5.4

Hello everyone!

So as far as five four goes, it’s just building off of five three’s ideas. To start it all off, let’s review:

Last section told us that log has a subscript 10 (common log) and that ln has a subscript e (natural log). Now that we have that covered, moving onto the…

Law of Logarithms.

1) log_a(uw) = log_au + log_aw

2)log_a(u/w) = log_au - log_aw

3)log_a(u^c) = c log_au

Easy enough, right? Now to le comparisons.

Common Logarithms	Natural Logarithms
log(uw) = logu + logw	ln(uw) = lnu + lnw
log(u/w) = logu – logw	ln(u/w) = lnu – lnw
log(uc) = c logu	Ln(uc) = c lnu

There are no differences in the way you play out these logarithms, just differences in the meanings.

BUT, BEWARE. THERE IS A SLIGHT CATCH. Log_a(u+w) does NOT translate to log_au + log_aw. Also, log_a(u-w) does NOT translate to log_au – log_aw. This isn’t like f(x) in those ways at all!

     ^ You can’t have a log of a negative number, that’s why -4 doesn’t work.

Onward again, 

So I think that’s about it, I hope this helped!

Have a great weekend!

P.s. COME TO FALL PLAY TOMORROW AT 2 PM. MR WILHELM WENT TONIGHT(sorry I didn't say hi to you) SO YOU SHOULD GO TOMORROW.

Kristy :]

Section 5.3 - Logarithms!

This section was on logarithms! Logarithms are basically the inverse of an exponential equation. Literally, it means ratios of numbers.

exponentiation is what is being done to X in this case.


To solve, we need to undo what is being done to X, which is a logarithm :)

Mr. Wilhelm's KEY TO EVERYTHING!

That is how to rewrite an exponential function as a logarithmic function. With this, we can solve exponential equations that would otherwise be unsolvable!

This is how you might apply a logarithm to an exponential function. As you can see, Y=1,000  A=10   and X=3

The key to everything is as easy as that :)

Special Logarithms!

When it is just written "log" with no subscript, it is implied that it is log10 -- COMMON LOG!

When it is written ln, it is implied that it is loge -- NATURAL LOG!

How to graph a logarithm!

A few other little tidbits to know:

since the domain of an exponential function is (-∞, ∞), so thats the RANGE of a logarithm

the range of an exponent is (0, ∞), so thats the DOMAIN of a logarithm. (unless it has been moved by the C value)

*Exponents and Logarithms are 1 to 1*

Sorry, there are no pretty logarithm pictures on the internet :(

Links and stuff:

http://www.themathpage.com/aprecalc/logarithms.htm

http://www.sosmath.com/algebra/logs/log1/log1.html

http://www.purplemath.com/modules/logs.htm

http://www.shodor.org/unchem/math/logs/index.html

http://www.youtube.com/watch?v=mQTWzLpCcW0

And, just for old time's sake :

Nonagon Song - http://www.youtube.com/watch?v=x5ohtlewREI

One Dozen Monkeys -http://www.youtube.com/watch?v=2qdql9vsWWM&feature=related

Hope this helped and made logarithms a little less unbearable :)

Bye, Katie 

3.7 Operations on Functions 10/18/11

In this section, we looked at defining functions using various operations and combining multiple expressions using new methods.

First, we will take a look at the basic operations of functions: sum, difference, product, and quotient.

Sum:

---->  


Difference:

  ---->  


Caution: Remember to distribute the negative when inserting the g(x) expression.

Product:


  ---->


Quotient:


    ---->  



A sample of the sum method:




Find
Since this^^ = 
Then:






Next, we began to look into compositie functions.

The composite function  f  ° g  of two functions f  and g is defined by:

(f  ° g)(x) = f(g(x))


The domain of  f  ° g is the set of all x in the domain g such that g(x) is in the domain of f.

An example of a composite function problem:

Let  and 

(f  ° g)(x) 

(f  ° g)(x)

(f  ° g)(x)

(f  ° g)(x)

Domain problems are also common in this section, so be ready to find the domain of functions like we did in previous chapters.

Hope this helped!

-julia

3.6: Quadratic FUNctions

QUADRATIC FUNCTIONS are those funky-looking curved graphs with equations that have multiple xs and confusing exponents in them.

A function f is a QUADRATIC FUNCTION if and only if it fits in the form where a, b, and c are real numbers with a not equal to zero.

