Ch 1.2 Graph of Functions

 

Even and Odd Functions

A graph has symmetry with respect to the y-axis if when ever points (x,y) and (-x,y) are both on the graph. This is usually a even function.

 

A graph has symmetry with respect to the origin if when ever points (x,y) and (-x,-y) are both on the graph. This is usually a odd function.

Ex. (1)

      

  

 

Even function, symmetry to the y-axis, (x,y) and (-x,y)

 



 

Ex. (2)

   

 



Odd function, symmetry to origin, (x,y) and (-x,-y)





 

 

 

A function f is increasing on an interval if, for any x1 and x2 in the interval,

 implies

 

 

A function f is decreasing on an interval if, for any x1 and x2 in the interval,

 implies

 

 

A function value f(a) is called a relative maximum of f if there exists n interval (x1,x2) that contains a such that



 implies

 

A function value f(a) is called a relative minimum of f if there exists n interval (x1,x2) that contains a such that

 

 implies





1.1 Difference Quotient

Chapter 1.1 (Cont.) Difference Quotient:

A difference quotient is defined as the ratio of change of y-values over the change in x-values.

The purpose of evaluating a difference quotient is to find the slope of the secant line. The secant line is defined as the line in which two points pass through.The formula for finding the slope of a secant line is represented below.

Point A=
 


Point B=

The Secant Line in this graph is the line that travels through point A at (x, f(x)) and through point B at (x+h, f(x+h)). 

For the graph above, due to the fact that the slope of the secant line is represented by the change in Y over the change in X, the equation should look like this. 

Simplified down to 

First Example:

Let

  

Find

First Step:

Distributive Property 

 Simplify

 Next Step: Evaluating a Difference Quotient

 

For 

Find 

Solution

Distributive Property

Simplify 

Factor out an h from the numerator 

Answer 

 

 

*Important Note: Notice how in the example above, h cannot equal zero. Therefore, you must restrict the domain of the simplified expression by adding 

so that the simplified expression is equivalent to the original expression.

Some Extra Practice:

1.)

 Answer: 2

2.)

 Answer:



3.)

Answer: