Hi everyone! Minako here and going to talk about Degree and Radian Measure.

Radian and Degree are two units for measuring angles.


Initial side- is the starting point of the ray.
Terminal side- is where an angle stops.

If the rotation is positive then it is in clockwise direction
If the rotation is negative then it is in counter-clockwise direction.


Whenever radius is equal to its arc length, then theta will always be equal to 1 radian.
Whenever radius is not equal to arc length, then theta will be greater or lesser then 1 radian.


To convert degrees to radian
multiply degrees to Π/180 degrees

example: 270 degrees

To convert radian to degrees
multiply radian to 180/Π

example: Π/6

COTERMINAL ANGLES
-Two angles are coterminal if they are drawn int he standard position and both have their terminal sides in the same location.




Finding the positive coterminal angles:
Just add 360 degrees to the angle given.
(add 2Π if the given is in radian)


Finding the negative coterminal angles:
Just subtract 360 degrees to the angle given.
(for degrees higher than 360, subtract another 360 until the angle turns negative)
(subtract 2Π if the given is in radian)

QUADRANTAL ANGLES:
-An angle with the terminal side on the x-axis or y-axis. 
That is, the angles of  0°, 90°, 180°, 270°, 360°, 450°, ... as well as –90°, –180°, –270°, –360°, ...

Relative Minimum and Relative Maximum Value

Hey People! This is Chester and I'm talking about Relative Minimum and Relative Maximum Value.
 
Maximum and Minimum Value

Quadratic Functions:
 
- either have a maximum value or minimum value. It depends on how the function opens up.
- if the graph opens upward the function has a minimum value.
-if the graph opens downward the function has a maximum value.




Relative Maximum and Relative Minimum 

Cubic Functions:

- do not have neither maximum or minimum value
- but they may have relative maximum or relative minimum value





Absolute Minimum and Maximum Values

Quartic Functions:
- either have an absolute maximum or an absolute minimum value
- within intervals of the function, they may also have relative maximum or relative minimum value.




Multiplicity of a Zero

-  Multiplicity can be thought as "How many times does the solution appears in the original equation"
. For example, in the polynomial function f(x) = (x – 3)^4(x – 5)(x – 8)^2, the zero 3 has multiplicity 4, 5 has multiplicity 1, and 8 has multiplicity 2. Although this polynomial has only three zeros, we say that it has seven zeros counting multiplicity.

- If a graph has real zeros of odd multiplicity the function will cross the x-axis
- If a graph has real zeros of even multiplicity the function will bounce off the x-axis

example: 

f(x) = (2x+1)^2 (x-1)

- at x = -1/2 (zero of even multiplicity), the sign of the function does not change and it bounces off the x- axis

- at x = 1 (zero of odd multiplicity) the sign changes and the function crosses the x- axis

GOOD LUCK ON THE TEST!! 

Polynomial Functions

                                                           Polynomial Functions

Hey guys! It's April and im going to talk about polynomial functions. This blog will include the

characteristics of a polynomial functions, the special names for its degrees and how to graph the 

function.

Characteristic of Polynomial Functions: 

     Polynomial functions are written in the form f(x)=anxn + an - 1xn - 1 + ... + a1x + a0, where (n) cannot be a negative integer. Each part of the expression is referred to as a "term", and a term has coefficient associated to with it. In polynomial functions, the term that has the highest power of x is called "leading term", while its coefficient is called the "leading coefficient". The power of x is called the "degree" of polynomial.

Special names: Degree of Polynomials

   Degree                        Functions                             Example
   
       0                             Constant                                    f(x) = 2
       1                             Linear                                        f(x) = 2x + 2
       2                             Quadratic                                  f(x) = 2x^2 - 4
       3                             Cubic                                        f(x) = x^3 +3x^2 + x - 8
       4                             Quartic                                      f(x) = x^4 = 6x^3 - 4x^2 + 2x + 6
       5                             Quintic                                      f(x) = x^5 - 7x^4 + 2x^3 - 3x^2 + 2x - 4

Constant Functions example:


Linear Functions example:


Quadratic Functions example:


Cubic Functions example:


Quartic Functions example:


Quintic Functions example:


How to graph Polynomial Functions:
In graphing ODD degree polynomials: -when the leading coefficient is positive (+) ;
left arm will go down and the right arm will go up
- when the leading coefficient is negative (-) ;
left arm will go up and right arm will go down

In graphing EVEN degree polynomials:
-when leading coefficient is positive (+) ;
both arms will go up
-when leading coefficient is negative (-) ;
both arms will go down

How to match Polynomial Function with its graph:
-Define the type of the function, as well as the degree; is it even or odd?
-The end behavior
-Determine the number of possible x-intercepts
-Find its maximum or minimum value
-What is its y-intercept

Important Graphing Rules!!!!
1. If each root of the given equation is repeated or the same, the curve crosses at the x-axis.
2. If you have 2 or more of the same root...
a) even number of the same root - curve bounces at the x-axis
b) odd number of the same root - curve crosses at the x-axis
3. The higher the degree of the roots, the more the graph will flatten out.

