Algebra 1 Common Core Student Edition, Grade 8-9

The graphs of m any rational functions are related to each other. C om pare the

graphs below of y = \ and y = J 33.

The graphs are identical in shape, but the second graph is translated 3 units right.

Notice that the graph of y = \ approaches both axes but does not cross either axis. The

axes in this graph function as asymptotes. A line is an asym p t ot e of a graph if the graph

gets closer to the line as x or y gets larger in absolute value. In the graph of y = ^33

above, the x-axis and the line x = 3 are asymptotes.

When the numerator and denominator of a rational function have no common factors

other than 1 , there is a vertical asymptote at each excluded value.

Using a Vertical Asym ptote

c

What is the vertical asymptote of the graph of y = x + 2? Graph the function.

Think

The numerator and

denom inator have no

common factors. To find

the vertical asymptote,

find the excluded value.

To graph the function,

first make a table of

values. Use values of

x near - 2 , w h e r e th e

asymptote occurs.

Use the points from the

table to make the graph

Draw a dashed line for

the vertical asymptote.

Write

X + 2 = 0

x = - 2

Th e vert ical asym pt ot e is t he line x = - 2.

f Y V

X -7 — 4 -3 -1 0 3 s

Vy -1 -2.5 -5^5 2.5^1 /

..>

W'Pr o b l em 2

& Got It? 2. What is the vertical asymptote of the graph of h{x) = 313? Graph the function.

706 Chapter 11 Rational Expressions and Functions