2.6 Rational Functions and Asymptotes

Objective – To recognize where holes and asymptotes occur in rational functions.

Discontinuities

Holes (removable discontinuity) • occur when there is a value of x that would make the denominator equal to zero,

but we can cancel out the factor

1. Set the canceled factor equal to 0. 2. Solve for x.

3. Plug the answer into the simplified equation to get y. 4. (x, y ) is the hole

Examples of functions with holes:

1. 2

2( )4

xf xx

+=

- 2.

2 2 15( )3

x xg xx- -

=+

Vertical Asymptotes (non-removable discontinuity) • occur when there is a value of x that would make the denominator equal to zero,

but we can’t get rid of the factor (Equation of VA: x = #)

Examples of functions with vertical asymptotes:

1. 2 6

xyx

=-

2. 2

104

yx

=-

3. 2

56 5

xyx x

+=

+ +

To put it simply: When it comes to the denominator if it cancels out à hole

if it doesn’t cancel out à vertical asymptote

f x_pwg x 5 xtTxt2MuwXx 2 glx 3

hole at 4 2hole at x 3

VA at X 2

y 4

yD

ys2x3Thw FEWht2Tw xtsTwntTuVA at 4 3 VA at 1 2

and x 2 hole at x 5

VA at x I

Horizontal Asymptote – occur in 2 out of the 3 cases

• to find a horizontal asymptote, see what happens to the function as x approaches infinity

Conclusion

Case 1: Denominator exponent is larger àEquation of HA: y = 0 (ALWAYS!)

Case 2: Exponents are equal à Equation of HA: y = ratio of leading coefficients

Case 3: Numerator exponent is larger à No Horizontal Asymptote

1. 12 5

yx

=+

2. 2

2

5 4 27

x xyx+ +

=-

3. 2 2

4x xy

x- +

=+

Slant Asymptotes (diagonal lines)

Steps: Use synthetic division to determine the equation of the slant asymptote

• occur only when the highest power in the numerator exceeds the highest power in the denominator by exactly one in the simplified function

Examples of functions with slant asymptotes:

1. 2

6xy

x=

- 2.

33 2 43

x xyx- +

=-

3. 23 5 7

2x xy

x- +

=-

compare degrees

degreetu

agreedeg Cw dlg 2 deg

2

NG Cu deg 2 deg 1

HA at y o HAaty 5 NO HA

deg 2 dl9 deg 2deg1 deg1 µg 4

yes 5A No SA yes SA where

X 23 2 5177or

I 3 5 7

ignored.fr

3xt

Graph the following. You can use a graphing calculator (put parentheses around your numerator and denominator). Identify the domain, hole(s), VA, HA, and intercepts for the following.

1. 5 34

xyx

-=

+

D:

Hole:

VA:

HA: y-intercept: x-intercept:

2. 2

23 10

xyx x

-=

+ -

D:

Hole:

VA:

HA: y-intercept: x-intercept:

3. ! = #$%&'()&

D:

Hole:

VA:

HA: y-intercept: x-intercept: