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:

7

5

4

6

7)()(

21

)()(

5)()(

3)()(

xxpd

xxhc

xxgb

xxfa

Example – Leading Term

Determine the leading term of

)5()3()12()( 32 xxxxf

Leading Term: (2x)2 (x)3 (x) = 4x6

Rational FunctionsA 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 and-2.

All real numbers except -4 and 4.

All real numbers.

Behavior Near Undefined Values

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

2)()(

2)()(

12

)()(

3

2

2

xx

xhc

xx

xgb

xx

xfa

x = cy

xx = c

y

x

Vertical Asymptotes

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.

Vertical asymptotes: x = -1 and x = 1

No vertical asymptotes

Vertical asymptote: x = -4

Find the vertical asymptotes, if any, of the graph of each rational function.

y = L

y = R(x)

y

x

y = L

y = R(x)

y

x

Horizontal Asymptotes

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

y

x

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.

Consider the rational function

Horizontal asymptote: y = 0

Horizontal asymptote: y = 2/3

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

Oblique asymptote: y = x + 6