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Horizontal and Vertical Asymptotes

Guidelines that graphs approach based on zeros and degrees in rational functions.

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Practice Horizontal and Vertical Asymptotes

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Asymtotes and Rational Functions

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Vocabulary

Complete the table.
 Word Definition Vertical Asymtote ___________________________________________________________ ___________ A horizontal line marking a specific value toward which the graph of a function may approach, but will generally not cross Oblique Asymtote ___________________________________________________________ ___________ function that is a ratio of two polynomials End Behavior ___________________________________________________________ ___________ A single point that is not possible on the graph of an otherwise continuous function

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Remember: Rational functions are defined as r(x)=p(x)q(x)\begin{align*}r(x)=\frac{p(x)}{q(x)}\end{align*} where p(x)\begin{align*}p(x)\end{align*} and q(x)\begin{align*}q(x)\end{align*} are polynomials

In your own words, what is an asymtote? ________________________________________

Asymtotes

How do you find asymtotes?

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Vertical

To find a vertical asymtote:

1. Set the denominator to 0

2. Solve for x

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This graph has two vertical asymtotes and one horizontal asymtote:

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Horizontal

To find a horizontal asymtote:

1. Put in standard form

2. Remove all terms but the one where x has the largest exponent in numerator and denominator

3. Three possibilities:

1. If the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote crosses the y\begin{align*}y-\end{align*} axis at y=0\begin{align*}y=0\end{align*}

2. If the degree of the denominator and the numerator are the same, the horizontal asymptote equals to the ratio of the leading coefficients

3. If the degree of the numerator is larger than the degree of the denominator, then there is no horizontal asymptote

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Horizontal asymtotes often indicate end behavior.  As the values of x\begin{align*}x\end{align*} get very large or very small, the graph of the rational function will approach (but not reach) the horizontal asymptote. If there is no horizontal asymtote the function will continue to ___________.
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Oblique

If the degree of the numerator is one greater than the degree of the denominator, then there is an oblique asymtote. To find this asymtote:

1. Divide the numerator and the denominator
2. Re-write that function in the form f(x)D(x)=Q(x)+R(x)D(x)\begin{align*}\frac{f(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}\end{align*} where Q(x)\begin{align*}Q(x)\end{align*} is the quotient and R(x)\begin{align*}R(x)\end{align*} is the remainder
3. Q(x)\begin{align*}Q(x)\end{align*} is the asymtote
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Oblique asymtotes are diagonal lines:

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Holes

Holes occur when one term in the numerator cancels with one term in the denominator. It is not an asymtote, it is merely a missing point.

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Practice

Identify the asymtotes and holes. Then sketch the graph.
1. f(x)=x+73x10\begin{align*}f(x) = \frac{x + 7}{3x - 10}\end{align*}
2. f(x)=x3+7x3x26x+13\begin{align*}f(x) = \frac{x^3 + 7x}{3x^2 - 6x + 13}\end{align*}
3. f(x)=x2+2x4x+8\begin{align*}f(x) = \frac{x^2 + 2x}{-4x + 8}\end{align*}
4. f(x)=x+14x10x2+x+2\begin{align*}f(x) = \frac{x + 14x - 10}{x^2 + x + 2}\end{align*}
5. f(x)=4x3+9x\begin{align*}f(x) = \frac{4}{x^3 + 9x}\end{align*}
6. y=4x3x2+3\begin{align*} y = \frac{4x^3}{x^2 + 3}\end{align*}
7. y=(3x)(x+8)x+2\begin{align*} y = \frac{(3x)(x + 8)}{x + 2}\end{align*}
8. f(x)=x3+4x2\begin{align*}f(x) = \frac{x^3 + 4}{x^2}\end{align*}
9. y=4x422x3\begin{align*}y = \frac{4x^4 - 2}{2x^3}\end{align*}
Factor the numerator and denominator, then set the denominator equal to zero and solve to find restrictions on the domain.
1. y=x2+3x10x2\begin{align*}y = \frac{x^2 + 3x - 10}{x - 2}\end{align*}
2. f(x)=x2+2x24x4\begin{align*}f(x) = \frac{x^2 + 2x -24}{x - 4}\end{align*}
3. f(x)=x212x+32x4\begin{align*}f(x) = \frac{x^2 - 12x +32}{x - 4}\end{align*}
4. y=x2+2120x+45\begin{align*}y = \frac{x^2 +\frac{21}{20}}{x + \frac{4}{5}}\end{align*}
5. y=x2+13x+42x+7\begin{align*}y = \frac{x^2 +13x + 42}{x +7}\end{align*}

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