1. Analyzing Graphs of Rational Functions
Sec. 2.7b

2. Directly in with some Practice Problems
Find the intercepts, asymptotes, use limits to describe the
behavior at the vertical asymptotes, and analyze and draw the
graph of the given rational function.
Numerator is zero when x = 1  x-intercept is 1
f (0) = 1/6  y-intercept is 1/6
Denominator factors as (x – 3)(x + 2)  V.A.: x = 3, x = –2
Degree of Numerator < Degree of Denominator  H.A.: y = 0
Graph:

3. Directly in with some Practice Problems
Find the intercepts, asymptotes, use limits to describe the
behavior at the vertical asymptotes, and analyze and draw the
graph of the given rational function.
Domain:
H.A.:
Range:
V.A.:
Continuity:
Continuous on D
Inc/Dec:
Dec:
Symmetry:
None
End Behavior:
Boundedness:
Unbounded
Local Extrema:
None

4. More Practice Problems
Find the intercepts and analyze and draw the graph of the given
rational function.
Numerator factors as 2(x – 1)(x + 1)  x-intercepts: –1, 1
f (0) = 1/2  y-intercept: 1/2
Denominator factors as (x – 2)(x + 2)  V.A.: x = –2, x = 2
Degree of Numerator = Degree of Denominator  H.A.: y = 2
Graph:

5. More Practice Problems
Find the intercepts and analyze and draw the graph of the given
rational function.
Domain:
Range:
H.A.:
Continuity:
Continuous on D
V.A.:
Inc/Dec:
Inc:
Dec:
End Behavior:
Symmetry:
y-axis (even)
Boundedness:
Unbounded
Local Extrema:
Local Max of 1/2 at x = 0

6. More Practice Problems
Find the end behavior asymptote of the given rational function.
and graph it together with f
in two windows:
(a) one showing the details around the V.A. of f,
(b) one showing a graph of f that resembles its end
behavior asymptote.
Denominator is zero at x = 1  V.A.: x = 1
Rewrite f using polynomial division:
End Behavior Asymptote:

7. More Practice Problems
Find the end behavior asymptote of the given rational function.
and graph it together with f
in two windows:
(a) one showing the details around the V.A. of f,
(b) one showing a graph of f that resembles its end
behavior asymptote.
First, graph these two functions in [–4.7, 4.7] by [–8, 8]
Now, change your viewing window to [–40, 40] by [–500, 500]

8. More Practice Problems
Find the intercepts and analyze and draw the graph of the given
rational function.
Use a calculator to find the x-intercept
= – 0.260
f (0) = y-intercept = –1
V.A.: x = 1
H.A.: none
E.B.A.:
Graph:

9. More Practice Problems
Find the intercepts and analyze and draw the graph of the given
rational function.
Domain:
H.A.:
None
Range:
V.A.:
Continuity:
Continuous on D
E.B.A.:
Inc/Dec:
Dec:
Inc:
End Behavior:
Symmetry:
None
Boundedness:
Unbounded
Local Extrema:
Local Min of 3 at x = 2