Degree and Radian Measure 1

Hi! My name is Jamie, to those who don't know me, I am the second person sitting in the row near Mr. P's desk. 

Last Thursday, Mr. P introduced to us the Degree and Radian Measure 1, which he also recap earlier in our afternoon class today. 

NOTE: "CIRCULAR FUNCTIONS" IS AN IMPORTANT UNIT of PRECAL40S and probably the hardest one. (maybe?)


An angle is determined by two rays:

                Initial side - the starting point of the ray 

                Terminal side - the position after rotation

We define an angle of rotation by  rotating a ray about its endpoint, creating a vertex.

Angles can be measure using different units, such as:

1. REVOLUTION
2. ROTATIONS
3. DEGREES
4. RADIANS
5. GRADIANS

NOTE:
     Angle measures without units are considered to be RADIANS.
     Angle measures in degrees MUST show the DEGREE SYMBOL.
     OBTUSE ANGLES measures GREATER than 90°.
    ACUTE ANGLES measures LESS than 90°.

*If the rotation of the ray is in COUNTERCLOCKWISE direction, the angle is considered POSITIVE.
   If the rotation is in a CLOCKWISE direction, the angle is considered NEGATIVE. *

 An angle in a coordinate plane is said to be in standard position if:

• Its VERTEX is at the ORIGIN
• Its INITIAL SIDE is on the positive x-axis (Quadrant one)

A REFERENCE ANGLE is the measure of the angle between the TERMINAL ARM and the x-axis

please excuse my messy notes :))

Radian Measure
Radian Measure is the first method that we're going to learn on how to measure the angles in a circles. 
When the arc of a circle has the same length as the radius of a circle, the measure of the central angle that intercepts the arc is 1 radian. 

To find the arc length of a circle, we will use the formula a=θr
               a = arc length
               r = radius
               θ = angle measured in radians

To convert degrees to radians, we multiply the degrees by π/180°.
To convert radians to degrees, we multiply the radian measure by 180°/π.

Translate revolution to degrees, exact radian measure and approximate radian measure.

Example 1: Draw each angle in standard position. Change each degree measure to radian measure and each radian measure to degree measure. Give answers as both exact and approximate measures (if necessary) to the nearest hundredth of a unit. 

A.) for letter a the given measure is a degree number, so we have to convert it into radian measure by multiplying the degree by π/180°.

                    30°  x  π/180° 
                    30°  x  π/180°6 = π/6
                                        π/6  = 0.523598776 
                                        π/6  = 0.52 

(same method for letter B)

π180π180C.) For letter c. the given measure is -60°. we will also do the same step, 

                   -60°  x  π/180° 
                   -60°  x  π/180°3 = - π/3
                                      - π/3  = - 1.047197551
                                      - π/3  = - 1.05

     but the difference in this example is the sign of the degree angle which shows negative, Therefore, the standard position will rotate in clockwise direction.  

*NO NEGATIVE ANGLES*
e.) In letter E, Mr. P gave example on how to measure without using formula.

g.) we are given a approximate radian measure, so we have to convert it into degree measure by multiplying  180°/π.

               1 radian  x  180°/π = 57. 29577951°

                                                 = 57.30°

I.) 5π/4 is an exact radian measure. It is the same step when converting radian measures to degree measures but with denominator. 

*Here's an easy method for measuring exact radian measure. 

take first the exact radian measure without the numerator and convert it to a degree.

after that, we can just simply multiply the degree with the numerator. 

Letter L is an example of Quadrantal Angle. It is when the angles of the standard position where the terminal side lies on the x or y axis. 

I hope I covered most of our first lesson in Circular functions unit. 
might as well share my art blog here :)
[ps. hope you guys don't mind my poor grammar -_- ]

Model and Solve Problems Involving Polynomial Functions

I try to draw cute, but that doesn't work out so well...

Good morning!

As for the reason why I'm up and about at 1:31 AM is beyond me, and I'm pretty sure I'm going to regret this later. So Mr. P, if I look like I'm going to pass out -- you will know why. Anyways, my name is Charina. For those of you who are still unaware of my apparent existence in your classroom, I'm the girl with the indigo travel mug with the nice-smelling tea and it's now my turn to teach all of you some math!

I had a hunch that I'll be doing the scribe, because the day before yesterday I randomly pulled out Word and started thinking about what I would say if I were to scribe. I wager it's because I'm on hiatus on my actual blog (yes, I blog) -- and the whole nostalgia that comes along with it.

Putting those aside, it's time to get serious. 

Yesterday, we finished what was left of the Polynomial Functions booklet which was "Model and Solve Problems Involving Polynomial Functions" which is, in layman's terms -- problem solving and graphing it. If you happened to be absent or didn't really understand what Mr. P was doing, then it would be wise for you to keep reading.

This sheet!

