rational function · Graph a Rational Function: n = m + 1. Determine any asymptotes and intercepts...

• 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

Step 1 Find the domain.

A. Find the domain of and the equations

of the vertical or horizontal asymptotes, if any.


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


Step 1 The zeros of the denominator

B. Find the domain of and the

equations of the vertical or horizontal asymptotes,

if any.


Step 2

The horizontal asymptote is y = 2.


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 2

Step 1


Step 3

Step 4


Answer: vertical asymptote at x = –5; horizontal asymptote at y = 0; y-intercept: 1.4; D = {x | x≠ –5, x };


B. For , determine any vertical and

horizontal asymptotes and intercepts. Then graph

the function and state its domain.


Step 2

Step 3

Step 1


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.


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


and intercepts for . Then graph the

function, and state its domain.

Factoring both numerator and denominator yields

with no common factors.

Step 1


Step 2

Step 3


Step 4 Graph the asymptotes and intercepts. Then find and plot points in the test intervals (–∞, –3), (–3, –2), (–2, 2), (2, 4), (4, ∞).


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


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 ∉ }.


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.


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) ≈


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

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 . Then

graph the function and state its domain.

Step 1


Step 2

Step 3


Step 4


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 .

• Stop here. Review any slides you did not understand.

rational function · Graph a Rational Function: n = m + 1. Determine any asymptotes and intercepts...

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