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Let’s learn together about: How to Determine the End Behavior of a Rational Function?
Question: How do you find the end behavior of a rational function?
Answer:
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Determining the End Behavior of a Rational Function
Step 1:
Take a look at the degrees of denominator and numerator. If the degree in the denominator exceeds the degree in the numerator, then there is a horizontal asymptote y=0, which indicates the end behavior of the function.
Step 2:
If the degrees of denominator and numerator are equal, then there will be a horizontal asymptote of y=a/b. Where a is leading coefficient for the numerator, while b is leading coefficient for the denominator. This is the end behavior.
Step 3:
If the degree of the numerator is greater than the degree of the denominator, then there is a slant/oblique asymptote (if the degree of the numerator is exactly one larger than the degree of the denominator), or the function is asymptotic to a polynomial. To find the quotient, you can divide the polynomials by length. Then y=q(x), where q(x) is the quotient that provides the end behavior.
Determining the End Behavior of a Rational Function – Vocabulary and Equations
Rational Function: Rational functions are oA rational function is a function made up of a ratio of polynomials.
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where p(x) and q(x) are polynomials, and q(x)≠0
End Behavior:
A graph of a function’s end behavior is the graph’s behavior as xx approaches or falls below infinity.
The end behavior of a function is equal to its horizontal asymptotes, slant/oblique asymptotes, or the quotient found when long dividing the polynomials.
Degree:
A polynomial’s degree is its highest exponent.
These steps, definitions and equations will be used to determine the final behavior of rational functions in these two examples.
Example 1: Determining the End Behavior of a Rational Function
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Step 1: The degree of the numerator is 3 and the degree of the denominator is also 3. We must move on to the next step.
Step 2: The leading coefficient for the numerator’s numerator is 4, and the leading coefficient for the denominator, 2. We have thus a horizontal asymptote that is:
y=4/2=2
The end behavior of the rational function is the horizontal asymptote y=2.
Step 3:The end behavior of y=2 found in step 2 means that the graph of the function
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will approach the graph of the horizontal line y=2 as
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as seen in the graph below:
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Example 2: Determining the End Behavior of a Rational Function
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Step 1: The degree of the numerator is 4, and the degree of the denominator is 3. We must move on to the next step.
Step 2: The degrees are not equal – move to the next step.
Step 3: We need to long divide the polynomials.
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The quotient is 1/2 X. We have a slant asymptote: y=1/2 x
meaning that as x→±∞, the graph of the rational function approaches the graph of the straight line y= 1/2 x
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Conclusion:
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