Solve the problem.
-Determine which rational function $R ( x )$ has a graph that crosses the $x$-axis at $- 1$, touches the $x$-axis at $- 4$, has vertical asymptotes at $x = - 2$ and $x = 3$, and has one horizontal asymptote at $y = - 2$.
A) $R ( x ) = \frac { - 2 ( x + 1 ) ( x + 4 ) ^ { 2 } } { ( x + 2 ) ^ { 2 } ( x - 3 ) } , x \neq - 2,3$

B) $R ( x ) = \frac { - 2 ( x + 1 ) ( x + 4 ) } { ( x + 2 ) ( x - 3 ) } , x \neq - 2,3$

C)$R ( x ) = \frac { - ( x + 1 ) ( x + 4 ) ^ { 2 } } { 2 ( x - 2 ) ^ { 2 } ( x + 3 ) } , x \neq 2 , - 3$

D) $R ( x ) = \frac { - 2 ( x - 3 ) ( x + 2 ) ^ { 2 } } { ( x + 4 ) ^ { 2 } ( x + 1 ) } , x \neq - 4 , - 1$

Question

Solve the problem.
-Determine which rational function $R ( x )$ has a graph that crosses the $x$-axis at $- 1$, touches the $x$-axis at $- 4$, has vertical asymptotes at $x = - 2$ and $x = 3$, and has one horizontal asymptote at $y = - 2$.
A) $R ( x ) = \frac { - 2 ( x + 1 ) ( x + 4 ) ^ { 2 } } { ( x + 2 ) ^ { 2 } ( x - 3 ) } , x \neq - 2,3$

B) $R ( x ) = \frac { - 2 ( x + 1 ) ( x + 4 ) } { ( x + 2 ) ( x - 3 ) } , x \neq - 2,3$

C)$R ( x )  = \frac { - ( x + 1 ) ( x + 4 ) ^ { 2 } } { 2 ( x - 2 ) ^ { 2 } ( x + 3 ) } , x \neq 2 , - 3$

D) $R ( x ) = \frac { - 2 ( x - 3 ) ( x + 2 ) ^ { 2 } } { ( x + 4 ) ^ { 2 } ( x + 1 ) } , x \neq - 4 , - 1$

Quizplus · Accepted Answer

The correct function must have factors that correspond to its x-intercepts and vertical asymptotes. For an x-intercept at -1, there should be a factor of $x + 1$; for touching the x-axis at -4, there should be a factor of $(x + 4)^2$ (a double root causes the graph to touch but not cross the axis); for vertical asymptotes at $x = -2$ and $x = 3$, there should be factors of $(x + 2)$ and $(x - 3)$ in the denominator. The horizontal asymptote at $y = -2$ suggests the leading coefficients of the numerator and denominator should result in a ratio of -2 when the highest powers of $x$ are considered. Option A meets all these criteria.

Solve the Problem R(x)R ( x )R(x) Has a Graph That Crosses The

Solve the Problem $R ( x )$ Has a Graph That Crosses The