Question Type 2: Given asymptotes and relations between variables, coming up with the rational equation Exercises

Find an expression for the rational function $R(x)$ that has a vertical asymptote at $x=0$, a horizontal asymptote at $y=3$, a numerator with a leading coefficient of $3$, and satisfies $R(1)=2$.

Determine the rational function $R(x)$ with vertical asymptote $x=5$, slant asymptote $y=2x+1$, and satisfying $R(0)=3$.

Find the rational function $R(x)$ with vertical asymptote $x=-3$, slant asymptote $y=3x+4$, and passing through the point $(-1,2)$.

Find the rational function $R(x)$ with vertical asymptote $x=3$, slant asymptote $y=x-2$, and passing through $(1,1)$.

Find the rational function $R(x)$ with vertical asymptote $x=3$, horizontal asymptote $y=2$, passing through the point $(1,1)$, and whose numerator has leading coefficient $2$.

Find the rational function $R(x)$ that has vertical asymptote $x=2$, horizontal asymptote $y=5$, numerator leading coefficient $5$, and passes through $(-1,1)$.

Find $R(x)$ given vertical asymptote $x=1$, horizontal asymptote $y=3$, $R(2)=2$, and numerator leading coefficient $3$.

Determine the rational function of the form $R(x) = \dfrac{ax+b}{x+c}$ with vertical asymptote $x=-2$, horizontal asymptote $y=-1$, and satisfying $R(3)=5$.

Determine the rational function $R(x)$ which has vertical asymptote $x=-2$, horizontal asymptote $y=1$, satisfies $R(0)=2$, and has numerator leading coefficient $1$.

Determine $R(x)$ with vertical asymptote $x=4$, horizontal asymptote $y=-2$, numerator leading coefficient $-2$, and passing through $(2,0.5)$.

Construct the rational function $R(x)$ with vertical asymptote $x=4$, horizontal asymptote $y=-1$, passing through $(-1,0)$, and numerator leading coefficient $-1$.

Find the simplest rational function $R(x)$ with vertical asymptotes $x=1$ and $x=-1$, horizontal asymptote $y=0$, and satisfying $R(2)=1$.