Any help would be greatly appreciated!
Thanks in advance!
The vertical asymptote is found by solving $\displaystyle px + t = 0$.
The horizontal asymptote is found by noting that $\displaystyle y = \frac{cx + d}{px + t} = \frac{c + \frac{d}{x}}{p + \frac{t}{x}}$ and then taking the limit $\displaystyle x \rightarrow + \infty$. Alternatively, $\displaystyle y = \frac{cx + d}{px + t} = \frac{c}{p} + \frac{d - \frac{c}{p} t}{px+t}$ (found using polynomial long division) and the horizontal asymptote can be seen by inspection.
To find the vertical asymptote, take the denominator and set it equal to zero:
$\displaystyle \begin{aligned}
px + t &= 0 \\
px &= -t \\
x &= -\frac{t}{p}
\end{aligned}$
To find the horizontal asymptote, note the degrees of the polynomials in the numerator and denominator. Since they are the same (1), take the ratio of the leading coefficients:
$\displaystyle y = \frac{c}{p}$
Thus the product would be
$\displaystyle \frac{-ct}{p^2}$ .
Edit: Beaten to it by mr fantastic!
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Sorry for my terrible handwriting but I did the long division polynomial method and as a final answer I get: -ct/p and that is not even one of the multiple choice!
In my country at least, the standard form of the rational function is:
f(x) = (a/(b(x-h))) + k
The standard form is the same for my country, too.
First of all, remember that asymptotes are equations of lines, not numbers. So it doesn't make sense to write them as an ordered pair like $\displaystyle \left(-t, \frac{c}{p}\right)$. Second, $\displaystyle y = \frac{c}{p}$ is the horizontal asymptote, but where did the $\displaystyle -t$ come from? If that's supposed to be the vertical asymptote, that's wrong. It's $\displaystyle x = -\frac{t}{p}$ as mr fantastic and I both explained.
When the problem asked to find the "product of the parameters that defines its vertical and horizontal asymptotes," I took it to mean that you multiply $\displaystyle -\frac{t}{p}$ by $\displaystyle \frac{c}{p}$ , getting the answer I stated above.
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