles are graphed as in a linear equation, passing through the invariant points and then getting bigger or smaller, but never touching the asymptotes. The equation of the asymptotes here will be y=o and x=0 because the x intercept is at 0. This function has two solutions.

This function will have three hyperboles, as well as 4 solutions. The equations of the asymptotes will be y=o, and from the graph we can find out that the x-intercepts are at 1.732 and -1.732.

This week in Math class we learned more about graphing, but instead of graphing equations, we learned how to graph inequalities. In order to understand this better, we can use the number line.

Let’s take the inequality:

Factoring it out, it would look like this: (x+5)(x-3)<0. We know that, according to the zero product law, that x will be equal to -5 and 3. These are known as the zeros of the inequality or the solutions.

In order to find the values for x, we will use number lines, in what is known as the sign chart:

Using two number lines will allow us to put in both values of the x ( because we have two solutions).

The first solution (x=-5), will have a zero value at -5, therefore that is the zero point. We know that before the -5, all the x-values will be negative, and after the -5, all the x-values will be positive. The second solution (x=3) will have a zero value at 3 on the number line: therefore all the x-values before that will be negative, and all the x-values after will be positive.

Later, as seen in the picture, when we combine the sings of the x values, we will have + – + values. We know that we want to find a value for x that is less than zero, so it will have t0 be a negative value. The negative sign resulting from the chart says that x will have a negative value between -5 and positive three. Therefore x will be greater than -5, but less than 3 (-5<x<3). 

To make sure we are correct, we can simply test a number between -5 and 3 and plug it in the equation. Let’s take x=0.

0+0-15<0

-15<0 …. this is a true statement.

Graphing:

Graphing inequalities is quite different from graphing inequalities.

We have to understand the symbols first. For example if we are given x>7, then x=7 would not satisfy the inequality, because it is not greater than 7. But if we are given x>7, then 7 would be a solution because x can be equal to or greater than 7. On a graph, if the sign is greater than or less than (<>), then the solutions are found by having x values larger or smaller than whatever value we are given, but that given value will never be part of the solution because x cannot equal the value. For these types of inequalities (which use <> symbols), when graphing them, the line will be broken. When the given value can also be equal to the x value, meaning that the symbols used will be:><, then the line will be solid.

Let’s graph a few equations to see this:

   This is an example of  linear equations. What we can notice here is that the line is broken because the inequality sign is < and >(greater than or less than) which means that on the line there will be no solutions in either of the equations. We also know that the y intercept is at +5 for both, and that the slope is 1/1.  

This is an example of a graph of two quadratic equations, which are the same, but the inequality sign is switched. The line here is solid because the inequality signs are <> (greater than or equal to; less than or equal to ) which means that there will be solutions on the line as well. 

  This is the equation given as an example in the beginning, and this is what it looks like when it is graphed. The line is broken because the inequality sign is (<) less than. 

Equivalent Forms of the Quadratic Function:

There are three ways we can write a quadratic equation to mean the same thing. These three forms are equivalent to each other – they mean the same thing. The advantage of writing an equation in more than one way is that each form will tell us something different about the equation:

1. The General Form: 

 Example:

2. The Standard Form:

 Example:

3. The Factored Form: 

 Example:

The examples of equations shown above are all equivalent to each other or the same equation put into three different forms.

Now, we will look at what each form of the equation lets us find out.

The General form lets us find the y intercept, which is simply term c. In the example above, the y intercept would be at 16.

The Standard form tells us the coordinates of the vertex. From the example above we can see that the vertex would be at (-4,0). *the 4 becomes negative because there is a negative sign in front of the q term in standard form.

The factored form lets us find the roots, or x intercepts. According to the zero product law, in order for the product to equal zero, one of the factors must be equal to zero.

If we take the first factor in the equation and equal it to zero, then x would be equal to -4:

x+4=0

x=-4

Since the second factor is exactly the same, we know that x will be – 4 again.

We know have enough information from these forms to graph the equation. In order to double check my work, I will desmos and graph all three equation. If I am right, all the equations will be graphed the same way, right on top of each other.

I now all the equations are equivalent, because they’ve all been graphed the same way.

Using these concepts, it is also possible to solve word problems, by modeling problems with Quadratic Function.

This week in Math class we learned more about graphing quadratic equations and properties of these.

if we make a table of values, like we learned in previous years, we can find out whether the equation is quadratic or linear by taking the difference of the y values. If we get the same difference the first time we subtract the y-value numbers, then it is linear. If; however we have different numbers the first time we subtract, but we have the same numbers when we subtract the second time, then it is a quadratic equation. Another way we can tell these two equations apart is to see the degree that x is at. If it is to the degree (exponent of 1), then it is a linear equation; if it is to the degree (exponent of 2) then it is a quadratic equation.

For example:

There are certain forms of the quadratic equation that will help us see where the equation is located on a graph.

For example:

a)

this means that the axis will be located exactly at 0, and since the coefficient in front of x is 1, the patter will be 1,3,5 (up 1 over 1, up 3 over1, up 5 over 1). since x is also positive, it open up.

b)

this means that the axis will be translated horizontally to the left 5 places. if the sign was negative then it would be moving horizontally to the right 5 places. This parabola will open up since the sign in front of the squared term is positive. if it was negative it would open down.

c)          or          

The numbers in this equations (+45 and-67) tell how many places up or down the parabola is moving vertically. The first one is moving 45 places up, and the second one is moving 67 places down.

When combining all these forms into a standard form, we come up with the following equation:

• a tells if the parabola opens up or down depending on the sign (+ or -), and if there is a coefficient in front of it, it tells whether it compresses or expands (the smaller the coefficient the more it expands)
• the -p tells how many places it is being translated horizontally 
•  the +q tells hoe many places it is being translated vertically
• The p and q values are the vertex values (p is x) (q is y)

Let’s describe the following graph without actually graphing it:

We know that the vertex is at 3 and 6. Three is positive because since it was negative in the equation, it means that it is now being translated to the right 3 places (from 0) which makes it positive. We also know that the axis of symmetry (where the parabola would touch the x axis ) is at x=3. This parabola opens up because a is positive. It is also congruent to because it follows the 1,3,5 pattern. It will have a minimum point at y=6. Its range is is y=>6 and its domain is x ∈ R. 

This is the graphed form of this equation: