rational functions

types of horizontal asymptotes

@ y = 0; when the degree of the numerator is LOWER than degree of denominator slant = when the degree of the numerator is HIGHER than degree of denominator same degree = simply divide numerator by denominator and you should get a y = value

How do you find x-intercepts?

Factor out the numerator and see which values for x will produce '0'; these x-values will be you intercepts ex. (x,0)

Two types of horizontal asymptotes:

if they are the same degree, divide the polynomial of the numerator by the polynomial of the denominator if the degree in the numerator is LOWER than degree in denominator the HA is y = 0 (x-axis)

GOAL when simplifying rational expressions/equations

in any part of problem (whether it be in the numerator or denominator) be sure to find a LCD so that you can combine all terms at the top. After that, divide your "sum" by the term you multiplied by. If this occurs in the numerator or denominator, you are now ready to simplify by cross multiplying

Least Common Denominator

lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions

you can solve/simplify rational expressions by

multiplying by the LCD

When does a vertical asymptote occur?

occurs for whatever value of x makes the DENOMINATOR 0

How do you find y-intercept?

plug in '0' for x in the function; the answer will be your y-value

when there is a hole for an x value...

there can be no vertical asymptote at that x-value, no x-intercept that includes that x-value

When there is a hole at (0, y)

there is no y-intercept

How do you find the equation of the line for a slant asymptote?

use long division (numerator/denominator)

What causes a hole?

when the numerator and the denominator both equal 0 *check for common factors in the numerator and the denominator

in most work rate problems, you set it up as

work done (in time)/your rate

when you factor out x in expressions both in the numerator and denominator...

you have a hole at 0

How do you find the point of discontinuity?

Once you find the x-value for the hole, plug that value back into equation where factors are canceled out (that way there will be no zeroes in your computation)

Why do you need an LCD?

So that you can add or subtract fractions

What causes a slant asymptote?

When the polynomial in the numerator is a higher degree than the polynomial in the denominator

What causes a horizontal asymptote?

When the polynomial in the numerator is a lower/same degree than a polynomial in the denominator