1 Long Run Behavior_Rational Functions.notebook
July 23, 2014
Long Run Behavior of
Rational Functions
Simplify each of these:
Long Run Behavior of
Rational Functions
Long Run Behavior of
Rational Functions
Simplify each of these:
Simplify each of these:
What does the function mostly behave like? Compare graphs of original function & simplified function.
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1 Long Run Behavior_Rational Functions.notebook
In order to use the properties from the graph of q(x), we have to write a rational expression in quotient form:
July 23, 2014
Quotient form of a
Rational Expression
REMAINDER
QUOTIENT
"the degree of r(x) < degree "the long run behavior"
of b(x)"
Observe graphically. What do we call y=1?
What would you describe the long run behavior of these functions? Can you simplify? Can you tell how it behaves overall?
We can determine properties of the graphs of these rational functions because for large values of x, q(x) has more of an effect on the properties of the graph than Evaluate these previous functions at really large numbers, say x=1,000 & x=10,000
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1 Long Run Behavior_Rational Functions.notebook
July 23, 2014
Long Run Behavior of
Rational Functions
any asymptotes?
So­­the leading terms of numerator & denominator seem to tell us a lot about the funky function as x gets really large.
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1 Long Run Behavior_Rational Functions.notebook
July 23, 2014
Long Run Behavior of
Rational Functions
Thus, the leading terms coefficient ratio gives us an approximation of the horizontal asymptote if the ratio is a constant.
Slant (Oblique) Asymptotes
At that's the end of part 1...let's take a break & mathercise a minute how about it?
• If the quotient q(x) is NOT a constant, then the long run behavior will not be a horizontal asymptote.
• q(x) could be a linear function producing a slant asymptote.
Mathercise E22.pdf
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1 Long Run Behavior_Rational Functions.notebook
July 23, 2014
Slant (Oblique) Asymptotes
Example:
Graph & describe the long run behavior.
Leading terms?
asymptotes?
zeros?
Slant (Oblique) Asymptotes
Example:
Graph & describe the long run behavior.
Leading terms?
asymptotes?
zeros?
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1 Long Run Behavior_Rational Functions.notebook
July 23, 2014
Slant (Oblique) Asymptotes
Example:
Graph & describe the long run behavior.
Leading terms?
asymptotes?
zeros?
Long Run Behavior of
Rational Functions
In summary, for rational functions:
• the Long Run Behavior (a.k.a. as ) is given by the ratios of the leading terms
• the Zeros are the zeros of the numerator
• the Vertical Asymptotes occur at the zeros of the denominator
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1 Long Run Behavior_Rational Functions.notebook
July 23, 2014
Write one function for each of the following descriptors:
1) Has a slant asymptote
2) Has vertical asymptotes but no horizontal asymptotes
3) Has two horizontal asymptotes
And consider this question:
Can graphs ever cross their asymptotes?
Investigate
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1 Long Run Behavior_Rational Functions.notebook
July 23, 2014
Continue thinking on those & now complete the handout:
Algebra Bootcamp for Limits
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Attachments
Mathercise E22.pdf