rational function · graph a rational function: n = m + 1. determine any asymptotes and intercepts...
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• rational function
• asymptote
• vertical asymptote
• horizontal asymptote
• oblique asymptote or slant asymptote
• holes
Vertical asymptotes always occur where the denominator equals zero!
Horizontal asymptotes exist when end behavior reaches a constant value
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
Step 1 Find the domain.
A. Find the domain of and the equations
of the vertical or horizontal asymptotes, if any.
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
Step 1 The zeros of the denominator
B. Find the domain of and the
equations of the vertical or horizontal asymptotes,
if any.
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
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 2
Step 1
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 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 }
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
Determine any vertical and horizontal asymptotes
and intercepts for .
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 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;
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
Determine any vertical and horizontal asymptotes
and intercepts for .
Graph a Rational Function: n = m + 1
Determine any asymptotes and intercepts for
. Then graph the function, and state
its domain.
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.
Step 1 The function is undefined at b(x) = 0, so the domain is D = {x | x ≠ –3, x ∉ }.
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 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, ∞).
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) ≈
Graph a Rational Function: n = m + 1
Answer: vertical asymptote at x = –3;
oblique asymptote at y = x – 2;
x-intercepts: and ;
y-intercept: ;
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
Graph a Rational Function with Common Factors
Determine any vertical and horizontal asymptotes,
holes, and intercepts for . Then
graph the function and state its domain.
Step 1
Graph a Rational Function with Common Factors
Answer: vertical asymptote at x = –2; horizontal
asymptote at y = 1; x-intercept: –3
and y-intercept: ; hole: ;
;
–4 –2 2 4
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)
Determine the vertical and horizontal asymptotes
and holes of the graph of .
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