Math 165 - Sections 4.4 and 4.5 Rational Functions

Math 165 - Sections 4.4 and 4.5 – Rational Functions

1)A rational function is a quotient of polynomial functions:

2)Explain how you find the domain of a rational function:

a)Write a rational function with domain x

b)Write a rational function with domain x

c)Write a rational function with domain all real numbers

3)The graphs of a rational function is shown below. Analyze the end and local behavior for each one. Write in symbolic form. Write the equations of the asymptotes.

Math 165 - Sections 4.4 and 4.5 – Rational Functions

4)The graphs of rational functions are shown below. Analyze the end and local behavior for each one. Write in symbolic form. Write the equations of the asymptotes.

5)Tables of a rational function are shown below. Is the table describing a local or an end behavior? Write in symbolic form and write the equations of the vertical and horizontal asymptotes, if any. Sketch a possible graph.

Math 165 - Sections 4.4 and 4.5 – Rational Functions

6)Tables of a rational function are shown below. Is the table describing a local or an end behavior? Write in symbolic form and write the equations of the vertical and horizontal asymptotes, if any. Sketch a possible graph.

Table 1Table 2Table 3

7)Analyze the graph of

Math 165 - Sections 4.4 and 4.5 – Rational Functions

8)Reflect on what we have done and summarize procedures for finding each of the following: (read the notes and/or book if necessary)

Vertical, horizontal asymptotes.
Oblique asymptotes
X-intercepts
Y-intercepts
Coordinates of holes

In the following problems sketch “a portion” of the graph of a function according to the given conditions.

9)as x → ∞, f(x) → 0

10)as x → - ∞, y → 5- (from below)

11)as |x| → 0, f(x) → ∞

12)as |x| → ∞, y → 2

Math 165 - Sections 4.4 and 4.5 – Rational Functions

13)Sketch a function with the following local and end behavior. Write the equations of the vertical and horizontal asymptotes, if any.

as x → 2+ (from the right), f(x) → ∞

as x → 2- (from the left), f(x) → ∞

as x → ∞, f(x) → 0+

as x → - ∞, f(x) → 0-

14)Sketch at least two graphs of a rational function satisfying the following conditions

The vertical asymptote is x = 2

The horizontal asymptote is y = 1

Graph 1Graph 2

Now for each of the graphs complete the following:

For Graph 1
As x → 2+then y →……..
As x → 2-then y →……..
As x → ∞then y →……..
As x → -∞then y →…….. / For Graph 1
As x → 2+then y →……..
As x → 2-then y →……..
As x → ∞then y →……..
As x → -∞then y →……..

15)Sketch the graph of a rational function satisfying the following conditions

The x-intercepts are –2 and 2

The vertical asymptote is x = 0

The horizontal asymptote is y = -5

Now complete the following:

As x → 0+then y →……..

As x → 0-then y →……..

As x → ∞then y →……..

As x → -∞then y →……..

Math 165 - Sections 4.4 and 4.5 – Rational Functions

16)Write the equation of a rational function that satisfies the following conditions:

x = 5 is the vertical asymptote

y = 0 is the horizontal asymptote

17)Write the equation of a rational function that satisfies the following conditions:

x = 1 and = -2 are the vertical asymptotes

y = 2 is the horizontal asymptote

18)Write the equation of a rational function that satisfies the following conditions:

the only x-intercept is x = 1

x = 2 is the vertical asymptote

y = 3 is the horizontal asymptote

19)Write the equation of a rational function that satisfies the following conditions:

x = -7 is the vertical asymptote

the graph has a “hole” at x = 2

there is no horizontal asymptote

20)Write the equation of a rational function that satisfies the following conditions:

there is no vertical asymptote

the horizontal asymptote is y = -1

21)Write the equation of a rational function that satisfies the following conditions:

there is no vertical asymptote

there is no horizontal asymptote

Math 165 - Sections 4.4 and 4.5 – Rational Functions