The semester is coming to an end, this class has been one of the most difficult but also rewarding classes I’ve ever taken as into has taught me not only the importance of C, D, P, E, U, but it has also taught me things about myself and my learning. Here are the top 5 things I’ve learned throughout the course of the last semester in Pre-calculus 11.

1. Factoring:

factoring is extremely important! This is possibly the most important skill to have/concept to understand in all of pre-calculus 11. It is by far the most relevent and repeated idea throughout the entire course. Factoring reappears in over half of the chapters as it is necessary for units like inequalities, Quadratics, as well as reciprocal functions etc…

2. Geometric vs Arithmetic, Sequence vs Series:

This unit was our very first unit that we completed in the class, I thought this unit was important to add to this post because of how different it is from any of the other units and thus slightly more difficult to remember. An Arithmetic sequence is a constant addition from on term to the next in a predictable pattern. An Arithmetic series is still dealing with addition (as it is Arithmetic) however it focuses on the sum of all of the terms within the pattern. A geometric sequence deals with constant multiplying as opposed to adding, and a geometric series one again focuses on the sum of all of the combined terms within the pattern.

3. Special Triangles:

I found special triangles to be quite interesting, I enjoyed their predictability and accuracy. We learned about 2 special triangles one being originally part of a square that had been cut in half diagonally (1:1: , the other from an equilateral triangle that had been cut directly down the Center (1:2:

4. Absolute Value:

At first I found understanding absolute value to be quite difficult, I had this concept explained to me in a few different ways and I think that the easiest way to understand it was to think of it on a number line, the absolute value is telling us how many units away from zero that value is located.

5. Completing the Square:

Completing the square is a “technique” used to convert standard equation forms into general (also known as vertex) form, this form of an equation gives you a very clear vertex point which can be very important when graphing. I personally found this completing the square method difficult at first, likely because there are a fair few steps to remember when applying this method.

Applications for Rational Equations are questions relating rational equations to real life situations and circumstances.

Hints/Steps:

1. Read through problem carefully to understand what is being asked

2. Decide where you will need to introduce a variable in place of an unknown number

3. Create an equation based on the given information

4. Solve the equation

5. Express answer in a full sentence and check that the solution makes sense for what was being asked

Note: there are 4 types of applications

1. Problems involving distance, speed, and time

2. Problems involving proportions

3. Problems involving motion

4. Problems involving work

Example:

GENERALLY

Trigonometry definition:

trigonometry can be defined as simply as the mathematics involving the relationships between side lengths and angles of a triangle.

New vocabulary:

terminal arm: the line found within any of the 4 quadrants, creating an angle

coterminal angles: angles that share the same initial and terminal sides

quadrantal angles: angles that lie directly on the x and/or y axis (ie. 0°, 360°, 90°, 180°, 270°)

Rotational angle: the angle between 0° and the terminal arm

reference angle: the angle measured between the closest seperation of the x-axis and the terminal arm.

LESSONS SUMMARY

6.1 General updates-

Calculating the reference angle:      ref angle= 180 – 92= 88°

calculating the coterminal angle: 

hypotenus=radius

sides= x & y

——>

6.2 Exact Trig Values

sin=y/r, cos=x/r, tan=y/x

To remeber in which quadrants each ratio is positive: CAST rule

Special Triangles

originally a square:

originally an equilateral triangle:

The Sine Law

this formula can be modified so that the variable for which you are solving (either an angle or side) can be solved in fewer steps.

The Cosine Law

To be used if there are no angles with corresponding side lengths given and thus sine law will not work

This week in chapter 7.5 we began solving Fractional Equations, when dealing with Fractional equations it’s important to remember that you must SOLVE as opposed to SIMPLIFYING, which is related to expressions as opposed to equations. This also means that after solving you should be left with a value that will be equal to x.

There are two types of Fractional equations, there are “simple” equations and there are “ugly” equations.

Solving Simple Equations:

When you are given two simple fractions you may CROSS MULTIPLY:  this is accomplished by multiplying each numerator by the opposite denominator this will get rid of the fraction and you will be left with a much simpler fraction-free equation, from there, you must isolate x.

Solving Ugly Equations:

when solving an “ugly equation, or an equation with either more than one fraction, or polynomial fractions etc… MULTIPLY BY THE DENOMINATOR: First determine the common denominator, then divide the common denominator by the original denominator from each fraction and then multiply the remainder by the numerator. Lastly, solve to isolate x (or any variables in the equation).

this method is suitable for any kind of Fractional equation (ie. it will always work).

This week we started the “Absolute value and reciprocal functions” unit. This first lesson focused mainly on the shape of the graph that will result from graphing an absolute value of both a line and a parabola.

When you graph the absolute value of an equation, any part of the line or parabola that may drop below the X axis (ie. is a negative value) will “flip up” to the positive section of the graph. When the previously negative section of the graph moves up into the positive section it should be COMPLETELY SYMMETRICAL to its previous position on the negative half of the graph and it resembles a “mirror image”. This is because the values are the same except positive, they are exactly the same distance from zero.

On a linear graph (has only a line, no parabola). There is one singular point where the graphs negative values are “flipped” so that they are positive, this point is referred to as the critical point, this causes the graph to look like a big “V” but this DOES NOT make it the same as a parabola, it is still a linear graph.

Similarly to graphing linear inequalities in 2 variables, the goal is to determine where on the graph is the solution “true”. When talking about linear 2 variable inequalities, the graph will be shaded one side of the line (the shaded side contains the solutions that make the equation “true”). However, since every quadratic equation will be graphed as a parabola instead of a line, the shaded are where the solutions will be found is the area found inside the curve of the parabola.

Steps:

1. Determine all that can be determined by the given inequality ie. opens up/down, congruent to___, the y intercept etc…

2. if the inequality is presented general form, Factor (to determine the axis of symmetry and thus the vertex), factor to find the zeros, then add the value of the zeros together and divide that number by 2, this is the axis of symmetry.

3. Graph the inequality.

4. Use a Test Point to establish weather the inside or the outside of the graph contains the solutions.

5. Shade the area where the solutions are found.

Example:

In an infinite geometric series, the series gradually and eventually diverges or converges, this means that there may not always be a determinable sum because the values will continue increasing and there will theoretically always be another value in the series; it’s never ending I.e infinite

Infinite vs Finite series:

In a Finite series, n will always have a determined or determinable value because it is converging. This value can be calculated using the formula . In an Infinite geometric series however, n is Infinite due to it’s converging or diverging nature and therefor cannot be determined using the same formula. Instead we use the formula as demonstrated below.

Converging:

12, 6, 3…

r=o.5

diverging:

2, 8, 32…

r=4

NO POSSIBLE SUM