What can you figure out from a QUADRATIC FUNCTION?

a: if positive, the funtion opens up; negative, the funtion opens down

it controls the width of the graph

b: controls how the vertex moves along the inversion of the function

c: the y intercept

VERTEX: The minimum or maximum value of the function (minimum if a>0; maximum if a<0)

The coordinate of the vertex of a parabola is

AXIS OF SYMMETRY: That dotted line down the middle of the parabola that runs through the VERTEX.

on the right is the STANDARD FUNCTION

of a parabola with Vertex V(h,k)

You can make your ugly QUADRATIC EQUATION into a fashionable STANDARD EQUATION by simply completing the square, as demonstrated on the right.

Functions

A function is relation that assigns exactly one value in the Range to each value in the Domain.

This does not mean you cannot have multiple values in the Domain assigned to one value in the Range.

The values in the domain are called elements, while the values in the range are called correspondents.



The Domain (or elements of the domain) can also be called:

• X
• Input
• Independent Variable

The Range (or correspondents of the range) can also be called:

• Y
• Output
• Dependent Variable

Y= f(x)

Note: Anytime you see "y-squared" it is not a function because it will have 2 x values for every y value. Also, a circle cannot be a function because it includes "y-squared".

Graphing functions can result in many different types of graphs such as:

Linear Function:

Hyperbola:
Parabola:

Trinomial:

and many others!

So... that's functions... I don't know what else to say so... yeah... I'm gonna stop typing now... and you're just gonna have to deal with it... yeah... well... YOU LOST THE GAME!

3.1 to 3.3

Hi. The stuff covered in 3.1 to 3.3 is pretty much all things we have already learned. The first concept covered is the Cartesian Cordinate system with a x-axis, a y-axis, and four quadrants pictured below.



The next thing covered is the distance formula . The distance between any two points (x₁, y₁) and (x₂, y₂) is



The distance formula can be proved drawing a right triangle between points and finding the length of the hypothenuse using the Pythagorean Theorem as shown below.



Next we learned about the Midpoint formula. The midpoint of the line segment from (x₁, y₁) to  (x₂, y₂) is

This can be proved because the x-cordinate of the midpoint is equal to the average of the x-cordinates of the endpoints. The same goes for the y-cordinate. That's pretty much everything in 3.1.

3.2 deals with the graphs of equations. We went over some basic terminology.
     Solution: And ordered pair that yeilds a true statement
     X-intercepts: The x-cordinates of the points where a graph intersects the x-axis
     Y-intercepts: The y-cordinates of the points where a graph intersects the y-axis
You can find the x-intercept of a function by substituting the y value with 0. You can find the y-intercept by substituting the x value with 0.

The next thing covered is graphs of circles. The standard equation of a circle with center (h,k) and radius r is

Keep in mind that circles are not a function because they don't pass the vertical line test. If you want to graph a circle on your graphing calculator (which only graphs functions) you can graph two semicircles. All you have to do is solve the equation for your circle for y, which will lead you to two answers (±). For y1 enter the positive value of the equation and for y2 enter the negative value.

3.3 goes over lines

The slope of a line is defined by the formula

If the line is parallel to the y-axis, then the slope is undefined. A line parallel to the x-axis has a slope of 0.

Another important equation is the Point-Slope Form. An equation for the line through point  (x₁, y₁) is 

y – y₁ = m(x – x₁)

The books also talks about standard form (ax+by=c) which Mr. Wilhelm said we don't have to know.

We also went over relationships between lines.

Two nonvertical lines are parallel if and only is they have the same slope and two lines with opposite reciprical slopes are perpendicular.

Well thats pretty much everything. Bye.

     -Corn Murphy

1.2- Exponents and Radicals

Yesterday we learned about the rules of exponents. These rules are as follows:

The quotient rule above is the example for when the larger exponent is in the numerator, but if the larger exponent is in the denominator you would put the base 'x' to the power of n-m all over one. That would look like this 

This next picture is an example of the work that lets you get from to , where .

       

We also learned how to change exponents into radicals and vice versa. But before I get into that I will go over the names of a radical's parts.

Now, I will go into making exponents into radicals, and radicals into exponents.

This shows where the exponent's numerator and denominator go when it's made into a radical.

This shows where the radical's index and the radicand's exponent go when it's made into an exponent.


The diagram below shows some examples of turning radicals into exponents:

Hope you like the post and sorry its so long.

Bye,

and Go Blue,

Alex Hackert