Rational functions are expressions that look like this: (\frac{P(x)}{Q(x)}). Here, (P(x)) and (Q(x)) are polynomials, which are just expressions made up of variables and numbers. It's important to know how these functions behave to help us draw their graphs and understand what they look like.
Vertical asymptotes are special lines where the function doesn’t exist. This usually happens when (Q(x) = 0).
For example, in the function (\frac{1}{x - 2}), there’s a vertical asymptote at (x = 2).
In general, if (Q(x) = (x - a)(x - b)), then the vertical asymptotes will be at (x = a) and (x = b).
Horizontal asymptotes show us how the function behaves when (x) gets really big (towards infinity) or really small (towards negative infinity).
Here’s how to find horizontal asymptotes:
If the degree (or highest power) of (P(x)) is less than that of (Q(x)), the horizontal asymptote is (y = 0).
If the degrees of (P(x)) and (Q(x)) are equal, the horizontal asymptote is (y = \frac{a}{b}). Here, (a) and (b) are the leading numbers (the coefficients) from (P(x)) and (Q(x)).
If the degree of (P(x)) is greater than that of (Q(x)), there is no horizontal asymptote, but there could be an oblique (or slant) one.
The end behavior of a rational function depends on the leading terms of (P(x)) and (Q(x)).
Take the function (f(x) = \frac{2x^3}{x^2 + 1}) as an example.
As (x) goes to infinity, (f(x)) goes to infinity too. This means the function keeps rising and doesn't stop.
By understanding these important features, we can better predict how the graph of a rational function will look and behave.
Rational functions are expressions that look like this: (\frac{P(x)}{Q(x)}). Here, (P(x)) and (Q(x)) are polynomials, which are just expressions made up of variables and numbers. It's important to know how these functions behave to help us draw their graphs and understand what they look like.
Vertical asymptotes are special lines where the function doesn’t exist. This usually happens when (Q(x) = 0).
For example, in the function (\frac{1}{x - 2}), there’s a vertical asymptote at (x = 2).
In general, if (Q(x) = (x - a)(x - b)), then the vertical asymptotes will be at (x = a) and (x = b).
Horizontal asymptotes show us how the function behaves when (x) gets really big (towards infinity) or really small (towards negative infinity).
Here’s how to find horizontal asymptotes:
If the degree (or highest power) of (P(x)) is less than that of (Q(x)), the horizontal asymptote is (y = 0).
If the degrees of (P(x)) and (Q(x)) are equal, the horizontal asymptote is (y = \frac{a}{b}). Here, (a) and (b) are the leading numbers (the coefficients) from (P(x)) and (Q(x)).
If the degree of (P(x)) is greater than that of (Q(x)), there is no horizontal asymptote, but there could be an oblique (or slant) one.
The end behavior of a rational function depends on the leading terms of (P(x)) and (Q(x)).
Take the function (f(x) = \frac{2x^3}{x^2 + 1}) as an example.
As (x) goes to infinity, (f(x)) goes to infinity too. This means the function keeps rising and doesn't stop.
By understanding these important features, we can better predict how the graph of a rational function will look and behave.