Questions: Write an equation for a rational function with: Vertical asymptotes at x=-2 and x=1 x-intercepts at x=-1 and x=-4 Horizontal asymptote at y=6 y=

Transcript text: Write an equation for a rational function with: Vertical asymptotes at $x=-2$ and $x=1$ $x$-intercepts at $x=-1$ and $x=-4$ Horizontal asymptote at $y=6$ \[ y=\square \]

Solution

To construct a rational function with the given properties, we need to consider the following:

Vertical asymptotes occur where the denominator is zero, so the denominator should have factors corresponding to $x = -2$ and $x = 1$.
$x$-intercepts occur where the numerator is zero, so the numerator should have factors corresponding to $x = -1$ and $x = -4$.
A horizontal asymptote at $y = 6$ suggests that the leading coefficients of the numerator and denominator should be adjusted to achieve this ratio.

Paso 1: Construcción del numerador

Dado que los interceptos en $x$ son $x = -1$ y $x = -4$, el numerador se puede expresar como: \[ \text{numerador} = 6 \cdot (x + 1)(x + 4) \]

Paso 2: Construcción del denominador

Las asíntotas verticales se encuentran en $x = -2$ y $x = 1$, por lo que el denominador se puede expresar como: \[ \text{denominador} = (x + 2)(x - 1) \]

Paso 3: Formulación de la función racional

La función racional completa se puede escribir como: \[ y = \frac{6 \cdot (x + 1)(x + 4)}{(x + 2)(x - 1)} \]

Respuesta Final

\[ \boxed{y = \frac{6 \cdot (x + 1)(x + 4)}{(x + 2)(x - 1)}} \]

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