Page 526 - Algebra 1

Rational functions are discontinuous functions. Discontinuous functions are functions that have a break or jump in the graph. A break or jump in the graph can be due to an asymptote. An asymptote is a boundary line that the graph of a function approaches but never touches or crosses.
Caution
Students may draw asymptotes as solid lines. However, asymptotes
are usually shown with dashed lines because asymptotes are not
part of the graph of the rational function.
Horizontal Asymptote: y=2
8
4
O
x
-8
-4
4
8
-4
Vertical Asymptote: x=3
-8
When a graph of a rational function has a vertical asymptote, then there is
an excluded value for the function. To determine where vertical asymptotes
occur, find the excluded values for the function. Rational functions also have
a horizontal asymptotes. When the function is in the form y = _x - b + c, the
vertical asymptote occurs at x = b and the horizontal asymptote occurs at y = c. Determining Asymptotes
Example
2
Identify the asymptotes. a. y=_2
Hint
When c = 0 in the rational a
functiony=_x-b +c, the horizontal asymptote is the x-axis.
x-8 SOLUTION
y = _2 a rational function in the form y = _a x - 8 where a = 2, b = 8, and c = 0 x - b
+ c, Since b = 8, the equation of the vertical asymptote is x = 8.
Since c = 0, the equation of the horizontal asymptote is y = 0. b.y=_4 +3
x + 10 SOLU_TION
Hint
Compare the equation of the given rational
a functiontoy=_x-b +c
to determine the correct sign of the asymptotes.
4_
y= +3 arationalfunctionintheformy= x-b +c,
_x + 1 0 w h e r e a = 4 , b = - 1 0 , a n d c = 3 4
y= x-(-10) +(3)
Since b = -10, the equation of the vertical asymptote is x = -10. Since c = 3, the equation of the horizontal asymptote is y = 3.
y
a
Lesson 78 511