• rational function
• asymptote
• vertical asymptote
• horizontal asymptote
• oblique asymptote or slant
asymptote
• holes
A rational function is a fraction for
which the numerator and
denominator are polynomials.
Vertical asymptotes always occur
where the denominator equals
zero!
Horizontal asymptotes
exist when end behavior
reaches a constant value
To find vertical asymptotes,
To find horizontal asymptotes,
To state the DOMAIN of a
ration function, the shortcut is
to find where the function is
undefined (denominator = 0)
and exclude those values.
Find Vertical and Horizontal Asymptotes
A. Find the domain of
and the equations
of the vertical or horizontal asymptotes, if any.
Step 1 Find the domain.
Find Vertical and Horizontal Asymptotes
CHECK
The graph of
shown supports
each of these findings.
Answer: D = {x | x ≠ 1, x
}; vertical asymptote at
x = 1; horizontal asymptote at y = 1
Find Vertical and Horizontal Asymptotes
B. Find the domain of
and the
equations of the vertical or horizontal asymptotes,
if any.
Step 1
The zeros of the denominator
Find Vertical and Horizontal Asymptotes
Step 2
The horizontal asymptote is y = 2.
Find Vertical and Horizontal Asymptotes
CHECK
You can use a table of values to support
this reasoning. The graph of
shown also supports each of these
findings.
Answer: D = {x | x
}; no vertical asymptotes;
horizontal asymptote at y = 2
Find the domain of
and the equations
of the vertical or horizontal asymptotes, if any.
A. D = {x | x ≠ 4, x
}; vertical asymptote at
x = 4; horizontal asymptote at y = –10
B. D = {x | x ≠ 5, x
}; vertical asymptote at
x = 5; horizontal asymptote at y = 4
C. D = {x | x ≠ 4, x
}; vertical asymptote at
x = 4; horizontal asymptote at y = 5
D. D = {x | x ≠ 4, 4, x
}; vertical asymptote at
x = 4; horizontal asymptote at y = –2
This is a big deal, guys!
Graph Rational Functions: n < m and n > m
A. For
, determine any vertical and
horizontal asymptotes and intercepts. Then graph
the function and state its domain.
Step 1
Step 2
Graph Rational Functions: n < m and n > m
Step 3
Step 4
Graph Rational Functions: n < m and n > m
Answer: vertical asymptote at x = –5; horizontal
asymptote at y = 0;
y-intercept: 1.4; D = {x | x≠ –5, x
};
Graph Rational Functions: n < m and n > m
B. For
, determine any vertical and
horizontal asymptotes and intercepts. Then graph
the function and state its domain.
Graph Rational Functions: n < m and n > m
Step 1
Step 2
Step 3
Graph Rational Functions: n < m and n > m
Step 4 Graph the asymptotes and intercepts. Then
find and plot points in the test intervals
determined by the intercepts and vertical
asymptotes: (–∞, –2), (–2, –1), (–1, 2), (2, ∞).
Use smooth curves to complete the graph.
Graph Rational Functions: n < m and n > m
Answer: vertical asymptotes at x = 2 and x = –2;
horizontal asymptote at y = 0.
x-intercept: –1; y-intercept: –0.25;
D = {x | x ≠ 2, –2, x
}
Determine any vertical and horizontal asymptotes
and intercepts for
.
A. vertical asymptotes x = –4 and x = 3; horizontal
asymptote y = 0; y-intercept: –0.0833
B. vertical asymptotes x = –4 and x = 3; horizontal
asymptote y = 1; intercept: 0
C. vertical asymptotes x = 4 and x = 3; horizontal
asymptote y = 0; intercept: 0
D. vertical asymptotes x = 4 and x = –3; horizontal
asymptote y = 1; y-intercept: –0.0833
Graph a Rational Function: n = m
Determine any vertical and horizontal asymptotes
and intercepts for
. Then graph the
function, and state its domain.
Factoring both numerator and denominator yields
with no common factors.
