Pc Functions End Behavior Of Polynomial And Rational Functio

End Behavior  • End behavior of polynomial functions • End behavior of rational functions 

 

 

End Behavior of polynomial Functions

For large values of x, either positive  or negative, the graph of the  polynomial 

resembles the graph of the power function.   

 

Example – End Behavior Determine the end behavior of the following:

(a) f ( x) = 3x

6

(b) g ( x) = −5 x 1 5 (c ) h( x ) = x 2 7 (d ) p ( x) = −7 x 4

 

 

Example – Leading Term Determine the leading term of f ( x) = (2 x −1) ( x + 3) ( x − 5) 2

3

Leading Term: (2x)2 (x)3 (x) = 4x6  

 

Rational Functions

A rational function is a function of the form

Where p and q are polynomial functions and q  is not the zero polynomial. The domain  consists of all real numbers except those for  which the denominator is 0.   

 

Find the domain of the following rational functions:

All real numbers except ­6 and­2.

All real numbers except ­4 and 4.

 

All real numbers.  

Behavior Near Undefined Values

x−2 (a) f ( x) = x +1 2 x (b) g ( x) = 2 x −2 3 x (c ) h ( x ) = x+2

Examine the behavior of  the function near the  undefined values and as  x goes to ±∞ (as x goes  to the extremes.

 

 

 

 

Vertical Asymptotes

x=c

y

x x=c

y

x  

 

Theorem Locating Vertical Asymptotes A rational function 

In lowest terms, will have a vertical  asymptote x = r, if x ­ r is a factor of the  denominator q.   

 

Find the vertical asymptotes, if any, of the graph  of each rational function. Vertical asymptotes: x = -1 and x = 1 No vertical asymptotes

Vertical asymptote: x = -4  

 

Horizontal Asymptotes

y y = R(x)

y=L x y y=L x  

y = R(x)  

If an asymptote is neither horizontal nor vertical it is called oblique. y

x

 

 

Consider the rational function

1. If n < m, then y = 0 is a horizontal asymptote of the graph of R. 2. If n = m, then y = an / bm is a horizontal asymptote of the graph of R. 3. If n = m + 1, then y = ax + b is an oblique asymptote of the graph of R. Found using long division. 4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote. End behavior found using long     division.

Find the horizontal and oblique asymptotes if  any, of the graph of

Horizontal asymptote: y = 0

Horizontal asymptote: y = 2/3  

 

Oblique asymptote: y = x + 6