Integral Zero Theorm and Factoring High-Degree Polynomials

Hi everyone,it's Ashriti
On Wednesday,we were taught the Integral Zero Theorem.
This theorem basically helps you know which integer values of a to try when determining if p(a)=0.Consider the polynomial x^3+3x^2-6x-8.If x=a satisfies p(a)=0,then x^3+3x^2-6x-8=0 or x^3+3x^2-6x=8.Factoring out the common factor in the left side of the equation gives the product x(x^2+3x-6)=8
Then,the possible integer values for the factors in the the product on the left side are factors of 8.They are +_1,+_2,+_4 and +_8.
So,if x-a is a factor of a polynomial function p(x) with integral coefficients,then  a is factor of the constant term p(x).
Example:
Factor   f(x)=x^3+3x^2-6x-8
f(x)=x^3+3x^2-6x=8
The possible integral zeros of the polynomial are:+-1,+-2,+-4,+-8
Now,first try it with the value of -1
f(-1)= (-1)(-1)(-1)+3(-1)(-1)-6(-1)-8
f(-1)= -1+3+6-8
f(-1)=0
So, (x+1) is one of the factors of f(x).
Now divide (x+1) by f(x)
On dividing it we will get x^2+2x-8
And then factor this again 
On factoring it we get:(x-2)(x+4)

So,f(x)=(x+1)(x-2)(x+4)

Factoring High-Degree Polynomials

When factoring high degree Polynomials we have to take the x-value common and move the constant over to the other side of the equation(on the right side of=).After this check for the possible values of the factor.It basically reduces the number of options for the factor.

Example:

Factor:
P(k)=k^5+3k^4-5k^3-15k^2+4k+12
P(k)=k^5 +3k^4-5k^3-15k^2+4k= -12
The possible factors are:+- 1,2,3,4,6,12
Now,P(1)=(1)(1)(1)(1)(1)+3(1)(1)(1)(1)-5(1)(1)(1)-15(1)(1)+4(1)+12
P(1)= 1+3-5-15+4+12
P(1)=0
So, (k-1) is a factor of f(k).
Now, divide f(k) by (k-1)
After dividing we will get k^4+4k^3-k^2-16k-12
Now,factor this again
(k+1) is one of the factors of above equation.
After factoring this we get (k+1)(k-2)(k+2)(k+3)
So, the given equation:
f(k)= (k-1)(k+1)(k-2)(k+2)(k+3)

I choose number 11 to blog next.

Hey guys, Steffi again. This time I'm going to show you how to divide polynomials by other polynomials in two ways. On Tuesday, we divided polynomials using the Long division method, and on Wednesday, we learned how to divide polynomials using the Synthetic Division.

Option 1: Long division

Ex. Divide x² – 9x – 10 by x + 1

Step 1: Set up the division; making sure that the variables in both the dividend and the divisor are arranged in descending powers.



Step 2:  Look at the leading x of the divisor and the dividend. Divide the leading x of the dividend (x^2) by the leading x of the divisor (x), then put the answer ( which is x) on top as part of the quotient.

  

Step 3: Multiply that x (newly found quotient) to the divisor using Distributive Law and put that under the dividend, and then subtract each term. 

(x^2 - x^2 = 0; -9x - (+1x) = -10x )

                        * When using Long Division, always SUBTRACT.

Step 4: Bring down the last term (-10) to form the new dividend.

Step 5: Repeat steps 2 and 3 until the remainder is a degree lower than the divisor.

      ex. x^3 is a degree lower than x^4; 10 (a constant) is a degree lower than x (a variable). 

