Rational Functions: Asymptotes Arrow Notation

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Asymptotes. Objectives. – Find domain of rational functions. – Use transformations to graph rational functions. Use arrow notation. – Identify vertical asymptotes.
Objectives

Rational Functions: Asymptotes

REVIEW Find the domain of f ( x) =

Rational Function A rational function is a function of the form

f ( x) =

– Find domain of rational functions. – Use transformations to graph rational functions. Use arrow notation. – Identify vertical asymptotes. – Identify horizontal asymptotes. – Identify slant (oblique) asymptotes.

P ( x) Q( x)

Where p and q are polynomials and p(x) and q(x) have no factor in common and q(x) is not equal to zero.

Solution: When the denominator x + 4 = 0, we have x = −4, so the only input that results in a denominator of 0 is −4. Thus the domain is (−∞, −4) ∪ (−4, ∞). REVIEW: The graph of the function is the graph of y = 1/x translated to the left 4 units.

Example

Arrow Notation

Find the domain of each rational function. x 2 − 25 a) f ( x) = x −5

x b) g ( x) = 2 x − 25

1 x+4

x+5 c) h( x) = 2 x + 25

Symbol

Meaning

x → a+

x approaches a from the right

x → a−

x approaches a from the left

x→∞

x approaches infinity; x increases without bound

x → −∞

x approaches negative infinity; x decreases without bound

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Vertical Asymptotes

Rational Functions Different from other functions because they have asymptotes. Asymptotes- a line that the graph of a function gets closer and closer to as one travels along that line in either direction.

Vertical Asymptote Occurs where the function is undefined, denominator is equal to zero. Form: x = a where a is the zero of the denominator **Graph never crosses **

Vertical asymptotes • Look for domain restrictions. If there are values of x which result in a zero denominator, these values would create EITHER a hole in the graph or a vertical asymptote. Which? – If the factor that creates a zero denominator cancels with a factor in the numerator, there is a hole. If you cannot cancel the factor from the denominator, a vertical asymptote exists.

• If you evaluate f(x) at values that get very, very close to the x-value that creates a zero denominator, you notice f(x) gets very, very, very large! (approaching pos. or neg. infinity as you get closer and closer to x)

Example Find the vertical asymptotes, if any, of each rational function.

a) f ( x) =

x x −1 2

b) g ( x) =

x −1 x2 − 1

c) h( x) =

x −1 x2 + 1

Try 2x − 3 this. f ( x) = x 2 − 4 • Determine the vertical asymptotes of the function. • Factor to find the zeros of the denominator:

• Thus the vertical asymptotes are the lines:

Horizontal Asymptotes Horizontal Asymptote Determined by the degrees of the numerator and denominator. Form: y = a (next slide has rules) ** Graph can cross **

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Example

Determining if Horizontal Asymptote Exists

Find the horizontal asymptote, if any, of each rational function.

Look at the rational function r ( x) =

P ( x) Q ( x)

a) f ( x) =

9 x2 3x 2 + 1

b) g ( x) =

9x 3x 2 + 1

c) h( x) =

9 x3 3x 2 + 1

• If degree of P(x) < degree of Q(x), horizontal asymptote y = 0. • If degree of P(x) = degree of Q(x), leading coeff P(x) a = n horizontal asymptote y = leading coeff Q(x) bn • If degree of P(x) > degree of Q(x), no horizontal asymptote

Example Find the horizontal asymptote:

6 x 4 − 3x 2 + 1 f ( x) = 4 9 x + 3x − 2

Oblique Asymptote Degree of p(x) > degree of q(x) To find oblique asymptote:

Remember: • The graph of a rational function never crosses a vertical asymptote. • The graph of a rational function might cross a horizontal asymptote but does not necessarily do so.

What is the equation of the oblique asymptote?

4 x 2 − 3x + 2 f ( x) = 2x +1

– Divide numerator by denominator – Disregard remainder – Set quotient equal to y (this gives the equation of the asymptote)

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Example Find the slant (oblique) asymptote for

2 x2 − 5x + 7 f ( x) = x−2

ASYMPTOTE SUMMARY Occurrence of Lines as Asymptotes • For a rational function f(x) = p(x)/q(x), where p(x) and q(x) have no common factors other than constants:

• Vertical asymptotes occur at any x-values that make the denominator 0. • The x-axis is the horizontal asymptote when the degree of the numerator is less than the degree of the denominator. • A horizontal asymptote other than the x-axis occurs when the numerator and the denominator have the same degree.

ASYMPTOTE SUMMARY (cont.) • An oblique asymptote occurs when the degree of the numerator is 1 greater than the degree of the denominator. • There can be only one horizontal asymptote or one oblique asymptote and never both. • An asymptote is not part of the graph of the function. (You should always include your asymptotes as dashed lines on graphs that you draw!!)

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