4.5: Rational Functions

Hello friends! Let's learn about rational functions.

Let's start with a definition:

The domain of a rational function consists of all real numbers EXCEPT the zeros of the

denominator. Because we all know what happens when there's a zero

in the denominator...

Anyway, the cool thing about rational functions is they make these graphs call

ed hyperbolas. A hyperbola is a graph that comes REALLY REALLY REALLY close to a line, but never hits it. This line is called an asymptote. This kind of makes me sad. Like, poor function...all it wants to do is intersect the asymptote, but it never will. If this confuses you, here's some ways to think about it:

<<< Mr. Wilhelm walking towards a wall

Make sense? Good. Okay so now we move on to types of asymptotes: vertical and horizontal.

Vertical asymptote: The line x = a is a vertical asymptote for the graph of a function ƒ if

ƒ(x) ➞ ∞ or ƒ(x) ➞ -∞ as x approaches a from either the left or the right.

Horizontal asymptote: The line y = c is a horizontal asymptote for the graph of the function ƒ if

ƒ(x) ➞ c as ƒ(x) ➞ ∞ or ƒ(x) ➞ -∞.

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Theorem on Horizontal Asymptotes:

Let...

where

.

1. If n

2. If n=k, then the line y=a_n/b_k, or the ratio of the leading coefficients is the horizontal asymptote for the graph of ƒ.

3. If n>k, the graph of ƒ ha no horizontal asymptote. Instead, either ƒ(x) ➞ ∞ or ƒ(x) ➞ -∞ as ƒ(x) ➞ ∞ or as ƒ(x) ➞ -∞.

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GRAPHING RATIONAL FUNCTIONS!

Guidelines to follow:

1. find the x-intercepts - find the real zeros of the numerator and plot them on the x-axis

2. find the vertical asymptotes - find the real zeros of the denominator, and for each real zero a, sketch the vertical asymptote x = a

3. find the y-intercept, if it exists - set the function to zero - ƒ(0) - and plot the point (0, ƒ(0)) on the y-axis

4. find the horizontal asymptote - apply the theorem on horizontal asymptotes, then sketch the horizontal asymptote

5. determine if the horizontal asymptote intersects the graph. the x-coordinates of the points of intersection are the solutions of the equation ƒ(x)=c, where y=c is the horizontal asymptote. if they exist, plot them

6. based on all of the clues you found above, finish the graph. if necessary, plug in a number for x to find where the graph is places in relation to the asymptotes. keep in mind that if one curve goes down, the one across from it will go up, and vice versa, unless there is an even multiplicity, in which case they will both go the same direction.

HOLES

Sometimes, it is impossible for a graph to include a specific point, even though it runs straight through it. This happens when parts of the numerator and denominator cancel out. For example,

In this case, even though there is a hole at x=1. To graph this, place an open dot on the coordinate where the hole occurs in the graph to show that it is not a solution.

OBLIQUE ASYMPTOTES

When the numerator has a degree ONE greater than the degree of the denominator, it has an oblique asymptote. This sounds a lot more complicated than it is. Because the function does not have a horizontal asymptote, it instead has an oblique one, which is usually like a diagonal line as an asymptote. To find the slope of this line, all you have to do is use long division to divide the numerator by the denominator.

Okay, that's pretty much all you need to know about section five!

I HOPE YOU ALL KICK ASYMPTOTE ON MONDAY'S TEST

- Olivia Darany