# How To Determine End Behavior Of A Rational Function

How To Determine End Behavior Of A Rational Function. Consider the rational function {eq}f(y) = \frac{y^4 + y^3 + y^2 + y}{y^4 + y^3 +y^2 + 2} {/eq}. Now as x → ± ∞, you can see that the terms 2 x 2 and 1 x 2 disappear, so we have.

Y = 1 1 = 1 y = 1 1 = 1 → y = 1 y = 1. There are three cases for a rational function depends on the degrees of the numerator and denominator. If the leading term is negative, it will change the direction of the end behavior.

### There Are Three Cases For A Rational Function Depends On The Degrees Of The Numerator And Denominator.

About press copyright contact us creators advertise developers terms privacy policy & safety how youtube works test new features press copyright contact us creators. The slant asymptote is found by using polynomial division to write a rational function $\frac{f(x)}{g(x)}$ in the form $$\frac{f(x)}{g(x)} = q(x) + \frac{r(x)}{g(x)}$$ The usual trick to find asymptotes as x → ∞ or x → − ∞ is to divide the numerator and denominator by the highest power of x that appears in the denominator.

### In Your Case, This Is X 2:

As x→ ∞,f (x)→ 0,and as x → −∞,f (x)→ 0 as x → ∞, f ( x) → 0, and as x → − ∞, f ( x) → 0. Now as x → ± ∞, you can see that the terms 2 x 2 and 1 x 2 disappear, so we have. Two divided by arbitrarily large numbers, whether they are positive or negative, that's going to go to zero.

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End\:behavior\:y=\frac {x^2+x+1} {x} end\:behavior\:f (x)=x^3. When the leading term is an odd power function, as x decreases without bound, $f(x)$ also decreases without bound; A rational function is the quotient of two polynomials.

### Horizontal Asymptotes (If They Exist) Are The End Behavior.

A rational function is one of the form The end behavior of the rational function f (x) is defined as the behavior of the graph of f (x) as x either approaches positive infinity or negative infinity. However horizontal asymptotes are really just a special case of slant asymptotes (slope$\;=0$).

### Find The End Behavior Of The Rational Function.

A vertical asymptote is a vertical line that marks a specific value toward which the graph of a function may approach but will never reach. As the values of x x approach infinity, the function values approach 0. This prepares students for subsequent lessons in which they graph rational functions, identifying zeros and asymptotes when suitable.