Tuesday, January 19, 2010

Polynomial Fuctions

Hey guys,
this post will cover just some of the basics of polynomial functions. This post will focus on the divisioin of polynomials through long division and Factor Theorum.

Using Long Divison:

As many of you should alrady know how to divide, dividing with exponents isn't that much harder. So lets start with this example first. Use long division to divide 36x^3 - 6x^2 + 3x - 33 by 3x + 1 (Note: BY 3x + 1 is what they want you to divide with). So how do we do it?

1. Write down the question first
2. Like you would do for normal divison, ask yourself how many times can 3x go into 36x^3? (Answer: 12x^2)
3. After finding that out, treat the answer you found in #2 like 12x^2(3x + 1) in order to progress.
5. Bring down the 3x from the question.
6. Repeat the same processas you did for 2. (how many times does 3x go into -18x^2? Answer: -6x)
7. Repeat steps 3, 4, 5, 6.
8. Once you have reached the remainder (-30R), you can now write down the division statement (Dividend) = (Divisor)(Quotient) + Remainder

Factor Theorum:

The next thing we'll look at is how to use factor theorum to solve for a polynomial (FYI: MUCH fast and WAY easier).

If you are given a question such as: Factor x^4 + 4x^3 - x^2 - 16x - 12
So how do you do it?

1. To find the first zero, find a number for x which can cause the polynomial to equal to zero (in this case -2)
2. Your new polynomial should look like (x+2)(x^3 + 2x^2 - 5x - 6) But your not done yet.
3. Repeat step one, your zero's should now become more clear. (x + 2) (x + 3) (x^2 - x - 2)
4. Finally, factor out x^2 - x - 2 (product/sum)

Well, that's about all you need to know about long division of polynomials and factor theorum. Hope this helped.

Review: Trigonometic Functions

Trigonometic Funtions Part 1

• to convert degrees to radians, multiply degree by (/180)

Special Angles

• you can use the unit circle to solve
• use special angles with unit circle

Equivalent Trigonometric Expressions

• use the cofunction identity sinx = cos (/2 - x) to determine equivalent trigonometric expressions

Compond Angle Formula

• apply equivalent trigonometric expressions to compund angle formula
• determine exact trigonometric ratios for angles expressed as sums or differences or special angles

Identities

• treat each side independantly and transform expression on one side into exact form of the other

Trigonometric Functions Part 2

Sine, Cosine, and Tan Graphs
• y=sinx and y=cosx graphs have amplitude of 1 and a period of 2 and can be transformedy=tanx has no amplitude and no maximum or minimum

Reciprocals

• reciprocals are different from it's inverse
• csc means 1/sinx
• sec means 1/cosx
• cot means 1/tanx

Mapping Rule

• g(x) = af [k(x-d)] + c
• a represents amplitude
• d represents phase shift
• c respresents vertical translation

Period

• 2/k

Amplitude

• max - min / 2

Rational Functions: Exam Review

So Exams are coming up and basically this post is going to cover some things you need to know for Rational Functions. Things such as key features and skills you need to have when dealing with rational functions.

Rational Functions

f(x) = p(x)/q(x), q(x) cannot equal 0

f(x) = 1/x this is a simple rational, it is also the reciprocal of a linear functions.
therefore...

f(x) = 1/x^2 is reciprocal of a quadratic f(x) = 1/x^3 is a reciprocal of a cubic and etc.

Special Skills Required

Things you would need to know when dealing with rationals are...
- Finding a Vertical and/or Horizontal Asymptote
- Find any y-intercepts, zero's, holes (if there are any)
- State the Domain and Range
- Graph a rational function
- Factor into factored form

Key Features
- As stated above, the denominator cannot equal 0, anything being divided by 0 causes problems.

Horizontal Asymptote (H.A.)
- if n < m, H.A. is y =0
- if n = m, H.A. is y = coefficient of x^n/ coefficient of x^m
- if n > m, there is no H.A.

Vertical Asymptote (V.A.)
- find the zero's of the denominator

Y-intercept
- Lets x=0

X-intercept
-Let y=0

Example

f(x) = x^2 - 3x - 10 / x^2 + 9x + 14

Factoring
f(x) = x^2 - 3x - 10/x^2 + 9x + 14

Quadratic formula on top and bottom
f(x) = (x-5) (x+2) /(x+7)(x+2)

the (x+2) cancels each other out
f(x) = (x-5)/(x+7) , x cannot equal 7

Horizontal Asymptotes
- n = m so...
y= coefficient of x^n/coefficient of x^m
y= 1/1
y= 1

Vertical Asymptotes
- solve for the zero's of the denominator
x= -7

Y-intercept
-let x = 0

y= (0-5)/(0+7)
y= -5/7

X-intercept
-let y = 0

0= (x-5)/(x+7)
0= x-5
x=5

Holes
x= -2

Graph

Domain and Range
D= {x| x cannot equal -7, xE R}
R= {y| y cannot equal 1,> yER}

Saturday, December 26, 2009

7.2 Solving Exponential and Log Equations

Hey Guys,
This post is going to cover chapter 7.2, which we covered in class on Wednesday Decmber 16th 2009.
Most of the lesson was based on Radioactive decay and powers of different bases and applying quadratic formula.

