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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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