Focuses on the process of finding vertical asymptotes of rational functions. This involves factoring, simplifying, and identifying the roots of the remaining denominator.
A vertical asymptote is a vertical line, \(x=c\), that the graph of a function approaches but never touches or crosses. The function is undefined at this x-value.
What is the procedure for finding the vertical asymptote(s) of a rational function?
1. Factor the numerator and the denominator completely.
2. Cancel any common factors (these correspond to holes).
3. Set the remaining factors in the denominator equal to zero and solve for \(x\). The resulting \(x=c\) values are the equations of the vertical asymptotes.
Find the vertical asymptote of \(f(x) = \frac{5}{x+2}\).
The denominator is \(x+2\). Setting it to zero gives \(x+2=0\), so the vertical asymptote is the line \(x=-2\).
Find the vertical asymptote(s) for \(g(x) = \frac{x-1}{x^2 - 7x + 10}\).
Factor the denominator: \(x^2 - 7x + 10 = (x-5)(x-2)\). There are no common factors to cancel. Setting the denominator's factors to zero gives \(x-5=0\) and \(x-2=0\). The vertical asymptotes are at \(x=5\) and \(x=2\).
Does the function \(f(x) = \frac{x-3}{x^2-9}\) have a vertical asymptote at \(x=3\)?
No. First, factor the denominator: \(f(x) = \frac{x-3}{(x-3)(x+3)}\). The factor \((x-3)\) cancels out, creating a hole at \(x=3\). The remaining factor in the denominator is \((x+3)\), which creates a vertical asymptote at \(x=-3\).
Can the graph of a function ever cross a vertical asymptote?
No. A vertical asymptote occurs at an x-value for which the function is undefined. Therefore, there is no point on the graph at that x-value, and the graph cannot cross the line.
Find the vertical asymptote(s) of \(h(x) = \frac{2x^2+1}{x^2+4}\).
Set the denominator to zero: \(x^2+4=0\) gives \(x^2=-4\). This equation has no real solutions. Therefore, the function has no vertical asymptotes.
What does the graph's behavior look like near a vertical asymptote?
As \(x\) approaches the asymptote's value from either side, the function's value \(f(x)\) will increase without bound (approaching \(+\infty\)) or decrease without bound (approaching \(-\infty\)).
