4.5 Graphs Of Sine And Cosine Functions

When using a calculator,make sure that you are in radians when graphing!!!!

Sine

When y=sin x

• The range of the function is [-1,1] 
• The domain of the function is negative infinity to infinity.
• the period is 2╥ 

The sine graph is reflected about the origin.  This is further proof that sine is an odd function.

Cosine

When y=cos x

 

• The range of the function is [-1,1] 
• The domain of the function is negative infinity to infinity.
• the period is 2╥

 The cosine graph is reflected about the y-axis.  This is further proof that cosine is an even function.

Changes To The Graphs

y=a sin(bx-c) +d
y=a cos(bx-c) +d

Changes In A

Changes in a affect the amplitude

The amplitude is half the distance between the maximum and minimum values of the function
The larger a is, the larger the amplitude will be.
When a is negative the graph is reflected about the x-axis

This is the graph of y=2 sin x

Changes In B

Changes in b affect the period

If  b is between 0 and 1  the period will be greater than 2╥

The graph is horizontally stretched.

If b is greater than one the period is less than 2╥ 

The graph is horizontally compressed.

If b is negative than we can use sin(-x)=-sin x and cos(-x)=cos x to figure out what the new equation and graph look like.

 

 This is the graph of y=(-4x) compared to the graph of y=sin x

Changes In C And D

Changes in c and d have to do with horizontally and vertically translating the graphs.

When c is positive the graph is horizontally translated to the left

When c is negative the graph is horizontally translated to the right

When d is positive the graph is vertically translated to the right

When d is negative the graph is vertically translated to the left

This graph shows how each variable changes the graph.

3.2 Logarithmic Functions

Since the graph of an exponential function passes the horizontal line test, it must have an inverse.  The inverse of an exponential function is a logarithmic function.

The logarithmic equation can be read as "y equals the log base a of x".  This means that a to the yth power equals x.

Logarithms have properties:

 

if,  then

The following features of exponential graphs are the corresponding features of logarithmic graphs.

Exponential                                   Logarithmic

Horizontal Asymptote                   Vertical Asymptote

Y-Intercept                                    X-Intercept

Domain                                          Range

Range                                            Domain

ln x is log base "e", also called the natural log.  "e" is simply a number equaling 2.7182....etc. 

Logarithmic functions can be transformed.

-a vertically stretches the graph.

-b is the base.  The bigger the base, the slower the graph grows.  If the base is less than one, the graph     flips over the x- axis.  The base can't be negative.

-c shifts the graph left and right.

-d shifts the graph up and down.

Have fun!!!!

3.1 Exponential Functions and Their Graphs

Hello all!
Over the chapters and through the tests, to Chapter Three we go...

To start out, let's be like Mr. Wilhelm and slip in some definitions. What are exponential and logarithmic functions? They are otherwise known as transcendental functions; they can't be expressed in terms of algebra.

An Exponential Function is denoted by;

BUT:
0<a 1 and x is any real number.

Graphing
For example purposes, the two functions we'll use are  and .

Note that as the base number gets larger, the graph increases more rapidly. (Sorry about the scale for the y-axis being different.)
So let's go over the basics for this find of function;
-Domain: 
-Range: 
-Intercept: 
-Increases
-The x-axis is the horizontal asymptote
-Continuous

But what happens if you do something crazy, like put -x as the exponent? Well, what a great question you have...
Let's use the same equations but make -x the exponent.
 and  

Woah! The graph goes the OPPOSITE way! Crazy! But the same general idea holds true; the larger the base, the more rapidly the graph decreases.
Just like regular algebra, a function with a negative exponent can be written as a fraction instead. For example;  can also be written as .
The basics for this kind of function are...
-Domain: 
-Range: 
-Intercept: 
-Decreasing
-The x-axis is the horizontal aymptote
-Continuous

What to do next...hmm...transforming the parent function sounds fun!

So many possibilities, let's go alphabetically.
When you change a, the graph has a vertical stretch(or compress depending on whether 1>a or a>1).
Changing b horizontally stretches/compresses the graph(unless you make it negative, in which case it flips over the x-axis).
If you were to change c, it would horizontally shift the graph left/right.
Lastly, the changing of d would result in a vertical shift of the graph.

Natural Base
We can't forget about our friend, e. This little guy is known as the natural base, and it's function is . This e is in fact an irrational number that is approximated to 2.71828... but luckily, your calculator should have this as a key so you don't have to type all of the number in. Just be sure you realize that e isn't a variable, e has value just like 1 and 2, just not as exact.You plug e into your calculator just like any other number.
As far as graphing functions of e, all of the ideas we already went over apply. Just instead of using a constant, put e in as the base. Everything else is the exact same.

Compound Interest
 This is fairly straightforward, as long as you know what the formulas and variables are. For n compoundings per year, the equation is  .
-A stands for balance in an account
-P is principal
-r means annual interest rate(expressed as a decimal)
-n is the number of compoundings per year
-t stands for the amount of time in years

For continuous compounding, the equation is .
The variables are all the same as in the previous formula, just don't forget that e is an actual number, NOT a variable.

I'm pretty sure that covers the section. If my post doesn't answer all of your questions, ask a buddy, be a book-licker or ask the all knowing, Mr.Wilhelm.

3.3 Properties of Logarithms

Hello everybody! It's Eleni.
Recently, we've been learning about logarithms.  There are three properties of logarithms.


1. Logarithm Coefficients

Example:

2. Addition

ln (bc) = ln b + ln c
 log_a (bc) = log_a b + log_a c

Example:

Proof of property   log_a (bc) = log_a b + log_a c

1. Make x and y variables

2. Use the key to everything

3. Multiply b and c together

4. Law of exponents

5. Take the log base a of both sides

6. Use the coefficient property of logarithms

7. 

8. Substitute x and y back into equation



3. Subtraction

Example: 

These properties (for math students) are helpful for combining and simplifying logarithmic expressions.  Remember these are the only three. Don't accidentally makeup new ones.  As Wilhelm mentioned in class, if you get confused which property is true, try plugging in some numbers that are easy to work with.