write an equation for the rational function whose graph is given below. the x - intercept is (-1,0) and the…

write an equation for the rational function whose graph is given below. the x - intercept is (-1,0) and the y - intercept is (0,1). the graph has one vertical asymptote at x = 2

write an equation for the rational function whose graph is given below. the x - intercept is (-1,0) and the y - intercept is (0,1). the graph has one vertical asymptote at x = 2

Answer

Explanation:

Step1: Determine the general form of the rational function

A rational function has the form ( y=\frac{f(x)}{g(x)} ). Given a vertical asymptote ( x = 2 ), then ( g(x)=(x - 2) ). Given an ( x )-intercept ( (-1,0) ), then ( f(x)=a(x + 1) ). So the function is ( y=\frac{a(x + 1)}{x - 2} ).

Step2: Find the value of ( a )

Use the ( y )-intercept ( (0,1) ). Substitute ( x = 0 ) and ( y = 1 ) into ( y=\frac{a(x + 1)}{x - 2} ). [ \begin{align*} 1&=\frac{a(0 + 1)}{0 - 2}\ 1&=\frac{a}{-2}\ a&=- 2 \end{align*} ]

Answer:

( y=\frac{-2(x + 1)}{x - 2})