Chapter 4.5, Problem 1AYU

To determine

To find: If the graph of a rational function $R$ has the vertical asymptote $x = 4$, the factor $x - 4$ must be present in the denominator of $R$. Explain why.

Answer to Problem 1AYU

The vertical asymptotes will be generated by the zeroes of the denominator, Therefore the vertical asymptote $x = 4$, the factor $x - 4$ must be present in the denominator of $R$.

Explanation of Solution

$R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$

An asymptote is a line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the roots of $q\left(x\right)$. Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters.

If you can write it in factored form, then you can tell whether the graph will be asymptotic in the same direction or in different directions by whether the multiplicity is even or odd.

Asymptotic in the same direction means that the curve will go up or down on both the left and right sides of the vertical asymptote. Asymptotic in different directions means that the one side of the curve will go down and the other side of the curve will go up at the vertical asymptote.

The vertical asymptotes will be generated by the zeroes of the denominator, Therefore the vertical asymptote $x = 4$, the factor $x - 4$ must be present in the denominator.