        To check: Quotient * Divisor

                         (x -10) (x+1)

                      =  x^2 +1x -10x -10

                      =  x^2 -9x -10  [Dividend]

        Quotient:  x-10

            Roots:  x = 10 and x = -1   (from the quotient and the divisor)

Example with a remainder other than 0: 

     To check:  Quotient * Divisor + Remainder = Dividend

                      (x+2) (x+1) + 3

                    =  x^2 + 1x +2x +2 +3

                    =  x^2 + 3x +5    [Dividend]

    Quotient:  x + 2

        Roots:  x = -2 and x = -1

 Remainder:  3

Option 2: Synthetic Division 

Ex.  dividend/ divisor in form of (x-a)

Step 1: Arrange the coefficients of the dividend in order of descending powers of x (write 0 as the coefficient for each missing power) inside the division bracket. And write the divisor in the form of "x-a". Example:  x+3;  a= -3, since x-a = x -(-3) = x +3 . Then, place "a" outside the division bracket. 

 if a = -2, then x -(-a) = x-(-2) = x +2 [divisor]

Step 2: Bring down the first coefficient of the dividend and multiply it by "a", then add the product to the second coefficient of the dividend. Then repeat until the last term has a product.

* When using Synthetic Division, use ADDITION.

Step 3: The last number in the 3rd row is the remainder, while the other numbers are the coefficient of the quotient, which is a degree less than the dividend.

* Since the dividend has x^4, the quotient must have              x^3 (1 degree less)

   Quotient:  2x^3 -4x^2 +5x -6

Remainder:  3

Thank You and I hope that you have learned from this post. See Ya!

Hey guys, Steffi here. Sorry for the late post. Today I will be explaining to you what we have learned last Monday.

Going Backwards with Transformation

In the previous posts, you have learned transforming y = f (x)   to   y = 2 f  (x-3) + 2.

Today, we will learn how to graph it backwards from y = 2 f (x-3) +2   to   y = f (x). And to do that, we will do all the steps in reverse. 

Example (Graph): Consider the transformed function below as y = 2 f (x-3) + 2. How would             y = f (x) be shown?

Step 1: Figure out the transformation performed on y = f (x) to get y = 2 f (x-3) + 2
     

       X-values                                                      

• add 3 to all x-values

       Y-values

• multiply all y-values by 2
• then, add 2 to all y-values 

Step 2: Reverse (do opposite) the transformations and its order, so that we could get f (x)

       X-values

• subtract 3 to all x-values

       Y-values

• subtract 2 to all y-values
• then, divide all y-values by 2 or multiply by 1/2

Therefore, f (x) would look like this:

To check if f (x) is right, get all the points of f (x) and then perform all the transformations ( in step 1) to get 2 f (x-3) +2.

* Since the values in the table for both y=f(x) and y=2f(x-3)+2 match their corresponding graphs, then the graphed f(x) must be right.


Example (Algebraically): If y=3f 2(x-1) -1 = (3,-4), then what is f(x)?

In this example, you are given coordinates for y=3f 2(x-1) -1, so you are trying to find the coordinates for f(x). The steps for this example are stated above.

Step 1:   x-values:                                          

• multiply by 1/2
• then, add by 1

             y-values:

• multiply by 3
• then, subtract by 1

Step 2: Do the Reverse and apply to get the coordinates for f(x)

             x-values:                                                    

• subtract by 1                 ------>   x=3  ---->  3-1 = 2(2) =  4
• then, multiply by 2

             y-values:

• add by 1                       ------>  y= -4  ---->  -4+1 = -3/3 =  -1 
• then, divide by 3

Therefore,   f(x) = (4,-1)

Thank you.. I hope this can be a help to you... 

PRE-CAL IS LIFE     jk!

Hi Classmates, I'm Jr. Malig

and today i will explain how Reflections and Stretches works.

there are 2 Stretches the Vertical and Horizontal

Vertical Stretches

we call "a" as Vertical Stretch

(x , y) translate to (1 x , y) or (b x , y)
                               a

Horizontal Stretches

we call "b" as Horizontal Stretch

(x , y) translate to (1 x , y) or (b x , y)
                               b

Reflections                                                
This is the graph of a function y = f (x) and how the graph of -f (x) reflected to the original set of axes.


                                               
















Inverse of a Function

Inverse function is when the y in the equation of a function y = f (x) reflected in the mirror line y = x.

Putting It All Together

 this formula can be applied to any function

*y = a f ( b ( x + c ) ) + d*        the " + " inside the formula will change into " - "      y = a f ( b ( x - c ) ) + d


The "a" is located outside the function and it affects the y- values of the graph and also the "d ". therefore the effect on the graph is just the same.

while the "b" and the "c" are located inside of the function and it will affects the x - values of the graph, but the effect on the graph will be opposite.

The main thing you have to remember here is:

Outside the function -----> affects the y - values ----> The effect is the same.

Inside the function ------> affects the x - values ----> The effect is the opposite


THANK YOU!