Pc Graph Rational Functns

Graphing Rational Functions

P( x) A rational function is a function of the form f(x) = , Q( x)

where P(x) and Q(x) are polynomials and Q(x) = 0.

1 Example: f (x) = is defined for all real numbers except x = 0. x 1 f(x) = x

x

f(x)

x

f(x)

2

0.5

-2

-0.5

1

1

-1

-1

0.5

2

-0.5

-2

0.1

10

-0.1

-10

0.01

100

-0.01

-100

0.001

1000

-0.001

-1000

As x → 0+, f(x) → +∞. As x → 0–, f(x) → -∞.

The line x = a is a vertical asymptote of the graph of y = f(x), if and only if f(x) → + ∞ or f(x) → – ∞ as x → a + or as x → a –. as x → a – f(x) → + ∞

as x → a – f(x) → – ∞

x

x

x=a

as x → a + f(x) → + ∞

x=a

x

x=a

as x → a + f(x) → – ∞

x

x=a

Example: Show that the line x = 2 is a vertical asymptote of the 4 graph of f(x) = 2 . y ( x − 2) x=2 x

f(x)

1.5

16

1.9

400

1.99

40000

2

-

2.01

40000

2.1

400

2.5

16

Observe that: x→2–, f (x) → – ∞ x→2+, f (x) → + ∞

f (x) =

4 ( x − 2) 2

100

x 0.5

This shows that x = 2 is a vertical asymptote.

P( x) A rational function Q( x) may have a vertical asymptote at

x = a for any value of a such that Q(a) = 0. Example: Find the vertical asymptotes of the graph of f(x) =

1 . 2 ( x + 4 x − 5)

Set the denominator equal to zero and solve. Solve the quadratic equation x2 + 4x – 5. (x – 1)(x + 5) = 0 Therefore, x = 1 and x = -5 are the values of x for which f may have a vertical asymptote. As x →1– , f(x) → – ∞.

As x → -5–, f(x) → + ∞.

As x →1+, f(x) → + ∞.

As x →-5+, f(x) → – ∞.

x = 1 is a vertical asymptote.

x = -5 is a vertical asymptote.

Example: ( x + 2) Find the vertical asymptotes of the graph of f(x) = 2 . ( x − 4) 1. Find the roots of the denominator. 0 = x2 – 4 = (x + 2)(x – 2) Possible vertical asymptotes are x = -2 and x = +2. 2. Calculate the values approaching -2 and +2 from both sides. x → -2, f(x) → -0.25; so x = -2 is not a vertical asymptote. x → +2–, f(x) → – ∞ and x →+2+, f(x) → + ∞. y So, x = 2 is a vertical asymptote. f is undefined at -2 A hole in the graph of f at (-2, -0.25) shows a removable singularity.

x=2

(-2, -0.25)

x

The line y = b is a horizontal asymptote of the graph of y = f(x) if and only if f(x) → b + or f(x) → b – as x → + ∞ or as x → – ∞. as x → + ∞ f(x) → b – y

as x → – ∞ f(x) → b – y

y=b

as x → + ∞ f(x) → b + y

as x → – ∞ f(x) → b + y

y=b y=b

y=b

Example: Show that the line y = 0 is a horizontal asymptote of 1 the graph of the function f(x) = . x

As x becomes unbounded positively, f(x) approaches zero from above; therefore, the line y = 0 is a horizontal asymptote of the graph of f. As f(x) → – ∞, x → 0 –. x

f(x)

10

0.1

100

0.01

1000

0.001

0

–

-10

-0.1

-100

-0.01

-1000

-0.001

y 1 f(x) = x

x y=0

Example: Determine the horizontal asymptotes of the graph of 2 x f(x) = 2 . ( x + 1) Divide x + 1 into x . 2

2

1 f(x) = 1 – 2 ( x + 1)

1 1 – – As x → +∞, 2 → 0 ; so, f(x) = 1 – →1 . 2 ( x + 1) ( x + 1)

Similarly, as x → – ∞, f(x) →1–. Therefore, the graph of f has y = 1 as a horizontal asymptote.

y y=1 x

Finding Asymptotes for Rational Functions Given a rational function: f (x) =

P(x) Q(x)

=

am xm + lower degree terms bn xn + lower degree terms

• If c is a real number which is a root of both P(x) and Q(x), then there is a removable singularity at c. • If c is a root of Q(x) but not a root of P(x), then x = c is a vertical asymptote. • If m > n, then there are no horizontal asymptotes. • If m < n, then y = 0 is a horizontal asymptote. • If m = n, then y = am is a horizontal asymptote. bn

Example: 3x 2 + 6 x + 3 Find all horizontal and vertical asymptotes of f (x) = 2 . x −x−2

Factor the numerator and denominator. The only root of the numerator is x = -1. The roots of the denominator are x = -1 and x = 2 . x=2 Since -1 is a common root of both, there y is a hole in the graph at -1 . Since 2 is a root of the denominator but not the numerator, x = 2 will be a vertical asymptote. Since the polynomials have the same degree, y = 3 will be a horizontal asymptote.

y=3

x

A slant asymptote is an asymptote which is not vertical or horizontal. 2x2 + x −1 Example: Find the slant asymptote for f(x) = . x+3 2 14 2x + x −1 = 2x − 5 + Divide: x+3 x+3 y 14 As x → + ∞, → 0+. x +3 14 As x → – ∞, → 0–. x +3

Therefore as x → ±∞, f(x) is more like the line y = 2x – 5. The slant asymptote is y = 2x – 5.

x = -3

y = 2x - 5

x