APPC - Rational functions

If the leading terms of the num. and denom. have the same degree…

you find the horizontal asymptote by dividing the two leading coefficients

If the rational function is bottom heavy

the horizontal asymptote is at y = 0.

If the rational function is top heavy (by 1) _____, if the rational function is top heavy (by more than 1) _____.

there is a slant asymptote, there is an oblique asymptote

Let f(x) = g(x)/h(x). Then we know…

g(x) has zeroes where g(x) = 0 (if no holes)
h(x) is undefined (vertical asymptote) where h(x) = 0

Find the vertical asymptote where…

The denominator values
Ex: 1/(x-3)
vertical asymptote is x=3

Holes

occur at x-values where both g(x) and h(x) are zero, resulting in a removable discontinuity.

Sum of cubes

a³+b³=(a+b)(a²-ab+b²)

Difference of cubes

a³-b³=(a-b)(a²+ab+b²)

Horizontal asymptote - end behavior

A line where y approaches forever but can never touch.

Lim(x →∞ f(x) = horizontal asymptote

Vertical asymptote - end behavior

A line where x approaches forever but cannot touch.

Lim(x → vertical asymptote) f(x) = ∞

Horizontal asymptote

a horizontal line that the graph of a function approaches as the input value approaches positive or negative infinity, indicating the end behavior of the function.

Vertical asymptote

a vertical line that the graph of a function approaches as the input value approaches a finite number, indicating the behavior of the function near that value.

Domain of rational function

all real numbers except x-values of holes and vertical asymptotes.

Range of rational function

all real numbers except horizontal asymptote and y values of holes.