Notes 3-7: Rational Functions, Study notes of Pre-Calculus

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Notes 3-7: Rational Functions

Warm Up

Find the zeros of each function.

1. f ( x ) = x^2 + 2 x – 15 2. f ( x ) = x^2 – 49

x ≠ ± 1

x ≠ 6

Simplify. Identify any x -values for which the expression is undefined.

x^2 – 8 x + 12 x^2 – 12 x + 36

x – 2 x – 6

x + 4 x – 1

x^2 + 5 x + 4 x^2 – 1

The rational function 𝑓 𝑥 = (^) 𝑥^1 can be transformed by

using methods similar to those used to transform other

types of functions.

II. Transformations of Rational Functions

a. g ( x ) = Because h = – 4, translate f 4 units left.

x + 4

b. g ( x ) = Because k = 1, translate f 1 unit up.

x

Ex 1: Using the graph of 𝒇 𝒙 = 𝟏𝒙 as a guide, describe

the transformation and graph each function. Identify the location of the vertical and horizontal asymptotes.

Vertical asymptote: x = - Horizontal asymptote: y = 0

Vertical asymptote: x = 0 Horizontal asymptote: y = 1

B. Horizontal Asymptotes

If the highest degree is in the denominator, the horizontal

asymptote is at y = 0.

If the degrees in the numerator and denominator are the

same, the horizontal asymptote is the ratio of leading

coefficients.

If the highest degree is in the numerator, there is no

horizontal asymptote.

𝑓 𝑥 =

𝑥 − 1 𝑥^2 + 1 : horizontal asympotote: y = o

𝑓 𝑥 =

3𝑥 − 1 2𝑥 + 1 : horizontal asympotote: y =

3 2

𝑓 𝑥 =

𝑥^2 − 1 𝑥 + 1 : no horizontal asymptote

Ex 1: Determine the discontinuities for the graph of

f ( x ) = ( x^2 + 7 x + 6). x + 3

( x + 6)( x + 1) x + 3

f ( x ) =

Step 1 Vertical asymptotes/holes.

No Holes; Vertical asymptote: x = – 3

The denominator is 0 when x =

- 3. (x + 3) is not in the numerator, so it is a vertical asymptote and not a hole.

Step 2 Horizontal asymptotes.

None: The exponent in the numerator is the largest, so there is no horizontal asymptote.

Remember

This is the same as the graph of y = x + 3, except for the hole at x = 2. On the graph, indicate the hole with an open circle. The domain of f is { x | x ≠ 2}.

Hole at x = 2

Ex 3: Determine the discontinuities for the graph of

.

Step 1 Vertical asymptotes/holes.

No Holes; Vertical asymptotes: x = - and x = 0.

The denominator is 0 when x = - 1 or 0. Since neither of those factors are also in the numerator, they are vertical asymptotes and not a holes.

Step 2 Horizontal asymptotes.

y = 0. The exponent in the denominator is the largest.

x – 2 x^2 + x

f ( x ) = x^ –^2 x ( x + 1)

f ( x ) =

C. Slant Asymptotes.

The graph of a rational function has a slant

asymptote if the degree of the numerator is exactly

one more than the degree of the denominator. Long

division is used to find slant asymptotes.

The only time you have an oblique asymptote is

when there is no horizontal asymptote. You cannot

have both.

When doing long division, we do not care about the

remainder.

Example 1

• Determine if the following function has a slant

asymptote. If it does, find the equation for it.

x

f x

x

n > d by exactly one, so no horizontal

asymptote, but there is an oblique

(slant) asymptote.

The slant asymptote is the line y = x + 1