Radioactive Decay is the process by which element transforms into a different element. This can be modelled by the following equation:
A(t) = A0(1/2)^t/h ,
where A(t) represents the mass of a substance, Ao represents the initial amount, t represents time, and h represents half life.

In order to solve this equation, 3 variables are needed, so you may isolate the 4th and missing variable.

For more help and for an example of this problem, check out pg. 370 of the textbook.

Powers with Different Bases
When solving powers with different bases you use an algebraic reasoning, where you take the logaritm of both sides and apply the lower law of logs to remove the varibles from the exponent.

Ex: 4^ x+1 = 64^2x
log (4^x+1) = log (64^2x)
(x+1) log 4 = 2x log 64
x+1 = 2x log 64
log 4

x+1 = 2x (3)
x+1 = 6x
1= 5x
1/5 = x

The third method we used to solve is using the quadratic formula

example:

2^x – 2^-x = 4

To use the quadratic formula to solve this equatation, you must multiply both sides by 2^x so that a quadratic equation is obtained in terms of 2^x

2^x(2^x – 2^-x) = 2^x(4)
2^x(2^x) - 2^x(2^-x) = 2^x(4)
2^2x – 2^0 = 2^x(4)
2^2x – 1 = 2^x(4)

Then apply the power of a power law to the term 2^x:

(2^x)^2 – 1 = 2^x(4)
And then write in standard form (az^2+bz+c = 0)
(2^x)^2 – 4(2^x) – 1 = 0

Then you apply the quadratic formula to determine the roots.
This will result in 2 + sqrt(5) and 2 - sqrt(5).
Alternatively, you could have set b = 2^x and then solved for b which would have yielded

2^x = 2 + sqrt(5) &

2^x = 2 - sqrt(5)

At time point you must use the logarithm of both sides by using the power law of logs, and then dividing both sides by log2. This will yeild 2.08 which would be one root of this equation.
The second equation however yeilds a number smaller than 0 which results in no answer since a power must always be a positive number.

Hope that helped ! :)

Monday, December 21, 2009

6.1 The Exponential Function and its Inverse

This is our second last unit of our Advanced Functions course. YAYYYY ^^
In the lesson, it will be separated into two parts. The first part will talk about the exponential functions and the second part will talk about its inverse.

Since this is the first lesson of the new logarithmic unit, most of the information presented should be learned by students.

PART 1- EXPONENTIAL FUNCTIONS

In part 1, we will learn how to solve for the exponent with different bases. The "log" function on your calculators will be used greatly in this unit.

Side note: the default base of the "log" key on your calculators is ALWAYS 10.

PART 2- Inverse

In part 2, we will be pretty much doing the same thing as grade 11. Switch x and y in the same equation and isolate for X again.

Exponential functions can be represented in 3 DIFFERENT ways.

1: General Form- the simplest form of representation of an exponential equation

y=5x or y=1/5x

2. Table of Values- a table which a list of the "x" values with the corresponding "y" values

REMEMBER: the first ratio of differences of the x and y values are always constant.

How do we find the first ratio of differences?

The values of : Y2-Y1 = answer.

3: Exponential functions- this is a graph representation of the exponential functions. However, just by looking at a graph cannot help you determine whether an exponential function is or not.

Rate of Change:

To determine the rage of change of an exponential function is really much similar to finding the rate of change of any other functions.

1) Pick the x values that are really close to the value you are given to find the IROC.

2) For example: if the value that is given is 2. You may go to the left of the value or right to the value which results in x = 2.0001 or x = 1.9999.

3) Sub these values back into the original equation and find the “y” values

4) To choose the two closet points that lead to the given value will give you the best and most accurate slope of secant that is required for the answer.

5) After having the “x” values and the “y” values. Put it into the general equation used to find slopes which is:

AROC [x value – x value] = y2-y1 / x2-x1

6) To conclude your answer, you must write, “Therefore, the IROC of the given value is approximately at “answer of AROC of the 2 x values”.

Inverse functions:

The graph of the inverse function of a logarithmic function is a reflection on the y= x function.

To find an inverse function of a logarithmic function, switch x and y from the equation and solve for x again.

By switching (x,y) -> (y,x) , these can be your new inverse values that can be used to apply transformations.

Differences and similarities between an inverse function and an exponential function

1) A exponential function will have an increasing slope while the inverse function will have an decreasing slope

2) A horizontal asymptote (y = a value) is present in an exponential function. A vertical asymptote (x = a value) is present in the inverse function

3) Depending on the transformations, the horizontal asymptote or the vertical asymptote is affected by the vertical or horizontal shifts.