Should you have done Assignment #2 of this unit, figuring this out should be a little easier. If I recall well, there was a word problem on the second page where you have to increase the size of an object -- if you had that worksheet, then this problem is basically the opposite of that very question. This time, the situation calls to decrease the size:

Bill is preparing to make an ice sculpture. He has a block of ice that is 3ft wide, 4 ft high and 5ft long. Bill wants to reduce the size of the block of ice by removing the same amount from each of the three dimensions. He wants to reduce the volume of the ice block to 24ft^3.

a.) Write a polynomial function.

b.) How much should be removed in each dimension?

The first part wants us to write a polynomial function, but before that, we must figure out what numbers are we dealing with. Bill has a block of ice: 3ft wide, 4ft high and 5ft long. Does this remind you of something? It should. It's volume: Length x Width x Height! Since Bill wants to decrease the volume to 24ft^3, it will now be our new volume to solve for.

Our main goal is to find out what X is, however, to do that we must come up with the polynomial function. Using what we're given and what we are trying to accomplish, we'll use these facts write out our factored form:

We write out our factored form as ( 5- x ) ( 4 - x ) ( 3- x ) because it said in the problem that "Bill wants to reduce the size of the block of ice by removing the same amount from each of the three dimensions" therefore, we have to subtract an unknown ( x ) amount to each dimension. With 24 being our goal volume, this equation is telling us that (5 - unknown)(4 - unknown)(3 - unknown) will equal to 24.

I'll be honest and say that Mr. P's method was really confusing, however, I developed another way to solve this problem -- using the way I solved the word problem for Assignment #2. It works, so why not? 

First, we need to get the polynomial. How do we do it? We FOIL. After the FOIL-ing is finished, you get your polynomial equation. Hence, voila. 

Still remember Integral Zero? Good for you, because you'll need it in my method. What you have to do is to get the factors of the constant as shown above, and substitute X for a factor. Instead of looking for a zero, we're going to look for a factor that when it's plugged into the equation, it gives us 24. Yes, it's trial and error, but lucky for you because the answer is 1.

See?

Now that we know what X is, we also now know by how much each dimension should be decreased by to get it down to 24ft! Each dimension should be decreased by 1 in order to get to 24ft^3 of volume. However, that is not all -- we still have to graph it. But how do we graph it? Our x-intercepts! Be certain not to forget: ( 5- x ) ( 4 - x) ( 3- x ). Your y-intercept is determined by multiplying 5 x 3 x 4 = 60.

This is my sexy graph, which John criticised.
How dare you, you don't even go here.

Before I wrap up this post, I will give you the full list of the table that needs to be filled -- that I hope you guys know how to do for the test on Monday.

Degree: Cubic / X^3  (due to having three intercepts + it's on the equation)

Leading Coefficient: Negative (negative sign at the front of the equation)

End Behaviour: Quadrants 2 & 4

Zeros/ X-Intercepts: 5, 4, 3 (see above)

Y-intercept: 60 (see above)

Intervals: +ve = (infinity, 3] , [4, 5]

                      -ve = [3, 4] , [ 5, infinity)

I must rest now, before I turn into a walking zombie. See you!

Graphing Polynomial Functions

 Hi, my name is Anton, hopefully Mr P had a fantastic birthday weekend :)

Today I will be showing you guys how to graph polynomial functions. There are a few characteristics for graphing polynomial functions. The type of function and the degree (even or odd), the end behavior, number of x and y intercepts, the maximum or minimum value.

To sketch the graph of  polynomial function, use the x and y intercepts, the degree of the function, and the sign of the leading coefficient.

The x intercepts are the roots of the functions.

Determine the zeros (x intercepts) of a polynomial function from the factors.

Use the factor theorem to express a polynomial function in factored form.

For Graphing Odd Degree Polynomials
Factor the equation and look for the x intercept

If the leading coefficient is positive, the left arm will go down and the right arm will go up.

If the leading coefficient is negative, the left arm will go up and the right arm will go down.

Plot the zeros and the y intercept on the graph and draw the line the connects the end behavior, zeros, and y intercept.

To look for the y intercept, simply multiply the numbers in the bracket.

In this case (x+2)(x-1)(x-3) ..... (2)(-1)(-3)=6

So the y intercept is 6

For Graphing Even Degree Polynomials
Factor the equation
In this case its already factored out.

y=(x+1)(x-4)(x+5)(x-2)

If the leading coefficient is positive, both arms will go up.

If the leading coefficient is negative, both arms will go down. 

Plot the zeroes and the y intercept on the graph and draw the line that connects the end behavior, zeros and y intercept.

 (1)(-4)(5)(-2)=40

 So the y intercept is 40

Important graphing rules

1. Each root of the equation is unique (i.e. There are no two or more of the same root) meaning there's only one root. The curve crosses at the x-axis at this point

2. If you have two or more of the same root:
 a) An EVEN number of the same root, the curve is tangent meaning it will bounce off at the x-axis.
      For example: (x+1)², (x-3)⁶, (x-4)⁸

 b) An ODD number of the same root, the curve will cross at the x-axis
      For example: (x-3)³, (x+2)⁷, (x+5)⁹

3. The higher the degree of multiple roots, the more the graph flattens out at the point

Here are some examples:

Hopefully everyone had a good long weekend and once again happy belated to Mr P (: