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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 ≠3

b) Write a rational function with domain x ≠−5 ,5

3) Analyze the graph of f ( x )=1x

a) You are familiar with the graph of this function; sketch itb) What is the domain?c) Evaluate the function for values “close” to ____________ and explain what you notice

Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1d) The equation of the vertical asymptote is _______________________e) Evaluate the function for x-values that are “very large” (in absolute value). Explain what you notice

Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1f) The equation of the horizontal asymptote is ____________________________g) Find the x- and y-interceptsh) Describe this “local behavior” in symbols.i) Describe this “end behavior” in symbols.j) Graph the function

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4) Analyze the graph of f ( x )=2 x+6x+1

a) What is the domain?b) Evaluate the function for values “close” to ____________ and explain what you notice

Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1c) The equation of the vertical asymptote is _______________________d) Evaluate the function for x-values that are “very large” (in absolute value). Explain what you notice

Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1e) The equation of the horizontal asymptote is ____________________________f) Find the x- and y-interceptsg) Describe this “local behavior” in symbols.h) Describe this “end behavior” in symbols.i) Graph the function

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5) Analyze the graph of f ( x )= x2+ x−12x+1


Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1c) The equation of the vertical asymptote is _______________________d) Evaluate the function for x-values that are “very large” (in absolute value). Explain what you notice

Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1e) The equation of the horizontal asymptote is ____________________________f) If there is no horizontal asymptote, find the oblique asymptote.g) Find the x- and y-interceptsh) Describe this “local behavior” in symbols.i) Describe this “end behavior” in symbols.j) Graph the function

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6) Analyze the graph of f ( x )= x2−1x+1


Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1c) The equation of the vertical asymptote is _______________________d) If there is no vertical asymptote, what is going on in this function?e) Evaluate the function for x-values that are “very large” (in absolute value). Explain what you notice

Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1f) The equation of the horizontal asymptote is ____________________________g) Find the x- and y-interceptsh) Describe this “local behavior” in symbols.i) Describe this “end behavior” in symbols.j) Graph the function

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7) Analyze the graph of f ( x )=(2 x+1 )( x+3)x ( x+3 )

a) What is the domain?b) Is there a hole in this graph? If so, what are the coordinates?c) The equation of the vertical asymptote is _______________________d) The equation of the horizontal asymptote is ____________________________e) Find the x- and y-interceptsf) Describe this “local behavior” in symbols.g) Describe this “end behavior” in symbols.h) Graph the function

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

a. Vertical asymptotesb. Horizontal asymptotes.c. Oblique asymptotesd. X-interceptse. Y-interceptsf. Coordinates of holes

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9) Graph some of the following functions

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10) Graph some of the following functions

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11) 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. What can you say about the degrees of numerator and denominator? Are they equal? If not, which one has larger degree?

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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 a possible 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 a possible graph.

Table 1 Table 2 Table 3

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14) 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-

15) Sketch at least two graphs of a rational function satisfying the following conditionsThe vertical asymptote is x = 2The horizontal asymptote is y = 1

Graph 1 Graph 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


16) Sketch the graph of a rational function satisfying the following conditionsThe x-intercepts are –2 and 2The vertical asymptote is x = 0The horizontal asymptote is y = -5

Now complete the following:


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

x = 5 is the vertical asymptotey = 0 is the horizontal asymptote

18) Write the equation of a rational function that satisfies the following conditions:x = 1 and = -2 are the vertical asymptotesy = 2 is the horizontal asymptote

19) Write the equation of a rational function that satisfies the following conditions:the only x-intercept is x = 1x = 2 is the vertical asymptotey = 3 is the horizontal asymptote

20) Write the equation of a rational function that satisfies the following conditions:x = -7 is the vertical asymptotethe graph has a “hole” at x = 2there is no horizontal asymptote

21) Write the equation of a rational function that satisfies the following conditions:there is no vertical asymptotethe horizontal asymptote is y = -1

22) Write the equation of a rational function that satisfies the following conditions:there is no vertical asymptotethere is no horizontal asymptote

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23) Solve the following word problems

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24) Solve the following word problems

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