Math 165 - Sections 4.4 and 4.5 – Rational Functions

1)Form of a rational function:

2)Domain of a rational function:

3)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 horizontal, vertical and oblique asymptotes.

4)Read page 219 of the book for the definitions of horizontal and vertical asymptotes.

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

5)as x → ∞, f(x) → 0

6)as x → - ∞, y → 5- (from below)

7)as |x| → 0, f(x) → ∞

8)as |x| → ∞, y → 2

9)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-

10)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 →……..

11)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 →……..

12)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 the graph.

13)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 the graph.

Table 1Table 2Table 3

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

x = 5 is the vertical asymptote

y = 0 is the horizontal asymptote

15)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

16)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

17)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

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

there is no vertical asymptote

the horizontal asymptote is y = -1

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

there is no vertical asymptote

there is no horizontal asymptote

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