Step 1
Graph a Rational Function: n = m
Step 2
Step 3
Graph a Rational Function: n = m
Step 4
Graph the asymptotes and intercepts. Then
find and plot points in the test intervals
(–∞, –3), (–3, –2), (–2, 2), (2, 4), (4, ∞).
Graph a Rational Function: n = m
Answer: vertical asymptotes at x = –2 and x = 2;
horizontal asymptote at y = 0.5;
x-intercepts: 4 and –3; y-intercept: 1.5;
Determine any vertical and horizontal asymptotes
and intercepts for
.
A.
vertical asymptote x = 2; horizontal asymptote y = 6;
x-intercept: –0.833; y-intercept: –2.5
B.
vertical asymptote x = 2; horizontal asymptote y = 6;
x-intercept: –2.5; y-intercept: –0.833
C.
vertical asymptote x = 6; horizontal asymptote y = 2;
x-intercepts: –3 and 0; y-intercept: 0
D.
vertical asymptote x = 6, horizontal asymptote y = 2;
x-intercept: –2.5; y-intercept: –0.833
Graph a Rational Function: n = m + 1
Determine any asymptotes and intercepts for
. Then graph the function, and state
its domain.
Step 1 The function is undefined at b(x) = 0, so the
domain is D = {x | x ≠ –3, x ∉ }.
Step 2 There is a vertical asymptote at x = –3.
The degree of the numerator is greater than
the degree of the denominator, so there is no
horizontal asymptote.
Graph a Rational Function: n = m + 1
Because the degree of the numerator is
exactly one more than the degree of the
denominator, f has an oblique asymptote.
Using polynomial long division, you can write
the following.
f(x) =
Therefore, the equation of the oblique/slant
asymptote is y = x – 2.
Graph a Rational Function: n = m + 1
Step 3 The x-intercepts are the zeros of the
numerator,
and
, or
about 2.37 and –3.37. The y-intercept is
about –2.67 because f(0) ≈
Step 4 Graph the asymptotes and intercepts. Then
find and plot points in the test intervals
(–∞, –3.37), (–3.37, –3), (–3, 2.37), (2.37, ∞).
Graph a Rational Function: n = m + 1
Graph a Rational Function: n = m + 1
Answer: vertical asymptote at x = –3;
oblique asymptote at y = x – 2;
x-intercepts:
y-intercept:
and
;
;
Determine any asymptotes and intercepts for
.
A.
vertical asymptote at x = –2; oblique asymptote at y = x;
x-intercepts: 2.5 and 0.5; y-intercept: 0.5
B.
vertical asymptote at x = –2; oblique asymptote at y = x – 5;
x-intercepts at
; y-intercept: 0.5
C.
vertical asymptote at x = 2; oblique asymptote at y = x – 5;
x-intercepts:
; y-intercept: 0
D.
vertical asymptote at x = –2; oblique asymptote at
y = x2– 5x + 11; x-intercepts: 0 and 3; y-intercept: 0
Holes occur whenever a factor in the
denominator divides out a factor in
the numerator.
Graph a Rational Function with Common
Factors
Determine any vertical and horizontal asymptotes,
holes, and intercepts for
graph the function and state its domain.
Step 1
. Then
Graph a Rational Function with Common
Factors
Step 2
Step 3
Graph a Rational Function with Common
Factors
Step 4
Graph a Rational Function with Common
Factors
–4
–2
2
4
Answer: vertical asymptote at x = –2; horizontal
asymptote at y = 1; x-intercept: –3
and y-intercept:
; hole:
;
;
Determine the vertical and horizontal asymptotes
and holes of the graph of
.
A. vertical asymptote at x = –2, horizontal
asymptote at y = –2; no holes
B. vertical asymptotes at x = –5 and x = –2;
horizontal asymptote at y = 1; hole at (–5, 3)
C. vertical asymptotes at x = –5 and x = –2;
horizontal asymptote at y = 1; hole at (–5, 0)
D. vertical asymptote at x = –2; horizontal
asymptote at y = 1; hole at (–5, 3)
• Stop here. Review any slides you did
not understand.