Graphs of Other TRIG Functions

Graph of the tangent function

Here is the graph of y=tan x. Note that there are vertical asymptotes where the denominator of tan x equals 0. So when cos x=0.

Graph of the Cotangent Function

Here is the graph of y=cot x. We now must consider where sin x equals zero because this is where you will put the vertical asymptotes.

Graph of the Reciprocal Functions (y= sec x)

First of all, here is the graph of y=cos x.

-Using reciprocal identities, we can draw y=sec x.

Graph of the reciprocal Functions (y=csc x)

The graph in purple is y=csc x. The graph in red is y=sin x.

2.5 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra:

If has a polynomial degree of , the number of zeros is also

Conjugate Pairs:

and

Finding a Polynomial with Given Zeros:

Example:

Combinations of Functions

Combinations of Functions:

All of these are Arithmetic combinations of functions

Let f(x) =2x+3
let g(x) =

Creating new functions - examples

(f+g)(x) =f(x)+g(x)

=2x+3+

(f-g)(x) =f(x)-g(x)

= (2x+3)-( )

= -+2x+3

(fg)(x) =f(x)*g(x)

= (2x+3)( )

= 2+3

(f/g)(x)= ------> don’t forget to find the domain after dividing!

Remember-denominator cannot be zero!

= (2x+3)/( )

Domain and Range of Functions, Difference Quotient

Domain = all possible x-values
Range = all possible y-values

Domain

When finding the domain, check for:

• negative square roots


• denominators that equal zero
  

Examples

1.) In equations with fractions, the domain is determined only by the denominator. It is important to make sure that it will not equal zero.

Start by setting the equation equal to zero and solving.

These are the values that will make the denominator equal zero, so the domain of the function is all numbers except 2 and -1.

2.) In equations with a square root, it is important to make sure that we are not finding the square root of a negative number.

To start, set the equation as greater than or equal to zero and solve.

This is the lowest x-value that can be used before the equation becomes negative. Therefore, the domain of the equation is all values equal to or less than three halves.

Range

To find the range, graph the equation to see all the different y-values.

In this example, we can see that the lowest y-value on the line is -1, and that the lines extend upwards indefinitely.

Therefore, the range is

Difference Quotient

http://www.youtube.com/watch?v=1O5NEI8UuHM

Ch1- Functions

Function- f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the doman ( or set of inputs) of the function f, and the set B contains the range ( or set of outputs).

Functions can be expressed Verbally, Numerically, Graphically, and Algebraically
Use the vertical line test to check if there is a function on a graph. If the line passes 2 points then there are two of the same x values, therefore it is not a function.


Function Notation-
y is a function of x
y = f (x) ↔ The point (x, y) is on the graph f

Example- f (x)= 3x+10
f (2)= 3(2)+10
f (2)=16