(a)

The rational functions are 

.

.

.

Find the horizantal asymptote.  

The function is .

To find horizontal asymptote, first find the degree of the numerator and degree of the denominator. 

Degree of the numerator and degree of the denominator . 

Since the degree of numerator is less than degree of denominator, the function has horizontal asymptote at . \ \

The function is .

Degree of the numerator and degree of the denominator .

Since the degree of numerator is less than degree of denominator, the function has horizontal asymptote at . \ \

The function is .

Degree of the numerator and degree of the denominator . 

Since the degree of numerator is less than degree of denominator, the function has horizontal asymptote at .  

\ Function \	Horizontal Asymptote \ \

(b).

The function is .

Graph :

Draw the coordinate plane.

Graph the function .

The function is .

Graph :

Draw the coordinate plane.

Graph the function .

The function is .

Graph :

Draw the coordinate plane.

Graph the function .

(c).

Identify Possible Rational Zeros:

It is not practical to test all possible zeros of a polynomial function using only synthetic substitution.

The Rational Zero Theorem can be used for finding the some possible zeros to test.

The rational function is .

Because the leading coefficient is , the possible rational zeros are the integer factors of the constant term . Therefore, the possible rational zeros of  are .

Factoring yeilds .

Therefore, the real zeros of the numerator of are and .

The rational function is .

Because the leading coefficient is , the possible rational zeros are the integer factors of the constant term . Therefore, the possible rational zeros of  are .

Synthetic Division :

The function is .

Perform the synthetic division method by testing .

Since ,  conclude that is a zero of .

Therefore, is a rational zero.

The remaing quadratic factor yields no real zeros.

Thus, the real zero of the numerator of  is .

The rational function is .

Because the leading coefficient is , the possible rational zeros are the integer factors of the constant term . Therefore, the possible rational zeros of  are .

Factoring yeilds or .

yeilds no real zeros.

Therefore, the real zeros of the numerator of are and .

Function	Real Zeros of the numerator \ \
	and .
	and .

(d).

When the degree of the numerator is less than the degree of the denominator and the numerator has at least on real zero, the graph of the function will have  as an asymptote and will intersect the asymptote at the real zeros of the numerator.

(a).

Function	Horizontal Asymptote

(b).

Graph the function .

Graph the function .

Graph the function .

(c).

Function	Real Zeros of the numerator
	and .
	and .

(d).

When the degree of the numerator is less than the degree of the denominator and the numerator has at least on real zero, the graph of the function will have  as an asymptote and will intersect the asymptote at the real zeros of the numerator.