8.3 Graph General Rational Functions

GRAPHS OF RATIONAL FUNCTIONS

Let p(x) and q(x) be polynomials with no common factors other than ±1.

f(x) =

1. The x-intercepts of the graph of f are the real zeros of _p(x)_.

2. The graph of f has a vertical asymptote at each real zero of _q(x)_.

3. The graph of f has at most one horizontal asymptote, determined by the degrees m and n of p(x) and q(x).

· If m < n, the line _y = 0_ is a horizontal asymptote.

· If m = n, the line y = is a horizontal asymptote.

· If m > n, the graph has _no horizontal asymptote_.

The end behavior is the same as y = x^{m - n}.

Example 1

Graph a rational function (m < n)

Graph y = State the domain and range.

The numerator has no zeros, so there is no _x-intercept_. The denominator has no real zeros, so there is no _vertical asymptote_.

The degree of the numerator, _0_, is less than the degree of the denominator, _2_. So, the line _y = 0_ (the x-axis) is a horizontal asymptote.

The domain is _all real numbers_, and the range is _0 < y £ 1.5_

Example 2

Graph a rational function (m = n)

Graph y =

a

m

The zeros of the numerator x² - 9 are _±3_, so _-3_ and _3_ are the x-intercepts. The zeros of the denominator x² - 4 are _±2_, so x = _2_ and x = _-2_ are vertical asymptotes. The numerator and denominator have the same degree, so the horizontal asymptote is

b

n

y = = _1_.

Plot points between and beyond the vertical asymptotes.

		x	y
To the left of x = -2		-5	_0.8_
		-3	_0_
Between x = -2 and x = 2		-1	_2.7_
		0	_2.3_
		1	_2.7_
To the right of x = -2		3	_0_
		5	_0.8_

Example 3

Graph a rational function (m > n)

Graph y =

The numerator factors as _ (x - 3)(x + 1)_, so the x-intercepts are _3_ and _-1_. The zero of the denominator x + 2 is _ -2 _, so the vertical asymptote is _x = -2__. The degree of the numerator, __2__, is greater than the degree of the denominator, __1__, so the graph has no horizontal asymptote. The graph has the same end behavior as the graph of

y = x = _x_.

Plot points on each side of the vertical asymptote.

		x	y
To the left of x = -2		-7	_-12_
		-6	_-11.3_
		-4	_-10.5_
		-3	_-12_
To the right of x = -2		-1	_0_
		0	_-1.5_
		2	_-0.8_