VERTICAL AND HORIZONTAL ( T U E S D AY ) ( W E D N E S D AY / T H U R S . )

COLLEGE ALGEBRA

MR. POULAKOSMARCH 2011

Asymptotes

Asymptotes of Rational Functions,

A Rational Function is:An Asymptote is, essentially, a line that a graph

approaches, but does not touch or cross. There are two types:

Vertical Asymptote Horizontal Asymptote

The Asymptote is represented on x-y coordinate system as a dashed line “- - - - - - - ”

Why?

- - - - - - - - -

- - - - - - - - - - - - - - - - horizontal

Vertic

al

Vertical asymptotes

The Vertical asymptote is a line that the graph approaches but does not intersect. Vertical asymptotes almost always occur because the denominator of a fraction has gone to 0, but the numerator (top) has not.

For example,

Note that as the graph approaches x=2. From the left, the curve drops rapidly towards negative infinity. This is because the numerator is staying at 4, and the denominator is getting close to 0.

x

=2

Horizontal Asymptote

The Horizontal asymptote is also a line that the graph approaches but does not intersect

In the following graph of y=1/x, the line approaches the x-axis (y=0) as x gets larger. But it never touches the x-axis. No matter how far we go into infinity, the line will not actually reach y=0, but it will keep getting closer and closer.

This means that the line y=0 is a horizontal asymptote.

The domain for y=1/x is all real numbers except 0

Horizontal asymptotes

Horizontal asymptotes occur most often when the function is a fraction where the top remains positive, but the bottom goes to infinity.

Going back to the previous example, y=1/x is a fraction. When we go out to infinity on the x-axis, the top of the fraction remains 1, but the bottom gets bigger and bigger. As a result, the entire fraction actually gets smaller, although it will not hit zero. The function will be 1/2, then 1/3, then 1/10, even 1/10000, but never quite 0.

Thus, y=0 is a horizontal asymptote for the function y=1/x

REMEMBER:

ASYMPTOTES ARE ALWAYS LINES.

THEY ARE LINES THAT A GRAPH APPROACHES BUT DOES NOT TOUCH (DOES NOT INTERSECT)

Finding Asymptotes

Vertical asymptotes

Remember, Vertical asymptotes almost always occur because the denominator of a fraction has gone to 0, but the top has not the zeroes of the denominator.

Therefore, set the denominator to zero and solve for the variable.

For example, x–7=0x=7 is the asymptote.Factor : x2–16=0 (x–4)(x+4) = 0 Solve: a) x–4=0 and b) x+4=0, Therefore, there are 2 asymptotes. a) x = +4 and b) x = –4

Rational Function Vertical Asymptote (s) is/are at …

x = 5

x = +4 and x = –4

x = – 4 and x = – 2

Vertical asymptotes – Sample Problems

Page

Horizontal asymptotes

Finding the Horizontal asymptote(s) are more challenging… Compare the degree of the numerator

(n) to that of the denominator (m). If n<m, then the horizontal asymptote is at y

= 0. If n=m (the degrees are the same), then the

asymptote is at y = 1st coefficient of numerator ÷ 1st coefficient of denominator

If n>m, then there are no Horizontal asymptotes.

Examples follow …

See page 338

Examples -- Horizontal Asymptotes

n < mAsymptote is at y=0

n = mAsymptote

is at y=an/bm

Asymptote is at…

n >mNo

Asymptote

y=6/3= 2

y=2/5

y=6/4= 3/2