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Reminders… Hug your folks. Clean up your room.

The best way to navigate this site is to use the categories.

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Reminders… Hug your folks. Clean up your room.

The best way to navigate this site is to use the categories.

I was interrupted a couple of times on the second one. I’ll redo it later.

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A rational function, in lowest terms, will have vertical asymptotes at the real zeros of the denominator.

Examples

This above example has the zero 3 in the denominator. The vertical line x = 3 is a boundary for this rational function. Mentally, do you see the line x = 3. It is a boundary, It is an asymptote. This graph also has a horizontal boundary or asymptote. More about that in a few moments.

This above example has two zeros in the denominator: -3 and 4. The vertical lines x = -3 and x = 4 are vertical asymptotes. The curve dares not to touch these vertical asymptotes.

This above example has two zeros in the denominator: 0 and -2. You did see that 0 is a double root, Right? The vertical lines x = 0 and x = -2 are vertical asymptotes. Do you see how the curve rises on both sides of the vertical asymptote x = 0. Because it was a double root, this is a characteristic of that. The curve will either run up or down each side of an asymptote that was a double root or more exactly a root that occurred an even amount of times.

We find the vertical asymptotes by solving a rational function denominator for zero. These also correspond to the domain restrictions for the function.

Vertical asymptotes are sacred ground, horizontal asymptotes are just useful suggestions. You can never touch a vertical asymptote, you can (and often do) touch and even cross horizontal asymptotes.

1. If the degree of the numerator is less than the degree of the denominator, the rational function is a proper rational function and will have the horizontal asymptote y = 0.

Examples

In the above example, the degree of the numerator is less than the degree of the denominator. When this happens, the rational function is proper and the horizontal asymptote is the horizontal line y = 0. The horizontal line y = 0 is a horizontal boundary and in this example it is not very strict — the middle curve crosses the horizontal asymptotes. It will sometimes do this with horizontal asymptotes, never a vertical asymptote. We also have the vertical asymptotes x = 0 and x = -4.

In the above example, the degree of the numerator is also less than the degree of the denominator. Again when this happens, the rational function is proper and the horizontal asymptote is the horizontal line y = 0. The horizontal line y = 0 is a horizontal boundary. We also have the vertical asymptotes x = 1 and x = -2.

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Is this a proper rational function? Why? Yes, the degree of the numerator is less than the degree of the denominator.

The horizontal asymptote is y = 0.

The vertical asymptote is x = -2. And do you see that the curve is rising on both sides of this asymptote. This is indicative of the fact that -2 was a double root. When that happens the curve will either rise or fall on both sides of the vertical asymptote.

The degree of the numerators in these examples is less than the degree of the denominators. When this happens, the rational function is proper and the horizontal asymptote is the horizontal line y = 0.

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2. If the degree of the numerator is greater than or equal to the degree of the denominator, the rational function is improper.

a. If the degree of the numerator is equal to the degree of the denominator, the quotient of the lead coefficients of the numerator and denominator is a horizontal asymptote.

Examples

In the above example, the degree for the numerator and denominator are both first. The horizontal asymptote is equal to the quotient of the lead coefficients. y = 2/1 = 2.

In the above example, the degrees for the numerator and denominator are both second. The horizontal asymptote is equal to the quotient of the lead coefficients. y = 1/1 = 1.

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b. If the degree of the numerator is exactly one more than the degree of the denominator, the rational function has a slant asymptote that you will derive by division techniques.

Examples

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In these two examples, the degree of the numerator is exactly one more than the degree of the denominator. Because of this, use division techniques, either long or synthetic division, to find the slant asymptote. Disregard any remainders.

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c. If the degree of the numerator is greater than one more than the degree of the denominator, the rational function has neither a horizontal nor a slant asymptote.

Since the degree of the numerator is more than one more than the denominator, there is no horizontal asymptote.

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Graphing Rational Functions.

Step 1: Graph R using your graphing utility.

Step 2: Find the domain of the rational function.

Step 3: Write R in lowest terms.

Step 4: Locate the intercepts. The y-intercept is R(0). The x-intercepts are the roots of the rational function’s numerator in lowest terms.

Step 5: Identify any symmetry.

Step 6: Locate the vertical asymptotes. The vertical asymptotes are the zeros of the rational function’s denominator in lowest terms. Most of the time the vertical asymptotes will be the vertical lines of the domain concerns. However, if the rational function reduces, you may lose a domain concern.

Step 7: Locate the horizontal or slant asymptotes using previous discussed procedures. Horizontal and slant asymptotes are dependent on the degrees of the numerator and denominator. a) If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. b) If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the line y = the quotient of the lead coefficients. c) If the degree of the numerator is exactly one more than the degree of the denominator, then there is no horizontal asymptotes, there is a slant one instead — use division techniques. and d) If the degree of the numerator is more than one more than the denominator, then there is no horizontal nor slant asymptote.

Step 8: Use these results and your graphing utility’s table feature to graph R by hand.

Example.

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Another Example

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Problem 53

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